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co ¢n (a) 00, then the Julia set J (¢) is totally disconnected. 0 i(p) = ¢J (p). Hence P is preperiodic. o 11 > 1}. We leave the details to the reader. 0 omits at least two values, but the omitted values may vary with and all u, wE
Proof See [95, Theorems 111.4.1, 111.4.2].
Remark 1.34. A quadratic polynomial has only one finite critical point. Thus Theorem 1.33 covers all cases, so we can divide the set of quadratic polynomials into two classes according to the behavior of their finite critical point. Every quadratic polynomial is linearly conjugate to a unique polynomial of the form ¢c(z) = z2 + c, so we define a portion of the c-plane by
M = {c
E C : ¢n (0) is bounded for n ::::: I}
= {c E C : J(¢c)
is connected}.
This set M is the famous Mandelbrot set, illustrated in Figure 1.2. It is a dynamically determined subset of the moduli space of quadratic polynomial maps.
1.5. Properties of PeriodicPoints
Figure 1.2: The Mandelbrot set
27
M.
1.5 Properties of Periodic Points The dynamics of a rational map is influenced not only by the behavior of its critical points, but also by the behavior of its periodic points. The next result describes some of the properties of the periodic points and periodic cycles ofa rational map.
Theorem 1.35. Let ¢( z) E C (z) be a rational function ofdegree d 2:: 2. (a) The map ¢ has at most 2d - 2 nonrepelling periodic cycles in ]p'1(C). If ¢ is a polynomial map, then it has at most d - 1 nonrepelling periodic cycles in C. (b) The Julia set .:J (¢) is equal to the closure ofthe repelling periodic points of ¢. (c) Let U C ]p'1 (C) be an open set such that .:J(¢) n U =I- 0. Then there is an integer n 2:: 1 such that ¢n(u n .:J(¢)) = .:J(¢). Proof (a) The sharp bound of2d - 2 for rational maps is due to Shishikura, see [43, Theorem 9.6]. Much earlier, Fatou gave a weaker bound that is sufficient for many applications, see [95, Theorem III.2.7]. The bound for polynomial maps is due to Douady, see [95, Theorem VI. 1.2]. (b) This important result is due independently to Julia and Fatou. See [43, Theorem 6.9.2] or [95, Theorem III.3.l]. D (c) See [95, Theorem 111.3.2]. The Fatou set F( ¢) is an open subset Of]p'1 (C), so it consists of one or more connected components. It is known that the number of components is equal to 0, I, 2, or 00; see [95, Theorem IY.1.2]. If U is a connected component of F(¢), then the open mapping property of ¢ implies that ¢(U) is also a connected component of F(¢). In this way we obtain a map
28
1. An Introduction to Classical Dynamics
Components (F( ¢)) ----. Components (F(¢)) ,
U f----7 ¢( U),
and we say that U is aperiodic domain if ¢n(u) = U for some n ~ 1, apreperiodic domain if some iterate ¢n(u ) is periodic, and a wandering domain otherwise. The following important result answers a long-standing question of Fatou and Julia. Theorem 1.36. (Sullivan's No Wandering DomainsTheorem) A rational map ¢ E C (z ) has no wandering domains. Proof See [43, Chapter 8], [95, Theorem IY.1.3] or [426].
o
The possibilityof wanderingdomains having been eliminated, the periodiccomponents of F (¢) are classified into several types. We begin with the necessary definitions, then state the classification theorem. Definition. Let U be a connected component of the Fatou set F( ¢), and assume that U is forward invariant, i.e., ¢(U) = U. • U is parabolic if the boundary of U contains a rationally neutral periodic point ( such that limn ---+oo ¢n(ex) = ( for every ex E U. • U is a Siegel disk if there is an analytic isomorphism f from the unit disk {w E C : Iw I < I} to U such that ¢f is a rotation of the unit disk. • U is a Herman ring if there is an analytic isomorphism f from some annulus {w E C : a < Iwl < b} to U such that ¢f is a rotation of the annulus.
Here a rotation is simply a map of the form w f--+ eiIJ w . If U is a periodic connected component of period n of F (¢ ), then U is forwardinvariant for ¢n and we say that U is parabolic, a Siegel disk, or a Herman ring if it is of the appropriate type for ¢n. Theorem 1.37. Let U be a periodic connected component of the Fatou set of a rational map ¢ E C(z) . Then U fits into exactly one ofthefol/owing categories: (a) U contains an attracting periodic point. (b) U is parabolic. (c) U is a Siegel disk. (d) U is a Herman ring. Proof See [43, Theorem 7.1] or [95, TheoremIV.2.1]. We mentionthat it is nontriv-
ial to prove that Siegel disks and Herman rings exist. In particular, the Fatou set of a polynomial map cannot contain a Herman ring. 0
1.6 Dynamical Systems Associated to Endomorphisms of Algebraic Groups In this section we study certain rational maps whose dynamical properties are comparatively easy to analyze due to the existence of an underlying group structure.
1.6. Dynamical Systems Associated to Algebraic Groups
29
These maps thus provide a class of examples on which to make and test conjectures, albeit with the caution that they are rather special and in some ways atypical. Additional material on this interesting collection of maps may be found in Chapter 6.
1.6.1 The Multiplicative Group The dynamics of the polynomial ¢( z) = zd is extremely easy to analyze, since n we have the explicit formula ¢n (z) = zd . However, there is a more intrinsic reason that zd is such a nice map; namely, it is an endomorphism of the multiplicative group iC* of nonzero complex numbers. In other words, the map
iC*
--+
C",
is a homomorphism. It is the existence of the underlying group iC* that makes zd relatively easy to analyze. Remark 1.3 8. In fancier language, the map ¢( z) = zd is an endomorphism of the algebraic group Gm , where for any field K, the set of points of Gm is the multiplicative group Gm(K) = K*. The full endomorphism ring of Gm is Z via the identification
In even fancier language, Gm is a group scheme G m = SpecZ[X, Yj/(XY - 1) called the multiplicative group scheme. It is characterized by the property that for every ring R, the group of R-valued points of G m is the unit group G m (R) ~ R*.
1.6.2 Chebyshev Polynomials The multiplicative group iC* has a nontrivial automorphism given by the reciprocal map z ~ z-l. We can take the quotient of'C" by this automorphism to obtain a new space that turns out to be isomorphic to iC via the map
[z]
~ Z+Z-l.
The inverse isomorphism takes w E iC to the equivalence class of either of the roots of z2 - zw + 1 = O. This inversion automorphism commutes with the cfh-power map, so when we take the quotient, the cfh_power map will descend to give a map on the quotient space. In other words, there is a unique map Td that makes the following diagram commute:
30
1. An Introduction to Classical Dynamics C*
C*
1
1
C*
C*
The map Td is characterized by the equation
Td(z + z-l) = zd + z-d
for all
z
E C*.
(1.5)
It is not hard to see that Td is a polynomial; indeed, it is a monic polynomial of degree d with integer coefficients called the d:" Chebyshev polynomial. Writing z = e i t with t E C, we see that the Chebyshev polynomial satisfies
Td(2cost) = 2cos(dt). Using these formulas, it is not hard to prove that the Julia set of Td is given by .J(Td) = [-2,2], the closed real interval from -2 to 2, and to derive many further properties of the Chebyshev polynomials. See Exercises 1.30 and 1.31, as well as Section 6.2.
Remark 1.39. Classically the Chebyshev polynomials are normalized to satisfy the identity Td(COS t) = cos(dt), in which case the Julia set becomes .J(Td) = [-1,1]. The two normalizations are related by the simple formula i; (w) = ~ Td (2w).
1.6.3 Rational Maps Arising from Elliptic Curves There are three different types of (connected) algebraic groups of dimension one. First is the additive group Ga(C) = C+, whose endomorphisms are the maps z ----t dz. The dynamical systems associated to endomorphisms of the additive group are relatively uninteresting. Second is the multiplicative group G m (C) = C*, whose endomorphisms z I-t zd have interesting, but relatively easy to analyze, dynamical properties. The same is true of the Chebyshev polynomials, which come from the restriction of z I-t zd to a quotient ofG m . The third type of one-dimensional algebraic groups consists of a family of groups called elliptic curves. We do not have space to go into the theory very deeply, so we are content to make a few brief remarks here and later expand on this material in Chapter 6. For further material on the analytic, algebraic, and arithmetic theory of elliptic curves, see for example (1, 198,254,410,420,412]. We note that the reader may omit this section on first reading, since the material is not used other than as a source of examples until Chapter 6. An elliptic curve E (C) is the set of solutions (x, y) to an equation of the form y2
= x 3 + ax
+b
1.6. Dynamical Systems Associated to Algebraic Groups
31
Q
P
E
Additionof distinct points
Addinga point to itself
Figure 1.3: The addition law on an elliptic curve. together with an extra point O. We also require that ~ = 4a 3 + 27b 2 =I- 0, which means that the cubic polynomial has distinct roots and ensures that the curve E(C) is nonsingular. Alternatively, E( C) is the curve in the projective plane jp'2(C) defined by the homogeneous equation
y 2Z = X 3
+ aX Z 2 + bZ3 ,
with the extra point being 0 = [0, 1, 0]. There is a natural automorphism on E (C) given by [- l ](X ,Y, Z) = (X , -Y, Z ). If P and Q are two points on E (C), then the line through P and Q will intersect E (C) at a third point R, and we define an addition law by setting P EB Q = [- 1]R. (If P = Q, we take the "line through P and Q" to be the tangent line to E (C) at P.) These operations satisfy
P EBO
= P,
P EB[-l](P ) = 0 ,
P EBQ
= QEBP,
(P EBQ)EB R = P EB (QtBR),
so they make E (C) into an abelian group. (The first three equal ities are obvious, while the associativity is somewhat difficult to prove .) An elementary calculation shows that the coordinates of P EB Q are given by rational functions of the coordinates of P and Q. In particular, if a and b are in some field K, and if P and Q have coordinates in K, then P EB Q will also have coordinates in K. Thus E(K) = E(C) n jp'2(K) will be a subgroup of E(C). Figure 1.3 illustrates the group law for points in E(lR.). Repeated addition in any abelian group gives an endomorphism of the group, so for each d 2': 1 we obtain a map
[d] : E (C)
--->
E (C) , d terms
called the multiplication-by-d map. Setting
[O](P) = 0
and
we obtain an (injective) homomorphism
[- d](P) = [- l ]([d]P ),
32
1. An Introduction to Classical Dynamics
Z
dt------t [d].
-----> End(E (C) ),
If this map is not onto, then E is said to have complex multiplication , or CM for short. For example, the curve E : y2 = x 3 + x has complex multiplication, since it has extra endomorphisms such as [i] : (x , y) t---+ (-x, iy). Most curves do not have CM. The multiplication map [d] clearly commutes with the involution [-1 ], so it descends to a map on the quotient space E (C)/{±l}. The quotient space is very easy to describe,
E( C)/{±l} ~ pI (C) ,
(x, y)
t------t
x,
so the multiplication-by-d map descends to give a rational map ¢E,d making both squares in the following diagram commute :
E(C)
[d] -------t
E(C)
[d]
E(C)
1
1
E (C)
{± 1}
-------t
(xtl
{±1}
1(xt
pI (C) ~ pI (C) The map ¢E,d is an example of a Lanes map. Thus ¢E,d is a rational function characterized by the formula
¢E,d(X(P) ) = x ([d]( P) )
for all P E E (C).
(1.6)
Notice the close similarity to the defining property (1.5) of the Chebyshev polynomials. More generally, it turns out that every endomorphism u E End(E ) commutes with [-1], even if E has complex multiplication, so every endomorphism u determines a rational function ¢E,u on P l( C).
Example 1.40. Let E : y2 = x 3 + ax + b be an elliptic curve as above. Then the duplication map [2] : E(C) --. E(C) leads to the rational function
x4 ¢E,2(X) =
2ax2 - 8bx + a2 4x3 + 4ax + 4b
-
Example 1.41. Let E : y2 = x 3 + x be the CM elliptic curve mentioned earlier. Then the map [1 + i] : E (C) --. E (C) defined by [1 + i](P) = P EB [i](P ) leads to the rational function
¢ E,l+i(X) =
;i
(x+
~) .
Proposition 1.42. Let E : y2 = x 3 + ax + b be an elliptic curve, let d ~ 2 be an integer, and let ¢E,d : PI (C) --. Pl (C) be the associated rational map as above. Then
33
1.6. Dynamical Systems Associated to Algebraic Groups
• • •
•
L
•
•
• Wl +W2
•
•
• • Figure 1.4: A lattice and fundamental domain associated to an elliptic curve.
PrePer(1)E,d) =
X
(E(C)tors) ,
where we recall that the torsion subgroup E(C)tors of E(C) is the set ofelements of finite order (cf Example 0.2). Proof We leave the proof as an exercise; see Exercise 1.32.
D
The dynamics of the rational function 1>E,d on ]P'l (C) can be analyzed using the group law on the larger space E(C). The utility of this approach is further enhanced by the analytic uniformization of E(C), which we now briefly describe. A lattice L in
The quotient space
A lattice L and a fundamental domain are illustrated in Figure 1.4. Notice that
1/JE :
-t
E(C).
Further, the isomorphism 1/JE respects the group structure,
34
1. An Introduction to Classical Dynamics
In particular, if we set P = 'l/JE(Z) in formula (1.6) and use the relation
[d]('l/Je(z)) = 'l/JE(dz) , then we obtain the useful formula
¢>E,d(X('l/JE(Z)))
= x([d]('l/JE(Z))) = x('l/JE(dz)).
To ease notation, we will let p( z) = x('l/JE(z)). The function p( z) is a slight variant of the classical Weierstrass p-function. It is meromorphic on C with double poles at the points of the lattice Land holomorphic elsewhere. Then our relation becomes (1.7) ¢>E,d(p(Z)) = p(dz), and iteration is given by the simple formula ¢>E d (p( z)) = p( dn z). The upshot of this discussion is that we ob~in a commutative diagram
CIL
z-s-tiz
~
1~
CIL
1~
that is of great assistance in studying the dynamics of the rational function ¢>E,d. For example, the map z f-t dz is clearly d2-to-l on CI L, so we see that ¢>E,d has degree d2 . Further, a point x = p( z) will be fixed by ¢> E,d if and only if z satisfies the congruence dz == ±z (mod L), so the fixed points of
(1.8)
But we know that p(z) is a meromorphic function on CI L, or equivalently, it is a meromorphic function on C with the periodicity property
p(z + w) = p(z)
for all wE L.
Differentiation gives the same relation p' (z + w) = p' (z) for the derivative. (Notice the analogy with the function e21ri z , which is holomorphic on C and invariant under the translations z f-t z + n for n E Z.) We evaluate at z = d(, use the fact that d( == ±( (mod L) to get p'(d() = p'(±() = ±p'((), and substitute into (1.8) to obtain the relation
35
Exercises ¢~ ,d(P(())p'( ( ) =
dp' (d( ) = ±dp' (().
Finally, canceling p' (O yields ¢~ d(P(( )) = ±d. In particular, (almost) every fixed point of ¢ E,d is expanding and th~s is in the Julia set. But much more is true. The relation (1.7) shows immediately that
so ¢'E,d = ¢E,dn. Thus the periodic points of ¢E,d of period dividing n are precisely the fixed points of ¢E,dn , and their multipliers are (almost) all equal to ±dn . This proves that (almost) every periodic point of ¢E,d is expanding , hence in the Julia set. Finally, we observe that the set of points in C j L satisfying
z ==
±~ z
(mod L )
for some n
= 1,2 ,3 , .. .
is dense in Cj L, so their image using the double cover p : Cj L -+ jp'l (C) is dense in jp'1(C). We have thus shown that .J (¢E,d) = jp'1(C). This proves the following theorem, which was published by Lattes in 1918 and provided the first known examples of rational maps with empty Fatou set. (See also [300, §6) for a discussion of earlier work by Schroder and Bottcher on elliptic functions and their associated rational maps, as well as an 1815 paper of Babbage in which he uses semiconjugacy to study the periodic points of certain maps.)
Theorem 1.43. (Lattes [260]) Let E be an elliptic curve, let d ~ 2 be an integer, and let ¢ E,d : jp'1(C) -+ jp'1 (C) be the rational map of degree d2 characterized by (1.6). Then and
Exercises Section 1.1. Rational Maps and the Projective Line
1.1. Define a mapping from the x y-plane (which we identify with C) to the unit sphere S2 in ]R3 by mapping z E C to the point z" E S2 such that the line through z and z" goes through the point (0, 0, 1). Notice that z" ---+ (0,0 ,1) as z ---+ 00, so if we set 00* = (0,0,1), we obtain a bijection jp'1(C) = CU {00} --->S2, Z f--> Z*, as illustrated in Figure I. I. Prove that with this identification, the chordal metric on jp'l (C) is given by p(z , w) = ~ I z* - w*l.
1.2. (a) Prove that the inversion map z ,... z -l is an isometry Ofjp'l (C) for the chordal metric (i) by a direct calculation using the formula for p; and (ii) by using the identification jp'1(C) ~ S2 described in Exercise I. I. (b) Let a, b E C satisfy lal 2 + Ibl2 = 1, and let j( z ) = (az - b)j (bz + a). Prove that P(J (z ),j(w)) = p(z ,w) for all z, w E jp'1(C) . (Hint. Although this can be proven directly, an alternative method is to show that j corresponds to a rigid rotation of S2, hence preserves chordal distances. For a nonarchimedean analogue, see Lemma 2.5.)
36
Exercises
1.3. Let ¢ : jp'l (C) -. jp'l (C) be a rational map. (a) Prove that there is a constant C (¢) such that ¢ satisfies a Lipschitz inequality for all a, (3 E jp'l (C).
p( ¢( a), ¢((3)) :::; C( ¢ )p(a, (3)
(b) Find an explicit formula for the Lipschitz constant C(f) in the case of a linear map
f(z)
= (az + b)/(cz + d) E PGL2(C).
1.4. Let (a, a', a") and ((3, (3', (3") be triples of distinct points in jp'l. Prove that there exists a unique automorphism f E PGL 2(C) satisfying
f(a)
= (3,
f(a')
= (3',
f(a")
= (3".
(Hint. First do the case that one ofthe triples is (0,1,00).)
Section 1.2. Critical Points and the Riemann-Hurwitz Formula
1.5. Let ¢( z) E q z) be a rational function. In the text we defined the ramification properties of ¢ at a point a E jp'l (C) provided that a -I 00 and ¢( a) -I 00. In general, let f E PGL2(C) be any linear fractional transformation, let ¢f = ¢ 0 I, and let (3 = I ( a ). If (3 -I 00 and ¢f ((3) -I 00, we say that ¢ is ramified at a if (¢f)' ((3) = 0 and we define the ramification index of ¢ at a to be
rIo
ea(¢)
r
= e{3(¢f).
(a) Assuming that none of a, (3, ¢(a), ¢f ((3) are equal to 00, prove that
Conclude that the ramification points of a rational map ¢ : jp'l -. jp'l are independent of the choice of coordinates, i.e., the vanishing of ¢' (a) is independent of the choice of f. (b) Show that the ramification index e a (¢) is well-defined, independent of the choice of f. (c) Under what conditions is it true that the derivative ¢' (a) is independent of the choice of f?
1.6. Let ¢( z) E
q z) be a rational function of degree d, say given by
¢(z) = F(z) = ao + alZ + + adzd. G(z) bo + bIZ + + bdzd (a) Prove that ¢ is ramified at z = 0 if and only if aobl = albo. (b) Prove that ¢ is ramified at z = 00 if and only if adbd-l = ad-Ibd. (c) Find a similar criterion for eo(¢) ::::: 3. Same question for Coo (¢) ::::: 3. 1.7. Let ¢ : jp'l -. jp'l be a rational function of degree d ::::: 2. Using the standard spherical measure on jp'l (C), normalized so total area is 1, prove that
1
1(¢n)'(z)1 dM(Z)
rv
n
d
as n -. 00.
1I'1(C)
Conclude that "on average," the map ¢ is expanding. 1.8. (a) Let C be a compact Riemann surface with 9 holes. Prove that if C is triangulated using V vertices, E edges, and F faces, then V - E + F = 2 - 2g. The number 9 is called the genus of C.
Exercises
37
(b) Let ¢> : C 1 ---> C2 be a finite map of degree d of compact Riemann surfaces, and let gi be the genus of C«. Give a topological proof of the Riemann-Hurwitz formula,
2g1 - 2 = d(2g2
-
2) +
L
ep(¢» - 1.
P EC ,
1.9. Let ¢> E iC( z ) be a rational map of degree d :::': 2. (a) Prove that 00 is a totally ramified fixed point of ¢> if and only if ¢> E iC[z]. (b) Let a E pI (iC) with a :f 00 . Prove that a is a totally ramified fixed point of ¢> if and only if (z - a)d / (¢>(z) - a) is in iC[z]. (c) Let a E pI (iC) be arbitrary. Prove that a is a totally ramified fixed point of ¢> if and only if there exists a linear factional transformation f E PGL2(iC) such that f-l (a) = 00 and ¢>f (z) E iC[z]. 1.10. Let K be a field of characteristic p > 0 and let ¢>(z) E K (z) be a rational function. (a) Show that the following are equivalent : (i) ¢>(z) ~ K(zP). (ii) The formal derivative ¢>' (z) is not identically zero. (iii) The field extension K (z)/ K (¢>(z)) is separable. If these conditions hold, we say that ¢> is separable. (b) Assume that ¢> is separable . Recall that the ramification index is defined by the formula e o (¢» = ordo (¢>(z ) - ¢>(a )) . Let T o (¢» = ordo (¢>' (z )) . If a is a critical point of ¢>, then we say that ¢> is tamely ramified at a if p f eo (¢» , and ¢> is wildly ramified at a ifp I eo (¢» . Prove that T o (¢» :::': eo (¢» - 1, with equality if and only if ¢> is either unramified or tamely ramified at a . (c) Prove that
2d - 2 :::':
L
To
(¢» :::':
oE II" (C)
L
(eo (¢» - 1).
o EI" (C)
(Hint. Mimic the algebra ic proof of Ihe Riemann-Hurwitz formula , Theorem 1.1.) (d) Deduce that a separable map has at most 2d - 2 critical points . (e) More generally, if the rational function ¢> is wildly ramified at t points , prove that ¢> has at most 2d - 2 - (p - l )t critical points.
1.11. Let K be a field of characteristic p > 0 and let ¢>(z) E K (z) be a rational function. (a) Suppose that ¢> is separable and that ¢>n E K[z] for some n :::': 1. Prove that ¢>2 E K[z]. Thus Theorem 1.7 is true in characteristic p for separable maps. (b) Let p = 2 and ¢>(z) = 1 + Z - 2. Prove that ¢>2 ~ K[z] and that ¢>3 E K[z]. Thus Theorem 1.7 need not be true in characteristic p for inseparable maps. (c) Suppose that ¢>2 ~ K[z] and that ¢>n E K[z] for some n :::': 3. Prove that ¢> has the form ¢> = (az q + b)/ (cz q + d), where q is a power ofp.
1.12. Let ¢> : pI ---> p I be a rational map of degree d :::': 2, and suppose that there is a point a E pI and an e > 0 such that
eo (¢>n ) :::': (1 +
fr
for infinitely many n :::': 1. Prove that a is a preperiodic point for ¢>, and that the eventual period of a is at most (2d - 2)/ f.
Exercises
38 Section 1.3. Periodic Points and Multipliers
1.13. We defined the multiplier of ¢ at a fixed point 0: to be the derivative ACt (¢) = ¢' (0:). This definition obviously must be modified if 0: = 00. In order to be compatible with Proposition 1.9, we must set Aoo(¢) = (¢f)'(f-l(oo)) for all f E PGL2(C). Show that taking f (z) = 1/z leads to the definition
Prove that Aoo (¢) is finite and may be computed without taking a limit by showing that the fraction on the righthand size is a rational function in iC(z) with no pole at z = 0, so it may be evaluated at z = o. 1.14. Let ¢ : jp'l ~ jp'l be a rational map. (a) Let P E jp'l bea point of exact period n for ¢ and suppose that ¢k(p) that n I k. (b) Prove that Per n (¢) is the disjoint union of Per;; (¢) over all min.
=
P. Prove
1.15. Let ¢ : jp'l ~ jp'l be a rational map of degree at least 2. (a) Let n > m ~ O. Prove that {p E jp'l : ¢n(p) = ¢m(p)} is a finite set. (b) Suppose that ¢l and ¢2 are rational maps of degree at least 2 and suppose that ¢l and ¢2 commute with one another, i.e., ¢l 0 ¢2 = ¢2 0 ¢l. Prove that PrePer(¢l)
= PrePer(¢2).
(Hint. Use (a) and apply Exercise 0.2.)
1.16. Suppose that 0: is a periodic point of ¢. Prove that the orbit of 0: contains a critical point of ¢ if and only if the multiplier ACt (¢) vanishes. 1.17. This exercise generalizes the multiplier sum formula described in Theorem 1.14. Let ¢(z) E iC(z) be a nonconstant rational map and let 0: E C be a fixed point of C. The residue fixed-point index of ¢ at 0: is the quantity
1 t( ¢, 0:) = -2.
1
1l"t
Iz-Ctl=-
z
_dz¢( )' z
where the integral is any sufficiently small loop around 0:. (a) IfACt (¢) =11, prove that 1 t(¢, 0:) = 1 _ ACt(¢)" (b) Prove that in all cases, the index t(¢, 0:) is independent ofthe choice oflocal coordinate at 0:. (c) Prove the index summation formula
L CtEFix(,p)
t(¢, 0:)
= 1.
39
Exercises (d) Suppose that An (¢J)
= ¢Jt (0:) = 1 and that ¢J" (0:) =I O. Provethat ~(¢J, o:)
=
¢J"' (o:) / 6 . (¢J"(0:) /2) Z
In other words, if the series expansionof ¢J( z) around z
¢J(z)
= 0: + (z -
= 0: has the form
0:) + A(z - o:) z + B( z - 0:)3
+ ...
with A
=I 0,
then ~ ( ¢J,o:) = BIA z .
(e) The polynomial¢J( z) = z + Az z + z3 has fixedpoints {O, -A, 00 }. The point at infinity is superattracting, so Aoo (¢J) = O. Use the formula in (d) and the summation formula to compute L A(¢J) , and then check your answer by computing LA (¢J) directly.
q z) be a rational function of degree d ~ 2. (a) Prove that #Pern(¢J) ~ d" + 1. (b) Prove that # Pern(¢J) --> 00 as n --> 00 . (c) Prove that Per~* (¢J) is nonempty for infinitelymany values of n. (d) * More precisely, prove that ifPer~* ( ¢J) is empty, then (n , d) is one of the pairs
1.18. Let ¢J( z) E
(n, d) E {(2, 2), (2, 3), (3, 2), (4, 2)}. 1.19. Let ¢J(z) E iC[z] be a polynomial of degree d ~ 2.
(a) Ifm I n, prove that the polynomial ¢Jm(z) - z dividesthe polynomial ¢In(z) - z. (b) Let f.l be the Mobius f.l function. Prove that for every n ~ 1, the rational function
~ (z)
=
II (¢Jm(Z) - zt
(n/ m)
min
is a polynomial. The polynomial ~( z ) is a dynamical analogue of the classical cyclotomic polynomial TImln (zm - l )JL(n/m). (c) Compute the first few polynomials ~ (z) for the quadratic polynomial ¢Jc(z) = ZZ+ c. (d) Let 0: be a root of
1.21. Let (S l, Pl) and (S z, pz) be metric spaces. A collection of maps from S , to Sz is said to be uniformly continuous iffor every € > 0 there exists a 8 > 0 such that
The family is said to be uniformly Lipschitz if there is a constant C PZ (¢J(O:), ¢J({3 )) ~ C· PI(0:, {3)
= C (
such that
for all 0:,{3 E s, and allrj> E .
Exercises
40
(a) If q, is uniformly continuous, prove that q, is equicontinuous at every point of 81. (b) If q, is uniformly Lipschitz, prove that q, is uniformly continuous. (c) Let q, = {4>n} n::o: 1 and suppose that for every point a E 8 1 , the limit
¢(a)
= n---;oo lim 4>n(a)
exists. If q, is equicontinuous at a E 81, prove that ¢ is continuous at a. Give an example to show that the equicontinuity assumption is necessary.
1.22. Give an example to show that equicontinuity is not an open condition. Thus in order to ensure that the Fatou set is open, the definition of F( 4» must include the requirement that it be open.
1.23. Let 4> : 8 --+ 8 be a surjective continuous map of a metric space with the additional property that 4> maps open sets to open sets. (a) Give a rigorous proofthat the Fatou and Julia sets of 4> are completely invariant, that is, 4>-1 (F) = F = 4>(F) and 4>-I(J) = J = 4>(J). (b) Is (a) true without the assumption that 4> is surjective? (c) Is (a) true without the assumption that 4> maps open sets to open sets? 1.24. Let 4>( z) E qz] be a polynomial map, and let F 00 be the connected component of F( 4» containing the point 00. Prove that 4>-1 (Foo ) = F oo = 4>(Foo ) . In other words, for a polynomial map, the connected component of 00 is completely invariant for 4>. 1.25. Let 4>( z) = 1 - 2/ Z2 be the rational map from Example 1.31. (a) Let fez) = II z. Prove that (4)1)' (0) = 0 and conclude that 00 is a critical point of 4>. (b) Let fez) = zl(z - 1). Prove that (1/)'(0) = 0 and conclude that 0 is a critical point of 4>. 1.26. Let 4>(z) E Q(z) be a rational map of degree d 2: 2 with coefficients in Q, and let J (4); Q) = J (4)) n jp'1(Q) be the set of points in the Julia set whose coordinates are algebraic numbers. (a) Is J( 4>; Q) Galois-invariant? (b) Is J(4); Q) infinite? More generally, is J(4); Q) dense in J(4))? (c) ** Does J (4); Q) contain points that are not preperiodic? More generally, does J (4); Q) contain infinitely many nonpreperiodic points Pi such that the orbits of the Pi are disjoint?
Section 1.5. Properties of Periodic Points 1.27. Let a be a periodic point ofa rational function 4> E C(z). (a) If a is attracting, prove that a E F( 4». (b) If a is repelling, prove that a E J (4>). 1.28. Let 4>(z) = Z2 - 2. Prove that the Julia set of 4> is the closed interval on the real axis between -2 and 2. Find the periodic points of 4>, compute their multipliers, and show directly that J(4)) is equal to the closure of Perre). (Hint. Write z = ei t + e- i t = 2cos(t), so 4>(z) = 2 cos(2t). Prove that 4>n(z) = 2 cos(2nt) and use this formula to study the dynamics of 4>.)
Exercises
41
q
1.29. Let 4>( z) E z] be a polynomial of degree d ;::: 1, and suppose that every a E Per n (4)) has multiplier Aq,(a) f. 1. Prove that the equation 4>n(z) = z has simple roots, and deduce that Per n (4)) contains exactly d" + 1 distinct points (including the point 00).
Section 1.6. Dynamical Systems Associated to AlgebraicGroups 1.30. This exercise describes algebraic properties of the Chebyshev polynomials Td (w). (a) Prove that T 2(w) = w 2 - 2, T3(w) = w 3 - 3w, and T4(W) = w 4 - 4w 2 + 2. (b) Prove that Td(Te(w)) = Tde(w) for all d, e;::: O. (c) Prove that Td(-w) = (-l)dTd(w). Thus T d is an odd function if d is odd and it is an even function if d is even. (d) Prove that the Chebyshev polynomials satisfy the recurrence relation
where we use the initial values To(w) = 2 and Tl (w) = w. (e) Prove that the generating function of the Chebyshev polynomials is given by 00
l: Td(W)X
d
=
d=O
2 -wX X X2' l-w +
(f) Prove that the dth Chebyshev polynomial is given by the explicit formula
1.31. This exercise describes dynamical properties of the Chebyshev polynomial Ts (w) for a fixed d ;::: 2. (a) Let U be an open subinterval of [-2, 2]. Prove that there exists an integer n ;::: 1 such that T:l(U) = [-2,2]. (b) Prove that limn~oo T:t(w) = 00 for all points wE iC not lying in the interval [-2,2]. (c) Prove that the Julia set .:J(Td) is the closed interval [-2,2]. (d) Find all of the periodic points ofTd and compute their multipliers. Observe that .:J(Td) is the closure ofthe (repelling) periodic points. (e) Let F(w) E C[w] be a polynomial of degree d ~ 2 with the property that the interval [-2,2] is both forward and backward invariant for F. Prove that F(w) = ±Td(W). 1.32. (a) Let A be an abelian group, let A tors = {P E A : nP = 0 for some n ~ I} be the torsion subgroup of A, let d ;::: 2 be an integer, and let 4> : A --+ A be the map 4>(P) = dP. Prove that PrePer( 4» = A tors. (This is Proposition 0.3, but try reproving it without looking back at our proof.) (b) Let E be an elliptic curve, let d ;::: 2 be an integer, and let 4> E ,d : ]p'l --+ ]p'l be the rational map associated to multiplication-by-d on E, as described in Section 1.6.3. Prove that PrePer(4)E,d)
= x(E(iC)tors).
Chapter 2
Dynamics over Local Fields: Good Reduction The study of the dynamics of polynomial and rational maps over ~ and C has a long history and includes many deep theorems, some of which were briefly discussed in Chapter 1. A more recent development is the creation of an analogous theory over complete local fields such as the p-adic rational numbers Qp and the completion C p of an algebraic closure of Qp. The nonarchimedean nature of the absolute value on Qp and C p makes some parts of the theory easier than when working over C or R But as usual, there is a price to pay. For example, the theory of nonarchimedean dynamics must deal with the fact that Qp is totally disconnected and far from being algebraically closed, while C p is not locally compact. In this chapter we begin our study of dynamics over complete local fields K by concentrating on rational maps ¢ that have "good reduction." Roughly speaking, this means that the reduction of ¢ modulo the maximal ideal of the ring of integers of K is a "well-behaved" rational map ¢ over the residue field k of K. Thus studying the dynamics of ¢ over k allows us to derive nontrivial information about the dynamics of ¢ over K. In Chapter 5 we take up the more difficult, but ultimately more interesting, case of rational maps with "bad reduction."
2.1 The Nonarchimedean Chordal Metric We begin by quickly recalling the definition and basic properties of absolute values, especially those satisfying the ultrametric inequality. Definition. An absolute value on a field K is a map I·I:K---+~
with the following properties:
• lal
20, and
lal =
0 if and only if a =
43
o.
44
2. Dynamics overLocalFields: Good Reduction
• la/31 = lal . 1/31 for all a, /3 E K . • la + /31 :::; lal + 1/31 for all a, /3 E K
(triangle inequality).
A valuedfield is a pair (K, 1 . I K) consisting of a field K and an absolute value on K, although we often omit the absolute value in the notation. A map of valued fields i : K ---- L is a field homomorphism that respects the absolute values,
for all a E K.
Definition. Let (K, estimate
I . I) be a valuedfield. If the absolutevalue satisfies the stronger la + /31 :::; maxjjc], 1/31}
for all a, /3 E K,
(2.1)
then the absolute value is nonarchimedean or ultrametric. The associated valuation is the homomorphism v : K*
----t
It satisfies
v(a +
v(a) = -log lal.
JR,
m~ mini v(a), v(/3)}.
The valuation v is discrete if v (K*) is a discrete subgroup of JR, in which case the associated normalized valuation (sometimes denoted ord.i) is the constant multiple of v chosen to satisfy ord, (K*) = Z. Example 2.1. The field Q has the usual real absolutevalue
lal oo = max{a, -a}. For each prime p it also has a p-adic absolutevalue definedas follows. Every nonzero rational number a has a unique factorization of the form a
=±
IT
pep(a)
with ep(a) E Z.
P prime
Then
lal p
= p-ep(a).
The p-adic absolute values are nonarchimedean. Two absolute values are said to be equivalent if there is a constant r > a such that lail = lal2 for all a E K. Ostrowski's theorem says that up to equivalence, the real absolute value and the p-adic absolute values are the only nontrivial absolute values on Q; see [78, 1.4.2,Theorem 3] or [249, 1.2, Theorem 1]. Example 2.2. Let k be a field and let K = k(T) be the field of rational functions. Then each a E k determinesan absolute value on K associatedto the valuation ord., that gives the order of vanishing of f(T) E k(T) at T = a. The degree map deg : k(T)* ---- Z is also a valuation. If k is algebraicallyclosed, this yields the complete set of absolutevalues on k(T) that are trivialon k, up to equivalence. More generally, if k is not algebraically closed, there is a valuation corresponding to each monic
45
2.1. The Nonarchimedean Chordal Metric
irreducible polynomial in k[T]. In this book we are primarily interested in the number field scenario, i.e., the field Q and its extensions and completions, but the reader should be aware that it is also interesting to study the analogous case of function fields , i.e. , the field k[T] and its extensions and completions. We now prove the elementary, but extremely useful, fact that if 10:1 the ultrametric inequality (2.1) is actually an equality.
I:-
Lemma 2.3. Let K be a field with a nonarchimedean absolute value let a, /1 E K. Then
Proof We suppose that 1/1lv
<
1/11, then
I . Iv
and
lal v > 1/1lv. The strict inequality
lal v = I(a
+ /1) - /1 lv ::; maxi la + /1l v, l/JIv}
implies that the maximum on the right is [o + /1lv, so we find that [o ], ::; la + /1lv' The opposite inequality is also true , since [o + /1l v ::; m ax{ la lv , 1/1l v} = lal v . 0 Recall that the chordal metric on by the formula
pI
(C), which we now denote by Poo, is defined
for points PI = [X I , YI ] and P2 = [X 2 , Y2] in p I (iC). In the case of a field K having a nonarchimedean absolute value I . Iv, it is convenient to use a metric given by a slightly different formula. Definition. Let K be a field with a nonarchimedean absolute value I . Iv, and let PI = [Xl, YI ] and P2 = [X2, Y2] be points in PI (K ). The u-adic chordal metric on pI (K ) is
It is clear from the definition that Pv(PI , P2) is independent of the choice of homogeneous coordinates for PI and P2. The first thing to check is that Pv is indeed a metric. In fact, it is an ultrametric; that is, it satisfies the nonarchimedean triangle inequality. Proposition 2.4. The u-adic chordal metric has the following properties. (a) 0 ::; Pv (PI , P2 ) ::; 1. (b) Pv(PI , P2) = 0 if and only if PI = P2. (c) Pv(PI ,P2) = Pv(P2,PI ). (d) Pv(PI ,P3 ) ::; max{PV(P 1 ,P2),Pv(P2,P3 )} .
46
2. Dynamics over Local Fields: Good Reduction
Proof The lower bound in (a) and parts (b) and (c) of the proposition are obvious from the definition. For the upper bound in (a), we use the nonarchimedean nature of v to compute IXIYZ
X zYIl v :::; max{I XIYzl v, IXzYIl v}
-
< max{IXIl v, IYIlv} m ax{IXzl v, IYzl v} . The proof of (d) requires the consideration of several cases. The following useful lemma makes the proof more transparent by allowing some freedom to change coordinates. It is the analogue of the fact that the classical chordal metric is invariant under linear fractional tran sformations that define rigid rotations of the Riemann sphere (cf. Exercise 1.2).
Lemma 2.5. Let
R = {a E K : lal v be the ring ofintegers ofK , and let tion ofthe f orm
f ([X YJ) ,
i.e., f
E
aX + bY cX +dY
=
f : pI
-t
:::;
I}
pI be a linear fractional transforma-
with a, b, e, d E R and ad - be E R* ,
PGLz(R ). Then Pv(J(PI ), f (Pz)) = Pv(PI , Pz)
(N.B. It is crucial that the quantity ad - be is a unit.) Proof Write each point as Pi = [Xi, Yi] with Xi , Yi E R and at least one of X i or Y, inR*. Then max{IXil v , IYilv} = 1,so
Pv(PI, PZ ) = IX1 YZ -XzYIl v. Further, the identities
d(aXi + bYi) - b(eXi e(aXi + bYi) - a(eXi
+ dYi) = (ad - be)Xi , + dYi) = - (ad - be)Yi,
and the fact that max{ IXi lv, IYi Iv} = 1 and lad - belv = 1 immediately imply that max{l aXi
+ bYilv, leXi + dYilv } =
1.
Thus
Pv(J(Pd , f (Pz ))
=
I(aX I + bY1) (eX
Z
+ dYz ) - (aXz + bYz )(eX 1 + dYdl v
= I(ad - be)(XIYz - XzYI )l v
= Pv(PI, Pz), where we have again made use of the fact that lad - bel v
= 1.
D
2.2. Periodic Points and Their Properties
47
We resume the proof of Proposition 2.4 and write each point as Pi = [Xi, Yi] with Xi, Yi E R and at least one of Xi or Yi in R*. Then
max{IXil v, IYilv} = 1 and Pv(Pi, Pj) = IXiYj - XjYilv, as usual. If IX2 1v > 1Y21v, we apply the map f = Y/X to the three points. This preserves the chordal distance (Lemma 2.5) and allows us to assume that
IX2 1v ::; 1Y21v = 1. Next we apply the map f = (Y2 . X - X 2 . Y)/Y to the three points. Lemma 2.5 again tells us that the chordal distance is preserved (note that 1Y21v = 1), and we are reduced to the case that P2 = [0,1]. Finally, we compute
IXI Y3 - X3 YI lv ::; max{ IXI Y3Iv, IX3 YIlv} ::; max{IXIl v, IX3Iv}, which is exactly the desired inequalitywhen P2 is the point [0, 1].
o
2.2 Periodic Points and Their Properties In Section 1.3 we described various properties of periodic points for rational maps definedover C. Virtuallyall of these definitions make sense, mutatis mutandis,when we work over any field with an absolute value. In this section we briefly recall the relevantmaterial. Let K be a field with an absolute value I . I and let ¢( z) E K (z) be a nonconstant rational map. The multiplier of ¢ at a fixed point 0: E K is the derivative
An (¢) = ¢' (0: ) . (See Exercise 1.13 for the case 0: = 00.) The multiplier is well-defined, independent of the choice of coordinateson pI; see Proposition 1.9. More generally, if 0: E pI (K) is a point of exact period n for ¢, then 0: is a fixed point of ¢n and we definethe multiplier of ¢ at 0: to be
The multiplier may be calculated using the chain rule as
The magnitude of the multiplier determines, to some extent, the behavior of ¢ in a small neighborhoodof a periodic point 0:. The periodic point 0: is called superattracting
if An (¢) = 0,
attracting
if IAn (¢) I < 1,
neutral (or indifferent)
if IAn (¢) I = 1,
repelling
if IAn (¢) I > 1.
The neutral periodic points are further divided into two types. The rationally neutral periodic points are those whose multiplier is a root of unity. The others are called irrationally neutral.
48
2. Dynamics over Local Fields: Good Reduction
2.3 Reduction of Points and Maps Modulo p One of the most important gadgets in the number theorist's toolbox is reduction modulo a prime. Thus when studying the number-theoretic properties of an object, we reduce it modulo a prime, analyze the properties of the hopefully simpler object, and then lift the information back to obtain global information. A typical example is provided by Hensel's lemma, which under certain circumstances allows us to lift solutions of a polynomial congruence f(x) == 0 (mod p) to solutions in Zp. Then, using information gathered from many primes, one is sometimes able to deduce results for a global field such as Ql. Our principal objects of study are maps ¢ : r 1 --t r 1 . In this section we study the behavior of such maps under reduction modulo a prime. We work over a discrete valuation ring, so we set the following notation:
K
I . Iv R p
R*
k
a field with normalized discrete valuation v : K* = cV(x) for some e
--*
Z.
> 1, an absolute value associated to v.
{a E K : v(a) :::: O}, the ring of integers of K. = {a E K: v(a) :::: I}, the maximal ideal of R. = {a E K : v(a) = O}, the group of units of R. = Rf», the residue field of R. =
reduction modulo p, i.e., R
--t
k, a
f---t
a.
Before studying reduction of maps, we consider the problem of reducing points modulo p. This is as easy in r N as it is in r 1 , so we look at the general case. Let
P = [XO,X1,'" ,XN] E rN(K) be a point defined over K. We cannot immediately reduce the coordinates of P modulo p, since some of the coordinates might not be in R. However, since the coordinates of P are homogeneous, we can replace them with
P
= [exo, eX1,"" eXN]
for any e E K*. Choosing e to be highly divisible by p, we can ensure that every ex, is in R. However, if we overdo this process and end up with every ex, in the prime ideal p, then when we reduce modulo p, we end up with [0,0, ... ,0], which does not represent a point of projective space. The trick is to "clear the denominators" of the Xi'S as efficiently as possible. To do this, we choose an element a E K* satisfying
v(a) = min{ v(xo), v(xt}, ... ,V(XN
n.
(2.2)
For example, a could be the Xj having minimal valuation. Then a- 1xi E R for every i, so we can reduce these quantities modulo p. We define the reduction of P modulo p to be the point
F= [~,~, ... ,a~] Er 1 (k).
2.3. Reduction of Points and Maps Modulo p
49
Note that P has at least one nonzero coordinate, since at least one of the numbers 0: - 1Xi is a unit. Hence P is a well-defined point in p I (k). We say that P = [xo , . . . , XN] has been written using normalized coordinates if
min{v(xo),v(xd, . .. , V (XN)} = 0, in which case
P is simply
[xo, ... , XN]'
iJ5]
E p I (Ql) . We can reduce P Example 2.6. Consider the point P = [I~ ' ; 5' ~~ , modulo 11 without any modification , since every coordinate is an l l-adic integer and not all coordinates vanish modulo 11. Thus P = [1,10,7,9] (mod 11). However, if we want to reduce P modulo 3, then we first need to divide all of the coordinates by 3,
P _ [3 -
9 24 27] _ [1 3 8 9] 14 ' 35 ' 49' 245 - 14' 35' 49' 245 '
and then P = [2,0,2,0] (mod 3). Similarly, in order to compute P modulo 7, we first multiply the coordinates by 49,
P _ [~ ~ 24 ~] _ [21 63 24 27] -
and then
P=
14 ' 35 ' 49 ' 245
-
2 ' 5 ' 1 ' 5
'
[0,0,3,4] (mod 7).
It may appear that the reduction P depends on the choice of 0: . We now check that this is not the case.
Proposition 2.7. Let P = [xo , . .. , XN ] E pendent of the choice of 0: satisfy ing (2.2).
pN (K).
Proof Suppose that 0: and {3 both satisfy (2.2). Then tion, so 0:{3- 1 E R*. This allows us to compute
[------ -----0:
- I
Then the reduction P is inde-
0:
and {3 have the same valua-
------ ] [------ ------ ------------ ------ ------ ] [------xo,{3------ ,{3------ ]
-I - 1 Xo , O: XI, ... , O: X N
_ -
_
-
- 1 - I -1 - 1 - 1 -I 0:{3 0: xo,o:{3 0: X I, .. . , o:{3 0: XN -1
{3
-I
XI, ...
- 1
XN
·
Hence the reduction of P modulo p is independent of the choice of 0: satisfying (2.2).
o We next prove an easy and useful lemma that relates reduction modulo p to v-adic distance .
Lemma 2.8. Let PI and P 2 be po ints
in ]P'I (K) .
Then
2. Dynamics over Local Fields: GoodReduction
50
Proof Write PI = [Xl, YI ] and P2 = [X2 , Y2 ] using normalized coordinates, so in particular Pv(pl, P2 ) = IXI Y2 - X 2 Yl lv ' Suppose first that PI = P2 • This means that there is a U E k* such that Xl = uX2 and Y"l = UY"2. In other words, there is a U E R* such that that Xl =' uX 2 (mod p) and YI =' uY2 (mod p). Hence
which completes the proof that Pv (PI, P2 ) < 1. Next suppose that Pv (PI , P2 ) < 1, which implies that X IY2 =' X 2YI (mod p). If X IX2 ¢. 0 (mod p), then Xl, X2 E k*, so we have
On the other hand, if X IX2 =' 0 (mod p), then the fact that the coordinates are normalized and the equality X I Y2 =' X 2 YI (mod p) imply that Xl=' X 2 =' 0 (mod p), 0 so PI = P2 = [0, 1]. This completes the proof of the lemma. As a first application, we show that fractional linear transformations in PGL 2(R) respect reduction modulo p.
Proposition 2.9. Let P, Q E pl(K) and J E PGL2(R). Then
P = Q if and only if J(P) = J(Q). Proof We combine Lemmas 2.5 and 2.8. Thus
P= Q
~
Pv(P, Q) < 1
from Lemma 2.8,
~
Pv(J(P),J(Q)) < 1
from Lemma 2.5,
~
~
~
J(P) = J(Q)
from Lemma 2.8 again.
o
Example 2.10. Let J = (~~), so J .,. PGL 2 (Z3). Consider the points P = [7,5] and Q = [4,2] in pI (Q3)' They satisfy P = Q = [1,2] in pI (IF3), but
J(P) = [45,75] = [3,5] =' [0,1] J(Q) = [24,36] = [2,3] =' [1,0]
(mod 3), (mod 3),
~
so J(P) f- J(Q). This shows the necessity of the condition J E PGL 2(R) in Proposition 2.9.
It is easy to see that if K is a field and PI, P2 , P3 are distinct points in pI (K), then there is an element ofPGL 2 (K) that moves them to the points 0, 1,00. (See Exercise 1.4.) The next proposition gives a stronger result for points whose reductions are distinct. It is especially useful because Lemma 2.5 says that the nonarchimedean chordal metric is invariant for maps in PGL 2(R), so changing coordinates via an element ofPGL 2(R) does not change the underlying dynamics.
2.3. Reduction of Pointsand Maps Modulo p
51
Proposition 2.11. Let PI , P2 , P3 E pI (K ) be points whose reductions PI , P2 , P3 are distinct. Then there is a linear fractional transformation f E PGL2 (R) such that
f(Pd = 0, Proof Write Pi = [Xi, Yi] with normalized coordinates. If v(X I ) > v(YI ) , we begin by applying the map f = Y/X E PGL2 (R) to each of the three points, so we may assume that v(X d ::; v(Yd . Since the coordinatesare normalized, this implies that v(YI ) = 0, so YI is a unit. We next apply the map
to the three points. Having done this, we see that PI = [0,1]. Next consider the point P3 . Since P3 i- PI = [0,1], we see that v(X3 ) = 0, so we can apply the map X /(Y3X - X 3 y) E PGL2(R) to the three points. This fixes PI and sends P3 to [1 ,0], i.e., P3 gets sent to 00. Finally, since P2 is equal to neither PI = [0,1] nor P3 = [1 ,0]' we see that v(X2 ) = v(Y2 ) = 0. Applying the map Y2X/ X 2Y E PGL2(R) to the three points then fixes PI and P3 and sends P2 to [1 ,1]. D Havinglookedat the reductionof a point, we next tum to the problem of reducing a rational map modulo p. Let ¢ : pI -; pI be a rational map of degree d defined over K, so ¢ is given by a pair of homogeneous polynomials of degree d,
F (X ,Y ),G(X,Y ) E K [X ,Y ]. Note that the map ¢(X, Y ) = [F (X ,Y ),G(X,Y )] does not change if F and G are each multiplied by a nonzero constant c E K *, since the coordinates are homogeneous.
Definition. Let ¢ : pI -; pI be a rational map as above and write
¢ = [F (X ,Y ),G(X,Y)] with homogeneous polynomials F, G E K [X, Y]. We say that the pair (F, G) is normalized, or that ¢ has been written in normalizedform, if F , G E R[X, Y] and at least one coefficient of For G is in R*. Equivalently, ¢ = [F, G] is normalized if
F(X, Y) = aoXd + aIXd-1y + ... + ad_lXyd- I
+ adyd
and satisfy min { v(ao) ,v(ad , . . . ,v(ad), v(bo), v(b1), .. . ,V(bd) } = 0.
(2.3)
Givenany representation ¢ = [F,G], it is clear that one can alwaysfind some e E K * such that reF, eGl is a normalizedrepresentation. Further, the element e is unique up to multiplicationby an element of R* .
2. Dynamics over Local Fields: Good Reduction
52
Writing ¢ = [F, G] in normalized form, the reduction of ¢ modulo p is defined in the obvious way,
¢(X, Y) = [F(X, Y), G(X, Y)] - y d,b-0 X d + b- 1 X d- 1y + ... + b-dy d] . - Xd + al - X d- 1y + ... + ad = [ao In other words, ¢ is obtained by reducing the coefficients of F and G modulo p, (If the prime ideal is not clear from context, we write ¢p or ¢ mod p.) The fact that at least one coefficient of F or G is a unit ensures that at least one of F and G is a nonzero polynomial, so the reduction ¢ gives a well-defined map ¢ : pI (k) ----. pI (k). Further, the reduced map ¢ is independent of the choice of F and G, a fact whose proof we leave to the reader (Exercise 2.4) since it is quite similar to the proof of Proposition 2.7. The mere existence of the reduction ¢ of a rational map ¢ does not imply that has good properties, as is shown by the following simple example.
¢
Example 2.12. Let a E K* and consider the rational map
°
If a E R*, then ii -=I- and the reduced map ¢a(X, Y) = liiX d, yd] is again a rational map of degree d. However, if v(a) > 0, then ii = 0, so ¢a(X, Y) = [0, yd] = [0, 1] is a constant map! Similarly, if v(a)
< 0, then a-I = 0, so
is again a constant map, but not to the same point!! To summarize, the reduction of the map ¢a(X, Y) = [aX d, yd] separates into three cases,
[iiX d,Y dj ¢a =
{
[0,1] [1,0]
ifv(a) =0, ifv(a) > 0, ifv(a) < 0.
Clearly it is only in the first case that the reduced map is interesting. Keep in mind that our goal is to use the dynamics of ¢ to help us understand the dynamics of ¢. In the above example, ifv(a) = 0, then it is easy to see that for any P E pl(K),
¢(p) = ¢(F),
and hence by induction,
¢n(p) = ¢n(F).
Thus the ¢ orbit of F yields valuable information about the ¢ orbit of P. However, ifv(a) -=I- 0, then ¢(F) is constant, independent of P, so the ¢ orbit of F contains no information. We formalize this notion in Section 2.5 after a preliminary discussion of the theory of resultants.
53
2.4. The Resultant of a Rational Map
2.4 The Resultant of a Rational Map A rational map c/J : !pI -; !pI is given by a pair of homogeneous polynomials c/J =
[F(X, Y) , G(X , Y)]
having no nontrivial common roots. However, if we reduce the coefficients of F and G modulo some prime , they may acquire common root s in the residue field. In order to understand this phenomenon, it is useful to have a tool that characterizes the existence of common roots in terms ofthe coefficients of F and G . This tool is called the resultant. Resultants and their generalizations are widely used , both theoretically and computationally, in number theory and algebraic geometry. We give in this section only a brief introduction to the theory of resultants . The reader desiring further information might consult [105, Section 3.3], [112, Chapter 3], [259, Section V 10], or [436, Sections 5.8, 5.9], while the reader interested in applications to dynamics may wish to peruse only the statements of Proposition 2.13 and Theorem 2.14 and return to the proofs at a later time .
Proposition 2.13. Let A (X , Y ) = aoX n + a l X n- 1y + B (X , Y ) = boX m + b X m- 1y +
+ an_ lXy n - 1 + any n , + bm_ 1Xym- l + bmy m
1
be homogeneous polynomials ofdegrees n and m with coefficients in afield K . There exists a polynomial
in the coefficients of A and B , called the resultant of A and B , with the following properties: (a) Res(A , B ) = 0 if and only if A and B have a common zero in!p 1 (K). (b) If aobo =I- 0 and if we f actor A and B as n
A = ao
II (X -
m
Qy)
i= l
then
B = bo
and n
Res(A , B ) = aob~
m
II II (Qi -
II (X - ,6jY) , j=l ,6j).
i=l j=l
(c) There exist polynomials
F l , G l , F2 , G2 E Z[ao, . .. , an, bo, ... , bm][X, Y ], homogeneous in X and Y of degrees m - 1 and n - 1, respectively, with the property that
Fl (X , Y )A(X , Y ) + G 1 (X , Y )B(X , Y ) = Res(A , B )x m+n- l , F (X ,Y )A(X , Y ) + G (X ,Y )B (X ,Y ) = Res(A , B )y m+n- 1 . 2
2
Notice that in the first equation. the variable Y has been eliminated, and similarly X has been eliminated in the second equation.
2. Dynamics overLocal Fields: Good Reduction
54
(d) The resultant is equal to the (m + n) x (m + n) determinant ao al a2 . . . an ao al a2 ... an ao al a2 ... an
Res(A, B) = det
m
ao al a2 . .. an
bo bl bz bm bo bi b2 . •. •..•. • bm bo bl b2 •. • •. • • .. bm
bo bl b2
••••••. ••
n
bm
In particular, Res(A, B) is homogeneous of degree m in the variables ao, ... , an and simultaneously homogeneous ofdegree n in the variables bo, ... , bm . Proof We begin by showing that the following three conditions are equivalent. (i) A(X, Y) and B(X, Y) have a common zero in jpl(k). (ii) A(X, Y) and B(X, Y) have a common (nonconstant) factor in the polynomial ring K[X, Y J. (iii) There are nonzero homogeneous polynomials C , DE K[X,Y ] satisfying
A(X,Y)C(X,Y)=B(X,Y)D(X,Y) with deg(C) ::::: m - 1 and deg(D) ::::: n - 1.
(2.4)
The equivalence of (i) and (ii) follows immediately from the fact that the greatest common divisor of A and Bin K[X, Y] vanishes at exactly the common zeros of A and B in jpl (k). (What we are really using here, of course, is the fact that the ring of homogeneous polynomials K[X, Y] is a principal ideal domain.) It is also clear that (ii) implies (iii), since if A and B have a common factor F, then we simply choose C and D using the formulas A = F D and B = FC. Finally, to prove that (iii) implies (i), we suppose that (2.4) is true. Ifwe factor both sides of (2.4) into linear factors in k[X, Yj, then A(X, Y) has n factors, while D(X, Y) has at most n - 1 factors. Therefore A(X, Y) shares at least one linear factor with B(X, Y) in k[X, YJ, and hence they have a common zero in jpl (k). We next multiply out equation (2.4), treating the coefficients of C and D as unknowns. This gives a system of m + n homogeneous linear equations in the m + n variables Co , ... , Cm - I , do, . .. , d n - I , and the matrix of this system is (up to changing the sign of some columns and transposing) equal to the matrix given in (d) . For example, if deg(A) = 3 and deg(B) = 2, then equating coefficients in (2.4) gives the system oflinear equations
2.4. The Resultant of a Rational Map coao COal COa2 COa3
55
= dob o,
+ ClaO = + Clal = + Cla2
dob l dob2
+ dlb o, + dlb l + d2bo, dlb2
=
Cla3 =
+ d2bl,
d2b2,
whose associated matrix
ao -bo al ao -b l -bo a2 al -b2 -b l -bo a3 a2 -b2 -b, a3 -b2 becomes equal to the matrix in (d) if we change the sign of the last three columns and transpose. The general case is exactly the same. To recapitulate, we have shown that equation (2.4) has a nontrivial solution if and only if the system of homogeneous linear equations described by the matrix in (d) has a nontrivial solution, which is equivalent to the vanishing of the determinant of the associated matrix. Hence if we take the determinant in (d) as the definition of the resultant Res(A, B), then the equivalence of (i) and (iii) proven above shows that Res(A, B) = 0 if and only if A and B have a common zero in JP'1 (k), which proves (a). In order to prove (c), we write and
xjyn-l-j
B
for
0
~
j
as a system of homogeneous equations,
x
ao al a2 ... an ao al a2 ... an ao al a2 ... an
+m - l X n +m - 2 y x n +m - 3y 2 n
ym-lA Xn-lB
ao al a2 ... an bm bo bl b2 bo b: b2 bm bo bl b2 . . • . . . • . . b-« bo bl b2
Xm-1A X m - 2YA X m - 3 y 2A
X n - 2YB X n - 3 y 2B
bm
xyn+m-2 yn+m-l
yn-lB
Notice that the matrix M appearing here is exactly the matrix in (d) whose determinant equals Res(A, B). We multiply on the left by the adjoint matrix Madj of M. (Recall that the entries of Madj are the cofactors of the matrix M, and that the product Madj M is a diagonal matrix with the quantity det (M) as its diagonal entries.) This yields the following matrix identity, where for convenience we write R(A, B) for Res(A, B):
56
2. Dynamics over Local Fields: GoodReduction
R(A, B)
o o
0 0 R(A, B) 0 0 R(A, B) ...
o o o
x n +m - 1 x
n + m - 2y
X n +m -
3y2
Xm-1A X m - 2YA X m - 3 y 2A
ym-1A Xn-1B X n - 2YB X n - 3 y 2B Xyn+m-2
o
o
o
yn+m-l
... R(A,B)
yn-1B
Examining the top entry on each side, we find that Res(A, B)xn+m-I on the lefthand side is equal to an expression of the form FI(X, Y)A(X, Y)
+ GI(X, Y)B(X, Y)
on the righthand side, where F I and G I are homogeneous polynomials of degrees m-l and n-l, respectively, whose coefficients are (complicated) polynomials in the coefficients of A and B. Similarly, the bottom entry shows that Res(A, B)yn+m-I is equal to an expression of the form F 2(X, Y)A(X, Y)
+ G2(X, Y)B(X, Y).
This completes the proof of part (c) of the proposition. Finally, we leave the proof of (b) as an exercise for the reader, or see [436, Section 5.9]. D We define the resultant of a rational map in terms of its defining pair of polynomials.
Definition. Let ¢ : lP'1
--t lP'I be a rational map defined over a field K with a nonarchimedean absolute value I . Iv. Write ¢ = [F, G] using a pair of normalized homogeneous polynomials F, G E R[X, Y]. The resultant of ¢ is the quantity
Res(¢)
= Res(F, G).
Since the pair (F, G) is unique up to replacement by (uF, uG) for a unit u E R*, we see that Res(¢) is well-defined up to multiplication by the 21fh-power of a unit. In particular, its valuation v(Res( ¢)) depends only on the map ¢. The resultant of a rational map ¢ provides an upper bound to the extent that ¢ is expanding in the chordal metric. In particular, a rational map is always Lipschitz with respect to the chordal metric, and if its resultant is a unit, then the map is nonexpanding. (See also Exercise 2.10.)
Theorem 2.14. Let ¢ : lP'I
--t
nonarchimedean absolute value
lP'1 be a rational map defined over afield K with a I . Iv. Then
2.4. The Resultant of a Rational Map
57
Proof Write ¢ = [F(X, Y), G(X, Y)] in normalized form. Proposition 2.l3(c) says that there are homogeneous polynomials F l , G l , F 2 , G 2 E R[X, Y] satisfying
Fl (X, Y)F(X, Y) + G l (X, Y)G(X, Y) F2(X, Y)F(X, Y) + G2(X, Y)G(X, Y)
= Res( ¢ )X = Res(¢)y
2d- l,
2d
-
l
.
Now let P = [x, y] E jp'l (K) be a point, which we assume written in normalized form. We substitute [X,Y] = [x, y] into the first equation and use the nonarchimedean triangle inequality to compute I Res( ¢)x 2d- ll v = 1F1 (x, y)F(x, y) + G l (x, y)G(x, y)Iv
::; max{ IFl (x, y)F(x, y)lv, IG l (x, y)G(x, Y)lv}
::; max{IF1(x,y)lv, IG1(x,y)!v}' max{IF(x,y)lv, IG(x,y)lv}
< max{ IF(x, y) lv, IG(x, y)Iv }. A similar calculation using the second equation gives the analogous estimate
IRes( ¢ )y2d-ll v < max{ IF(x, y)lv, IG(x, y) l.}. Since P is normalized, i.e., max{lxl v , Iylv} = 1, we find that I Res( ¢ )!v ::; max{ IF(x, y)lv, IG(x, y) Iv}.
(2.5)
Notice that this estimate bounds the extent to which F(x, y) and G(x, y) can be simultaneously divisible by high powers of p. Returning to the proof of the theorem, we write P, = [Xl, Yl], P2 = [X2, Y2], and ¢ = [F(X, Y), G(X, Y)] in normalized form. Then the distance from P, to P2 is
while we can use the inequality (2.5) (applied to both P, and P2 ) to estimate
max { IF(Xl, yd lv, IG(Xl, Ydlv} . max{ IF(X2, Y2)lv, IG(X2, Y2) Iv}
< IF(Xl,Yd G(X2'Y2) - F(X2'Y2)G(xl,Ydl v. -
I Res(¢m
To complete the proof, we observe that the polynomial
vanishes identically if Xl Y2 = X 2 Yl . It follows that it is divisible by the polynomial X 1Y2 - X 2Yl in the ring R[X l , Yl , X 2 , Y2 ], so we can write
58
2. Dynamics over Local Fields: Good Reduction
for some polynomial H E R[X1 , Y1 , X 2 , Y2 ] . Then
(
"'(P ) "'(Po )) < I(XIY2 -x2 yr)H (XI,YI,X2, Y2)lv l,'f/ 2 IRes (¢) I ~
P V 'f/
< !XIY2 - X2Yl iv
-
IRes ( ¢) I ~
= Pv(P1 , P2 )
0
IRes ( ¢)I ~ .
2.5 Rational Maps with Good Reduction As we saw in Example 2.12, the reduction ¢ of a rational map ¢ may bear little resemblance to the original map. Indeed, even the degree of the map may change. In this section we characterize maps for which deg(¢) = deg (¢ ). These maps are the dynamical analogue of varieties that have good reduction, and they share many of the same properties. See [410, Chapter VII] , for example, and compare the results of this section with the properties of elliptic curves that have good reduction. Theorem 2.15. Let ¢ : pi ....... pl be a rational map defined over K and write ¢ = [F, G ] in normalizedform. The following are equivalent: (a) deg(¢ ) = deg (¢ ). (b) The equations F (X, Y )
= G(X , Y) = 0 have no solutions [a,13j E p l (k).
(c) Res(¢ ) E R*. (d) Res CF, G) =I- 0. Proof The equivalence of (b), (c), and (d) is immediate from the basic properties of the resultant given in Proposition 2.13, once we observe that Res (F, G) = Res (F, G ). This equality follows from the fact that the resultant is simply a polynomial in the coefficients of F and G. To complete the proof, we observe that the degree of ¢ is equal to the degree of ¢ minus any cancellation that occurs in F(X, Y)jG(X, Y) . In other words,
d 1 = d '" _ (Nu~ber of com~on roots eg 'f/ eg'f/ of F (X , Y ) = G(X, Y) =
°' )
where the roots are counted with appropriate multiplicities in pI (k). In particular, deg ¢ = deg ¢ if and only if P and G have no common roots, which proves the equivalence of (a) and (b). 0 Definition. A rational map ¢ : p i ....... pi defined over K is said to have good reduction (modulo p) if it satisfies anyone (hence all) of the conditions of Theorem 2.15.
59
2.5. Rational Maps with Good Reduction
Remark 2.16. There is a fancier, but useful, characterization of good reduction in the language of schemes. The rational map ¢ is a morphism ¢ : -t over Spec(K), so it induces a rational map Pk -t P k over Spec(R). Then ¢ has good reduction if and only if this rational map over Spec (R) extends to a morphism . In other words, good reduction is equivalent to the existence of an R-morphism ¢ R : Pk -t P k whose restriction to the generic fiber is the original -t In this setting, the reduction ¢ is then simply the restriction map ¢ : of ¢ R to a morphism of the special fiber ¢ : pl -t Plover Spec(k). See Exercise 2.15.
Pk
Pk
Pk
Pk.
As a first application of the notion of good reduction, we use Theorem 2.14 to prove the somewhat surprising result that maps with good reduction have empty Julia sets. Later, in Chapter 5, we will prove that rational maps always have nonempty Fatou set. This is exactly opposite to the situation that holds over the complex numbers C, where the Julia set is nonempty, but the Fatou set may be empty.
Theorem 2.17. Let ¢ : p l -t pI be a rational map that has good reduction. (a) The map ¢ is everywh ere nonexpanding,
(b) The map ¢ has empty Julia set. Proof (a) This is immediate from Theorem 2.14 and the fact that good reduction is equivalent to Res( ¢) E R*. (b) It is clear from the definition of equicontinuity that a nonexpanding map is equicontinuous . Indeed, the iterates of a nonexpanding map are uniformly continuous, and indeed, even uniformly Lipschitz (cf. Section 5.4 and Exercise 5.9). 0
As their name suggests, rational maps with good reduction behave well when they are reduced.
Theorem 2.18. Let ¢ : p I -t p I be a rational map that has good reduction. (a) ¢(p ) = ;(P) l or all P E PI (K ). (b) Let 'l/J : pI -t p I be another rational map with good reduction. Then the composition
1J 0 'l/J has good reduction, and
--
1Jo 'l/J
=
- -
1Jo'l/J.
Proof (a) Write 1J = [F(X, Y ), G(X , Y )] in normalized form with homogeneous polynomials F , G E R[X,Y ], and write P = [a,,8] in normalized form with Ct,,8 E R. The good reduction assumption tells us that at least one of F(Ct, ,8) and G (Ct, ,8) is in R*, so the point
1J(P) = [F(Ct ,,8),G(Ct,,8)] is already in normalized form. Hence
60
2. Dynamics over Local Fields: Good Reduction
rj>(P) = [F(a,13), G(a , 13)] = [F(a,,B),G(a,,B)] =
J(P),
where the second equality simply reflects the fact that the reduction map R ----t k is a homomorphism. (b) Write rj> = [F( X, Y), G(X ,Y )] and ?/J = [f( X ,Y) ,g(X ,Y )] in normalized form with homogeneous polynomials F, G , i, 9 E R[X ,Y]. Then the composition is given by
(rj> 0 ?/J)(X , Y)
[A(X ,Y ), B (X ,Y )] = [F(j(X ,Y) ,g(X ,Y )), G(j(X,Y ),g(X ,Y ))].
=
Clearly A(X ,Y ) and B (X ,Y ) have coefficients in R. Suppose that their reductions A and B have a common root [a, 13] E p I (f). This means that
F(](a, 13),g(a, 13)) = 0
G(](a, 13) ,g(a,13))
and
= 0,
so F and Ghave the common root [j(a ,13),g(a, 13)]. But rj> has good reduction, so F and 6 have no common root in pI (f) , and hence we must have
j(a ,13) = g(a,13)
= o.
But this contradicts the assumption that ?/J has good reduction. This proves that the polynomials A(X , Y ) and B(X , Y ) have no common root in pl (f ), and therefore rj> o ?/J = [A, B] has good reduction. We have also shown that the pair (A, B ) is normalized, so
o
which completes the proof of the theorem .
Remark 2.19. (a) The good reduction assumption in both parts of Theorem 2.18 is essential. See Example 2.12 and Exercise 2.13. (b) For an alternative proof of Theorem 2.18(b) that uses formal properties of resultants and provides additional information about the reduction of the composition of two maps, see Exercise 2.12 . (c) It turns out that the converse of Theorem 2.18(b) is false. In other words, a composition rj> 0 ?/J may have good reduction, while both rj> and ?/J have bad reduction. For example, let rj>([x, y]) = [x 2, py2] and ?/J([x, y]) = [p2 x2 , y2]. Then ¢ = [1,0] and -J; = [0,1] are constant maps, so rj> and e have bad reduction. However,
so ?/J 0 rj> has good reduction. One might object to this example by noting that there is a change of variables such that rj>(z) = Z2/ P has good reduction, and similarly for ?/J( z ) = p2z2. Thus if j (z) = pz, then
rj> f (z) = (j -l
0
rj> 0 f)(z) = (j - l
0
rj»(pz) = j-l (pZ2) = Z2
61
2.5. Rational Maps with Good Reduction
has good reduction. However, it is not difficult to modifythis example so that ¢ and 1/J have bad reduction for all possible changes of variable; see Exercise 2.14. (d) If we use the scheme-theoretic definition of good reduction as described in Remark 2.16, then both parts of Theorem2.18 are clear. For example, if ¢ and 1/J are rationalmaps with good reduction, then they extendto maps ¢R and 1/JR over Spec(R), and the commutativity of the diagram jp'l
k
;j,
---+
1 jp'l R
r K
---+
k
1
1/JR
---+
jp'l
¢
jp'l
jp'l R
r
1/J - - - + jp'l
K
jp'l k
1
cPR - - - + jp'l
R
r
cP - - - + jp'l
K
immediately gives ~ = ¢ 0 ;j;. Similarly, a point P E jp'l (K) corresponds to a unique morphism (i.e., a section) PR : Spec(R) --+ jp'k, from which the equality ;;;(p) = ¢(P) is immediate using the fact that the composition of R-morphisms ¢R 0 PR behaveswell when restrictedto the special fiber ofjp'k. In other words, the following diagram commutes: Spec(k)
1 Spec(R)
P
---+
PR
---+
jp'l k
1
jp'l R
¢
---+
jp'l
k
1
cPR - - - + jp'l
R
An easy, but important, consequence of the theorem on good reduction is that periodic points behavewell under reduction. Corollary 2.20. Let ¢ : jp'l --+ jp'l be a rational map with good reduction. Then the reduction map sends periodic points to periodic points and preperiodic points to preperiodic points:
Per(¢) ---- Per(¢)
and
PrePer(¢) ---- PrePer(¢).
Further, if P E Per( ¢) has exact period ti and if P E Per( ¢) has exact period m, then m divides n. Proof Supposefirst that P is periodic of exact period ti, so P = ¢n (P). Reducing
both sides modulo p and using Theorem2.18 yields
which showsthat P is periodic.Let m be the exactperiodof P and write n = mk + r with a :S r < m. Then
2. Dynamics over Local Fields: GoodReduction
62
P = J>n(p) = J>r 0
------Jr
0··· 0
cfr(P) = J>r(p).
k iterations
The minimality of m implies that r = 0, and hence m divides n. This proves the assertion about periodic points. Similarly, if P is preperiodic, say cfi(p)
2.6 Periodic Points and Good Reduction Corollary 2.20 tells us that if ¢ has good reduction, then its periodic points reduce to periodic points of J>. In this section we analyze the reduction map Per(¢) --+ Per(J» and use our results to study Per(¢). We start with the following theorem, which is an amalgamation of results due to Li [266], Morton-Silverman [312, 313], Narkiewicz [325], Pezda [355], and Zieve [454].
Theorem 2.21. Let ¢ : ]P'I(K) --+ ]P'I(K) be a rational function of degree d ~ 2 defined over a local field with a nonarchimedean absolute value I . Iv' Assume that ¢ has good reduction, let P E ]P'I(K) be a periodic point of ¢, and define the following quantities: n m
The exact period of P for the map ¢. The exact period of Pfor the map
= ((jr)'(p)
J>.
r
The order of A1 (P ) unity.)
p
The characteristic ofthe residue field k ofK.
in k*. (Set r
= 00 if A1(P ) is not a root of
Then n has one ofthefollowingforms: n
=m
or
n
= mr
or
n
= mrp",
Remark 2.22. Let E / K be an elliptic curve defined over a local field, and assume that E has good reduction. Then one knows [410, VII.3.1] that the reduction map E (K) --+ jj; (k) is injective except possibly on p-power torsion, where p is the characteristic ofthe residue field k. This is very similar to the statement of Theorem 2.21. Inthe case ofelliptic curves, it is also possible to bound the power of p in terms of the ramification index of pin K. We discuss below (Theorem 2.28) analogous bounds in the dynamical setting. Proof We make frequent use of Theorem 2.18, which tells us that
Recall that we used this relation in Corollary 2.20 to prove that the ¢-period of P is divisible by the J>-period of P, which, in our current notation, says that m divides n.
63
2.6. Periodic Points and Good Reduction Replacing ¢ by ¢m and m by 1, we are reduced to th~ that
F is a fixed point
of ¢. Having done this, we note that AJ,(F) is equal to ¢'(P). If ¢(P) = P, then ti = m and we are done. We thus assume that ¢(P) -I- P. To simplify notation, we use Proposition 2.11 to find a transformation f E PGL2(R) with f([O, 1]) = P. Replacing P and ¢ with f-1(p) and ¢I = 0 ¢ 0 f, respectively, we may assume that P = [0,1]. Dehomogenizing z = X/Y, we write ¢ in the form
r:'
1 d ¢(z) = aoz + a1 zd- + bozd + b1zd- 1 +
+ ad-1 z + ad + bd-1Z + bd
with coefficients aQ' ... , b« E R and at least one coefficient in R*. The fact that [0, 1] is a fixed point of ¢ says that
¢(O) = ad/bd == 0 (mod p), so ad E p and bd E R*. (We are also using the fact that ¢ has good reduction, of course.) Multiplying numerator and denominator by bd1, we may thus write ¢( z) in the form
¢(z)
=
ad + ad-1 Z + 1 + bd- 1Z +
+ a1 zd- 1 + aozd + b1zd- 1 + bozd
The first couple of terms of the Taylor expansion of ¢ around z case may be obtained by simple long division, look like
¢(Z)=p,+AZ+
=
0, which in this
A(z) 2 B( )z l+z z
(2.6)
with
A(z), B(z)
E
R[z],
A = ¢'(O),
and
p, =
ad E p.
A simple induction argument using (2.6) shows that
(2.7) In particular, since ¢n(o) = 0 and p, E p, we find that 1 + A + A2
+ ... + An-1 == 0
(mod p),
(2.8)
The analysis now splits into two cases. First, suppose that r ::::: 2, or equivalently,
A¢. 1 (mod p). Then formula (2.8) implies that An == 1 (mod p), so we find that r divides n. If n = r, the proof is complete. Otherwise, we replace ¢ by ¢T and n by ii]«. By an abuse of notation, we continue to write
¢(Z)=p,+AZ+
A(z) 2 B( )z l+z z
with the understanding that the values of p" A, A(z), and B(z) may have changed. The principal effect of replacing ¢ by ¢T is that we are now in the situation that
64
2. Dynamics over Local Fields: Good Reduction
>. == 1
(mod p),
i.e., the new value of r is 1, which brings us to the second case that we need to consider. To recapitulate, we have a rational function ¢ satisfying
¢n(o)
= 0,
p, = ¢(O)
== 0 (mod p),
>. = ¢'( O) == 1 (mod p).
and
We are further assuming that ¢(O) =1= 0 (otherwise, we are done), so (2.8) becomes
n
== 1 + >. + >. 2 + ... + >. n - I == 0
(mo d p).
Thus n is divisible by p. We replace ¢ by ¢P and n by n ip. Ifnow ¢(O) = 0, we are done. If not, the same argument shows that n is again divisible by p. Repeating, we continue dividing n by p until finally we reach n = 1. This concludes the proof that the original period n has one of the forms n
for some e 2 1.
=m
or
n
= mr
or
n
= mrp" 0
We have seen that maps with good reduction are nonexpanding. This implies that their periodic points are nonrepelling. If the reduction ¢ is separable , we can say even more. (Recall that ¢(z) E k(z) is separable ifit is not in k(zP). See Exercise 1.10 for details.) Corollary 2.23. Let ¢ : lP'I -> lP'I be a rational map that has good reduction. (a) Every p eriodi c point of ¢ is nonrepelling. (b) If the reduction ¢ is separable, then ¢ has only finitely many attracting periodic points. Proof (a) Theorem 2.18 tells us that ¢n has good reduction . Let P be a periodic point of ¢ of exact period n. Using Lemma 2.5, we can make a change of coordinates so that P = [0,1]. Then we can write ¢n (z ) in normalized form as
The fact that ¢ n has good reduction implies that bo E R*, since otherwise z = 0 would be a common root of P and G. Hence
so I>'p (¢) Iv :S 1, which shows that P is a nonrepelling point for ¢. (b) Again let P be a periodic point for ¢ of exact period n, and let m be the period of the reduced point P. Then we have equivalences
2.6. Periodic Points and Good Reduction
P is attracting
65
¢:=:}
IAp(¢)lv = 1(¢n)'(p)lv < 1
¢:=:}
(¢n)'(p)
= 0
¢:=:}
(¢n),(p)
= 0
¢:=:}
¢'(P) . ¢'(¢P) . ¢'(¢2 P)
¢:=:}
(¢'(P) . ¢'(¢P) . ¢'(¢2 P)
¢:=:}
(¢m)'(p)n/m = 0
¢:=:}
OJ, (P) contains a critical point.
¢,(¢n-I P)
= 0
¢,(¢m-I p))n/m = 0
The fact that ¢ is separable implies that a version of the Ri~mann-Hurwitz formula (Theorem 1.1) is valid; see Exercise 1.1O. Hence the map ¢ _has finitely many (precisely, at most 2d - 2) critical points, and a fortiori, the map ¢ has only finitely many periodic orbits containing a critical point. In particular, there is a finite list of possible periods for P. Further, we know that the multiplier AJ, (P) = (¢m)'( P) of P is 0, so Theorem 2.21 tells us that n = m. There are thus only finitely many possibilities for the period of P, and since ¢ has finitely many points of any given period, we 0 conclude that ¢ has only finitely many attracting periodic points.
Remark 2.24. The separability assumption in Proposition 2.23 is necessary, as is shown by the example ¢( z) = z", all of whose periodic points are attracting. See also Exercise 2.16. E Z2[Z] be a polynomial of degree d 2:: 2 whose leading coefficient is a 2-adic unit. Then ¢ has good reduction. Let P E lP'1 (Q2) be a periodic point of exact period n 2:: 2. In the notation of Theorem 2.21, n = mr2 e , where m is the period of P in lP'1 (IF 2) and r is the order of AJ, (P) in lF~. But lP'1 (IF 2) has only three points, and the fact that ¢ is a polynomial means that the point at infinity is not in the orbit of i; so either m = lor m = 2. Similarly, we note that F has only one element, so r = 1. It follows that n = 2 8 for some s 2: O. Similarly, if ¢(z) E Z3[Z] is a polynomial of degree d 2: 2 with leading coefficient a 3-adic unit, and if P E lP'1 (Q3) is a periodic point of exact period n 2: 2, then we find that n = mr3 e with 1 ~ m ~ 3 and 1 ~ r ~ 2. Thus n = 2 t . 3U for some o ~ t ~ 2 and some u 2:: O. Finally, let ¢(z) E Z[z] be a polynomial of degree d 2: 2 whose leading coefficient is relatively prime to 6. Then ¢ has good reduction at 2 and 3, so the period n of a periodic point P E lP'1 (Q) satisfies both n = 2 8 and n = 2t . 3U with t ~ 2. This proves that n is either 1, 2, or 4. The examples ¢(z) = Z2 with z = 0 and ¢( z) = Z2 - 1 with z = 0 show that n = 1 and n = 2 are possible. Can you find an example with n = 4? See Exercise 2.20 for a stronger version of this example.
Example 2.25. Let ¢(z)
z
The above example illustrates how the local result given in Theorem 2.21 can be used to derive strong bounds for the periods of periodic points defined over number fields by applying the theorem to two different primes. We now use the same argument to give a general result that, although not the strongest possible bound using these methods, is sufficient for many applications.
2. Dynamics over Local Fields: Good Reduction
66
Corollary 2.26. Let K be a numberfield, let ¢ : pI -+ pI be a rational map defined over K, and let p and q be primes of K such that ¢ has good reduction at both p and q and such that the residue characteristics of p and q are distinct. Then the period n ofany periodic point of ¢ in pI (K) satisfies
n::; (Np2 - 1)(Nq2 - 1), where NP and Nq denote the norms of p and q respectively. In particular, the set Per( ¢, K) of K -rational periodic points is finite. (For an alternative proofofthe finiteness ofPer( ¢, K) using the theory ofheight functions, see Theorem 3.12.) Proof Using the obvious notation, we have m p = (periodof¢(F) mod p)::; #pl(JF p ) = Np
+ 1,
r p = (period of AJ(F) mod p) ::; #JF; = Np - 1, and similarly for m q and r q- Let p and q denote the residue characteristics of p and q, respectively. Then Theorem 2.21 says that
Since p and q are distinct primes, it follows that n ::; m p . r p . m q . r q
::;
(Np + l)(Np - l)(Nq + l)(Nq - 1),
which is the first part of the corollary. The finiteness ofPer( ¢, K) then follows from the fact that ¢ has good reduction at almost all primes of K and the fact that it has only finitely many periodic points of any given period n. 0
Remark 2.27. The bound for rational periodic points in Corollary 2.26 depends only weakly on ¢ in the sense that the bound is solely in terms of the two smallest primes of good reduction for ¢. There are many results in the literature using local and/or global methods that describe bounds for rational periodic points that depend in various ways on the rational map. See for example [52, 87, 90, 91, 162, 171,312, 325, 326, 328, 332, 353, 355, 358, 359, 361, 454]. However, none of these articles achieves the uniformity predicted by a conjecture that we discuss in Chapter 3 (Conjecture 3.15). This conjecture asserts that for a number field K of degree D and a rational map ¢ E K(z) of degree d 2 2, the number of K-rational preperiodic points of ¢ should be bounded solely in terms of D and d. If K is a discrete valuation ring of characteristic 0, then it is possible to bound the exponent e appearing in the formula n = mrp" in Theorem 2.21.
Theorem 2.28. (Zieve [454], see also Li [266] and Pezda [355]) We continue with the notation and assumptions from Theorem 2.21. Wefurther assume that K has characteristic and we let v : K* --* Z be the normalized valuation on K. If the period n of P E pI (K) has the form n = mrp", then the exponent e satisfies
°
2.6. Periodic Pointsand Good Reduction p
e- l
67
2v(p)
< - -. - p -1
(2.9)
Further, ifp = 2, then the upper bound may be replaced with v(p)/(p - 1). (Note that v(p) is the ramification index ofp in K.) Remark 2.29. Let R be the local ring of K and let F be a formal group defined over R. (See, e.g., [410, chapter IV] for basic material on formal groups.) Theorem 2.28 is a close analogue of the fact [410, IY.6.1] that the torsion in the formal group F (R) consists entirely of p-power torsion, and that if a E F (R) has exact order p" , then pe- l :::; v(p) /(p - 1).
o,
Example 2.30. Let q be a power of an odd prime p, let ( E be a primitive qth root of unity, let K = Qp((), and let ¢( z) = 1 + (z - z", The maximal ideal of K is P = (1 - (), so ¢ = 1 + z - zq has good reduction. The point a = 1 is a point of exact period q for ¢, since cP (1) = ( j, while a is clearly a fixed point of ¢. Further, ¢'(a) = 1. Hence in the notation of Theorem 2.21, we have n = q, m = 1, r = 1, and e is determined by q = p" , On the other hand, the extension K / Qp is totally ram ified, so the normalized valuation v : K * ->t Z satisfies v(p)
= [K
: Qp]
= q(l -
l /p )
= pe-l(p -
1).
Thus the inequality (2.9) in Theorem 2.28 becomes pe-l :::; 2pe- l, which shows that the power of p cannot be improved. We do not give the proof of Theorem 2.28, but are content to prove the following special case, which serves to indicate some of the combinatorial issues that arise.
Theorem 2.31. Let p ;::: 5 be a prime, let K = Qp, or more generally, an unramified extension of Qp, let ¢ : 1P'1 ---+ IP'I be a rational map defined over K with good reduction, and let P E 1P'1 (K) be a periodic point with the property that and
¢'(F) =
1.
Then ¢(P) = P . (In the notation of Theorem 2.21, this theorem asserts that ifm = = 1, then e = 0 and n = 1.)
r
Proof As in the proof of Theorem 2.21, we begin by moving the periodic point to the origin and dehomogenizing cjJ. Recall that during the proof of Theorem 2.21, we used a first-order Taylor expansion (2.6) for ¢(z) around z = O. In order to bound the exponent e, we need to use the second-order expans ion
cjJ(z) = J.l + AZ + uz
2
A(z)
3
+ l+ z B (z ) z ,
(2.10)
where
A(z), B(z) E R[z],
J.l E p,
A = cjJ' (O) == 1 (mod p) ,
and
v E R.
2. Dynamics overLocal Fields: Good Reduction
68
(In general, to prove a sharp estimate for the exponent as in Theorem 2.28, one needs to consider a longer Taylor expansion. But for the unramified case that we are considering, the second-order expansion (2.10) suffices.) Using the results already proven, it suffices to show that if f-l -f:. 0, i.e., if ¢(O) -f:. 0, then ¢P(O) -f:. O. This will then imply by induction that ¢Pe (0) -f:. 0 for all e 2: 1, contradicting the assumption that 0 is a periodic point of ¢. During the proof of Theorem 2.21 we gave a simple formula (2.7) for ¢i(O) modulo f-l2. In a similar manner we use (2.10) to find a formula modulo f-l3. To derive this formula, we write ¢k(O) = f-lak + f-l2 vbk, substitute into (2.10), and do some algebra to obtain f-lak+1
+ f-l2vbk+l = ¢(f-lak + f-l2 vbk) == f-l + A(f-lak + f-l2 vbk) + vf-l2a~ (mod f-l3) == f-l(1 + Aak) + f-l2v(a~ + Abk) (mod f-l3).
This yields the recurrences and Starting from al = 1 and b1 = 0, it is now a simple matter to find formulas for ak and bk and check them by induction. The end result is
1'(0)
=~(~ >,;) + 1'2v(~ >,H-'
(to
>,;
r)
(2.11)
».
We are going to apply (2.11) with k = p. Consider the sum L We know that A == 1 (mod p), and we are assuming that p is unramified in K, so A = 1 + cp for some c E R. Assuming that c -f:. 0, i.e., that A -f:. 1, we compute
Note that the final congruence is true because every binomial coefficient (~) with 1 ::; i < p is divisible by p (and we are assuming that p -f:. 2). We also observe that the congruence is true even for A = 1, although the intermediate calculation is incorrect. We perform a similar calculation for the more complicated sum in (2.11), but this time we are interested only in the value modulo p, so we can replace A by 1:
(2.13) Note that for the last step we are using the assumption that p 2: 5. Substituting (2.12) and (2.13) into the iteration formula (2.11) (with k yields
= p)
69
2.7. PeriodicPoints and Dynamical Units
¢P(O) == j.t(p + ap2) + j.t2vbp
(mod j.t3),
where a and b are in R . The fact that p is unramified in K means in particular that p divides u, so ¢P(O) == j.tp (mod j.tp2). Now using our assumption that j.t the proof of the theorem.
i- 0, we deduce that ¢P(O) i- 0, which completes 0
2.7 Periodic Points and Dynamical Units Let ( be a primitive pth root of unity. Then the expression
is a unit in the cyclotomic field iQ((), a so-called "cyclotomic unit." Similarly, if (1 and ( 2 are roots of unity of relatively prime orders, then the difference (1 - (2 is an algebraic unit. The crucial fact underlying these constructions is that distinct roots of unity remain distinct when they are reduced modulo primes . It follows that their differences are not divisible by any primes, and hence that they are units . Theorem 2.21 can be used to deduce conditions under which distinct periodic points remain distinct when reduced modulo primes , so it can be used to construct units in a similar fashion. These constructions can be done either using different points in a single periodic orbit or using points of different periods. The following proposition provides us with the information needed to construct units of various kinds . In fact, since Lemma 2.8 tells us that the v-adic chordal metric satisfies
the proposition actually says something stronger than the simple assertion that certain pairs of points have distinct reduct ions.
Proposition 2.32. Let ¢ (z ) E K (z) be a rational fun ction of degree d ?': 2 with good reduction. (a) Let P E Pl(K) be a point ofperiod n for o. Then
Pv(¢ip,¢jp) = Pv(¢i+kp, ¢i +kp ) forall i , j ,k
E Z,
wherefor i < 0 we use the periodicity ¢nP = P to define ¢i P. (b) Let P E pI (K) be a po int of exact period n for ¢. Then
Pv(¢iP, ¢i P ) = Pv(¢P, P ) for all i ,j
E Z satisfying gcd (i -
i . n) = 1.
(c) Let Pl , P2 E p l (K) be periodic points for ¢ ofexact periods nl and n2. respectively. Assume that nl n2 and n 2 nl. Then
t
t
70
2. Dynamics over Local Fields: Good Reduction
Proof (a) Proposition 2.14 and the good-reduction assumption imply that for all Q, R E pI (K).
Pv(Q, R) 2: Pv(¢Q, ¢R) Applying this repeatedly yields
Pv(Q, R) 2: Pv(¢Q, ¢R) 2: Pv(¢2Q, ¢2R)
2: Pv(¢3Q,¢3R) 2: ... 2: pv(¢nQ,¢nR). If we now make the further assumption that Q and R are points of period n, then pv(¢nQ, ¢nR) = Pv(Q, R), so all of the inequalities must be equalities. This proves that if ¢n( Q) = Q and ¢n(R) = R, then
Pv(Q, R) = Pv(¢kQ, ¢k R)
for all k E Z.
Substituting Q = ¢i P and R = q) P for the given point P of period n completes the proof of (a). (b) We know from (a) that the distance Pv(¢iP,q)P) depends only on the difference i - j. We use this fact and the nonarchimedean triangle inequality (Theorem 2.4(d» to estimate
Pv(P, ¢k P) :S max{pv(p, ¢P), Pv(¢P, ¢2P), ... , p(¢k-I P, ¢k pn =Pv(P,¢P).
(2.14)
In a similar fashion we obtain the estimate
Pv(P, ¢P) :S max{pv(P, ¢k P), Pv(¢kP, ¢2kP), Pv(¢2k P, ¢3kP), ... , pv(¢(m-I)k P, ¢mkP), pv(¢mkP, ¢P)} = max{pv(P,¢kp),pv(¢mkp,¢Pn.
(2.15)
If we now make the further assumption that gcd(k, n) = 1, then we can find an integer m satisfying mk == 1 (mod n). Then ¢mkP = ¢P, so (2.15) becomes
Pv(P, ¢P) :S max{pv(P, ¢k P), Pv(¢P, ¢pn = Pv(P, ¢k P).
(2.16)
Combining (2.14) and (2.16) yields
Pv(P, ¢P)
= Pv(P, ¢k P)
In particular, if gcd( i - j, n)
for all k satisfying gcd(k, n)
= 1.
= 1, we can use ttis formula and (a) to compute
Pv(¢ip,cpJp)
= Pv(p,¢i-jp) = Pv(P,¢P).
(c) Suppose that Pv(PI, P2) < 1, or equivalently from Lemma 2.8, suppose that = P2 . Let m be the exact period of and let r be the order of its multiplier AJ,(PI ) , Then Theorem 2.21 tells us that both nl and n2 are in the set
PI
A
{m, mr, mrp, mrp2, mrp3, .. .}.
It follows that either nl divides n2, or that n2 divides nl, contradicting the assump0 tion on nl and n2. Therefore Pv(PI, P2) = 1.
71
2.7. Periodic Points and Dynamical Units
It is a simple matter to use Proposition 2.32 to construct units from periodic points. We begin with the easier case of a polynomial mapping.
Theorem 2.33. (Narkiewicz [325]) Let ¢(z) E R[z] be a polynomial of degree d 2: 2 whose leading coefficient is a unit in R. Let a E K be a periodic point of ¢ of exact period n (with n 2: 2), and let i, j E 7l, be integers satisfying gcd(i - j, n) = 1. Then ¢i a - ¢Ja R* ----E . ¢a-a Proof The assumption that ¢(z) is a polynomial with unit leading coefficient implies that ¢ has good reduction, since
Res(aoX d + aIXd-Iy + ... + ady d, yd)
=
ago
We next observe that every periodic point of ¢ is integral over R, since it is a root of an equation ¢n (z) - z = O. Finally, we note that the distance between integral points a, (3 E R is given by
pv(a,(3)
= pv([a, 1], [(3, 1]) = la - (3lv'
Using this formula and Proposition 2.32(b), we compute
¢ia - ¢Ja I I ¢a-a v
= I¢ia - ¢Jal v = Pv(¢i a, ¢Ja) = 1. I¢a-al v
pv(¢a,a)
o
We create units from periodic points of arbitrary rational maps using a cross-ratio construction.
Definition. Let PI, P2 , P3 , P4 E pI (K), and choose homogeneous coordinates Pi = [Xi, Yi] for each point. The cross-ratio of PI, P2 , P3 , P4 is the quantity
Notice that K(PI, P2 , P3 , P4 ) is independent of the choice of homogeneous coordinates for the points.
Theorem 2.34. (Morton-Silverman [313, Theorem 6.4(a)]) Let ¢ E K(z) be a rational map of degree d 2: 2 with good reduction. Let P E pI (K) be a periodic point for ¢ ofexact period ti, and let i and j be integers satisfying
gcd(j, n) = gcd(i - 1, n) = gcd(i - j, n) = 1. Then
72
2. Dynamics over Local Fields: Good Reduction
Proof Comparing the definition of the cross-ratio to the definition of the chordal metric, we see that
The assumptions on i and j and Proposition 2.32(b) tell us that
Hence
o
which proves that the cross-ratio is a unit.
We can also construct units using periodic points of different periods. This may be compared with the two different types of cyclotomic units, and where ( n indicates a primitive nth root of unity and the condition gcd( m , n ) ensures that (m - ( n is a unit.
=
1
Theorem 2.35. Let
Proof Since we have taken normalized homogeneous coordinates, the chordal metric is given by
Pv(PI,P2) = IX I Y2 -X2Yllv' The assumptions on nl and n2 and Proposition 2.32(c) tell us that pv(n, P2) and hence Xl Y2 - X2YI is a unit.
=
1, 0
Remark 2.36. For further information on the geometric and arithmetic properties of periodic points and the fields that they generate, see Sections 3.9, 3.11, and 4.1-4.6. In some sense, periodic and preperiodic points play a role in arithmetic dynamics analogous to the role played by torsion points in the arithmetic of elliptic curves or abelian varieties, albeit the dynamical setting has considerably less in the way of helpful structure . Example 2.37. We use the polynomial ¢(z) = compute
Z2
+ 1 to construct
units. First we
2.7. Periodic Points and Dynamical Units
¢(z) - z
73
z2 - Z + 1, Z4 + 2z2 - Z + 2 = (z2 - Z + 1)(z2 + z + 2), z8 + 4z 6 + 8z4 + 8z 2 - Z + 5
=
¢2(Z) - Z = ¢3(z) - Z =
= (Z2 - Z + 1)(z6 + z5 + 4z 4 + 3z3 + 7z 2 + 4z + 5). It is not surprising that ¢(z) - z divides both ¢2(z) - z and ¢3(z) - z, since any root of the former is clearly a root of the latter two; cf. Exercise 1.19(a). We have
Per~'(¢) =
{I
± ~}
and
where recall that Per~' (¢) denotes the set of points of exact period n for ¢. Further, the six roots of the polynomial
constitute a set of the form
since we know that the roots of <1>3 (z) are permuted by ¢ into two periodic cycles of period 3. In particular, the splitting field K (a, (3) of <1>3 is a Galois extension of K of degree at most 18. Applying Theorem 2.33 with i = 2 and j = 1 gives a unit Ul
We can check that
Ul
=
¢2(a) - ¢(a) ¢(a) - a
= a2
+ a + 1 E R[a].
is a unit in R[a] by computing
Similarly, we can apply Theorem 2.33 with i
= 2 and j = 0 to obtain a second unit
¢2(a) - a
U2
2
= ¢ () = a + a + 2. a -a
And replacing a with ¢( a) or with ¢2 (a) in
Ul
and
U2
gives four more units,
= ¢(a)2 + ¢(a) + 1 = a 4 + 3a 2 + 3, u~ = ¢2(a)2 + ¢2(a) + 1 = a 4 + 2a 2 + a + 2, u; = ¢(a)2 + ¢(a) + 2 = a 4 + 3a2 + 4, u~ = ¢2(a)2 + ¢2(a) + 2 = a 4 + 2a 2 + a + 3. U~
(Note that in doing these computations, we make use of the fact that ¢3(a)
= 0.)
74
Exercises ll~~~~~~~~a~~~~~~~m~
computation yields
This is not surprising, since one can check that the polynomial ct>3( x) is irreducible over Q and that all of its roots are complex. Thus the global field Q( a) is a totally complex extension field of degree 6, so its unit group has rank 2. We can use Theorem 2.35 to create more units. Taking nl = 2 and n2 = 3 and letting 'I = ~(-1 + R), we have for 0 :::; i :::; 1 and 0 :::; j :::; 2. The extension Q(a, 'I) /Q is totally complex of degree 12, so has five independent units. We leave as an exercise for the reader to compute the number of independent units in the set {Ui,j}.
Exercises Section 2.1. The Nonarchimedean Chordal Metric 2.1. Let K be a field that is complete with respect to an absolute value v and let ¢(z) E K [z] be a polynomial. The filled Julia set K( ¢) of ¢ is the set
K(¢)
= {a E K
: l¢n(a)lv is bounded forn
= 1,2,3, .. .}.
(Note that this definition applies only to polynomials, not to more general rational maps.) (a) Prove that the filled Julia set K( ¢) is a closed and bounded subset of K. Note that if K is locally compact, for example K =
= 00,
(2.17)
where the limit is in the v-adic topology. Equivalently, prove that it is the set of P satisfying lim pv (¢n (P), 00) - . 0 or, writing P = [a, 1], the set of points satisfying l¢n(a)lv -. 00. The set determined by (2.17), which from (a) is an open set, is called the attracting basin of 00. (c) Prove that the Julia set.J( ¢) of ¢ is the boundary ofthe filled Julia set K( ¢). (d) Prove that the filled Julia set K( ¢) is completely invariant, i.e., prove that it satisfies
¢-I(K(¢))
= K(¢) = ¢(K(¢)).
Section 2.3. Reduction of Maps Modulo a Prime 2.2. Let
f =
(~~) E PGL 2(K) with
a, b, c, d E R, at least one of a, b, c, din R*, and
ad == be (mod p). Prove that there exist points P, Q E
P=Q
and
jp>1 (K)
j(P) #
f(Q).
Thus Proposition 2.9 is false in general if f ~ PGL2(R).
satisfying
Exercises
75
2.3. GeneralizeProposition2.11 to jp'2 as follows. Let Ql
= [1,0,0],
Q2 = [0,1,0],
Q3
= [0,0,1],
Q4
= [1,1,1].
Suppose that PI, P2, P3 , P4 E jp'2(K) are points whose reductions PI,... ,P4 are distinct and have the property that no three of them lie on a line in jp'2( k). Prove that there is a transformation f E PGL3(R) such that f(P;) = Q; for alII:::; i :::; 4. Formulateand prove an analogousstatement for jp'N.
2.4. Let ¢ = [F, G] = [F', G'] be two normalized representations of the rational map ¢ : jp'l -> jp'l. Prove that there is a constant u E k* such that
uF=F' Deducethat the reducedmap ¢ ; jp'l (k) representation.
uG=G'.
and -> jp'l (k)
does not dependon the choice of normalized
Section 2.4. The Resultant of a Rational Map 2.5. (a) Let A(X, Y) = aoX + al Y be a linear polynomialand let B(X, Y) be an arbitrary homogeneouspolynomial. Prove that Res(A, B) = B( -ai, ao). (b) Let A(X, Y) = aoX 2 + alXY + a2y2 and B(X, Y) = bOX 2 + blXY + b2y 2 be quadratic polynomials. Prove that
= (2aob2 -
4 Res(A, B)
alb l
+ 2a2bo)2 - (4aoa2 - ai)(4bob2 - bi).
(Of course, in characteristic 2 one needs to cancel 4 from both sides before using this formula!) 2.6. Withnotation as in the statementof Proposition2.13, prove that the resultant of A and B is related to the roots of A and B by n
Res(A, B)
= a~b~
'rn
i=] j=l
2.7. Let
n
II II (a; - (3j) = a~ II et«; 1) = (_I)""nb~ II A((3j, 1).
A(X, Y) = aoX n + alXn-ly + B(X, Y) = boX"" + blX""-ly +
i=l
j=l
+ an_lXy n- l + any n, + b""_IXy""-1 + b""Y"" ,
be homogeneous polynomials of degrees nand m and let a, (3", fI E K be arbitrary. (a) Prove that
Res(A(aX
+ (3Y, ,X + flY), B(aX + (3Y, ,X + flY)) = (afl - (3,)""n Res (A(X, Y), B(X, Y)).
(Hint. Use Proposition2.13(b).) (b) Suppose that m = n = d. Prove that
Res ( aA(X, Y)
+ (3B(X, Y), ,A(X, Y) + ss;x, Y)) = (afl - (31')d Res (A(X, Y), B(X, Y)).
(Hint. Use Proposition2.13(d).)
Exercises
76 (c) Continuing with the assumption that m las
= n = d, define new polynomials by the formu-
= M(aX + /3Y,1X + 8Y) - /3B(aX + /3Y,1X + 8Y), = -1A(aX + /3Y,1X + 8Y) +aB(aX + /3Y,1X + bY).
A*(X, Y) B*(X, Y) Prove that
Res(A*, B*)
= (a8 -
/31)d
2+d
Res(A, B).
(d) Suppose that m = n = d and that Res(A, B) =1= 0, so ¢> = [A,B] is a rational map ¢> : 1P'1 ----> 1P'1 of degree d, and suppose further that a8 - /31 =1= 0, so the map
f
= [aX + /3Y,1X + 8Y]
is a linear fractional transformation, Let ¢>* = [A*, B*]. Prove that ¢>* Use (c) to prove that ¢>* : 1P'1 ----> 1P'1 is a rational map of degree d.
=
r:'
0
¢>
0
f.
2.8. Let f E PGL 2(K) be a linear fractional transformation, and write
f(z) in normalizedform, (a) Prove that Res(f) (b) Prove that
= ad -
=
az + b ez+d
be.
p(J(P),f(Q))::; IRes(f)1-
1p(p,Q)
for all P, Q E 1P'1(K).
(Notice that this strengthens Theorem 2.14 for maps of degree 1.) 2.9. Let f E PGL2(K) be a linear fractional transformation, let ¢>(z) E K(z) be a rational map, and let ¢>f = ¢> 0 f. (a) If Res (f) E R*, prove that Res(¢>f) = Res(¢». (b) If Res(f) is not a unit, find a formula or an inequality relating the valuations of the three quantities Res(f), Res (¢», and Res (¢>f).
rio
2.10. Let ¢> : 1P'1 ----> 1P'1 be a rational map defined over a field K with a nonarchimedean absolute value I . Iv, and consider the statement
sup Pl,P2EI"1(K) PFP P2
pv(¢>(P1 ) , ¢>(P2)) Pv(P1 , P2)
1
(2.18)
(a) Prove that (2.18) is true for the map ¢>(z) = az", where a E R. This example shows that Theorem 2.14 cannot be improved in general. (b) Prove that (2.18) is not true for the map
over the field K 2.11. Let ¢> : 1P'1 (C)
= Qp by computing both sides. (Youmay assume that p =1= 2.) ---->
1P'1 (C) be a rational map of degree d.
77
Exercises (a) Prove that there is a constant C(d), depending only on d, such that
(b) Find an explicit value for the constant C (d). Note that this exercise provides an archimedean counterpart to Theorem 2.14, whose proof may, mutatis mutandis, be helpful in doing this exercise. Section 2.5. Rational Maps with Good Reduction
2.12. Let F(X, Y) and G(X, Y) be homogeneous polynomials of degree D, let !(X, Y) and g(X, Y) be homogeneous polynomials of degree d, and let A(X, Y)
= F(j(X, Y),g(X, Y))
and
B(X, Y)
= G(j(X, Y),g(X, Y))
be their compositions. (a) Prove that the resultants satisfy
Res(A, B)
= Res(F, G)d . Res(j, g)D
2
.
(b) Use (a) to give an alternative proof ofTheorem 2.18(b). (c) For any rational map ¢ : lP'1 ~ lP'\ define ov(¢) = v(Res(¢))j deg(¢), so ov(¢) is a kind of normalized resultant of ¢. Prove that Ov satisfies the composition formula
2.13. Show that the good-reduction assumptions in Theorem 2.18 are necessary by constructing the following counterexamples: (a) Find a rational map ¢ : lP'1 ~ lP'1, which will necessarily have bad reduction, and a point P E lP'1(K) such that i(P) -I- ¢(?). (b) ~ratio~al ~aps ¢ : lP'1 ~ lP'1 and'l/J : lP'1 ~ lP'1 such that ¢ has good reduction and
¢ 0 'l/J -I- ¢ 0 'l/J. (c) Same as (b), except now 7/J is required to have good reduction and bad reduction.
rP is allowed to have
2.14. Let p 2: 5 be a prime and define rational maps ¢(z) =
2
2 2
3
+p Z p3z2 + P Z
and
4
+P p3 Z + 1
'l/J(z) = p z
(a) Prove that ¢f has bad reduction modulo p for all! E PCL2(Qp). (N.B. We are allowing! to have coefficients in Qp and/or the determinant of! to be divisible by p.) (b) Prove that 'l/Jf has bad reduction modulo p for all ! E PCL 2 (Qp). (c) Prove that the composition 'l/J 0 ¢ has good reduction at p.
2.15. Let ¢ : pI ~ lP'1 be a rational map defined over K. (a) Prove that the map ¢ has good reduction if and only if there is an R-morphism ¢R: lP'k ~ lP'k whose restriction to the generic fiber is equal to the original map ¢ : lP'k ~ lP'k. (b) Assume that ¢ has good reduction. Prove that the reduction ¢ : lP'~ ~ lP'~ is the restriction of rPR to the special fiber oflP'k.
78
Exercises
2.16. Let K IQp be a p-adic field. (a) Prove that every periodic point of the map rjJ(z) = zP is attracting. (b) More generally, suppose that 'ljJ(z) E K(z) has good reduction, and let rjJ(z) Prove that every periodic point of rjJ is attracting. (c) Is (b) true without the assumption that 'ljJ has good reduction?
= 'ljJ(zP).
2.17. Let KIQp be a p-adic field and let rjJ(z) E K[z] be a polynomial with good reduction. Let a E K be a critical point of rjJ, i.e., rjJ' (a) = 0, and suppose that a E Per (¢). More precisely, suppose that the reduced point a satisfies ¢m (a) = a. Prove that there is an attracting periodic point (3 of rjJ of period m such that lim rjJnm(a) = (3. n~oo
Generalize to the case of a rational map rjJ( z) E K (z) of good reduction. 2.18. Let rjJ(z) E K(z) be a rational map of degree d ~ 2 and suppose that there is a point P E JIll! (K) and an integer m ~ 1 such that the following limit exists:
T
=
lim rjJmn(p).
(2.19)
n~oo
Prove that rjJm(T) = T, i.e., T E Perm (rjJ). (This is true for any complete field K, so for example, it holds for Qp,Cp, JR,
2.19. Let a, b, c E IZ and let rjJ(z) = (az2 +bz+c) I Z2. Suppose that P E JIll! (Q) is a periodic point for rjJ of exact period n. (a) If gcd(c, 6) = 1, prove that n = 1,2, or 3. (b) Give examples to show that it is possible for n to take on each ofthe values 1,2, and 3. (c) Show that if rjJ has a rational periodic point of exact period 3, then it has no other rational periodic points. (d) Give a similar description of possible rational periodic points under the assumption that gcd(c, 10) = 1. (e) Same as (d), but under the assumption that gcd(c, 15) = 1. 2.20. Let rjJ(z) E lZ[z] be a polynomial of degree d ~ 2 whose leading coefficient is odd and let P E JIIl!(Q) be a periodic point of exact period n. Prove that n E {1,2,4}. This strengthens Example 2.25. 2.21. Let rjJ(z) E lZ[z] be a polynomial of degree d ~ 2 whose leading coefficient is relatively prime to 15 and let P E JIll! (Q) be a periodic point for rjJ of exact period n. What can you say about the possible values of n? 2.22. Let R = IFp [T] be the ring of formal power series in one variable with coefficients in the finite field IFp' Prove that for every e > 0 there exists an element c E R with c == 1 (mod T) such that the polynomial rjJ(z) = zP + cz has a periodic point a == 0 (mod T) of exact order pe . Thus Theorem 2.28 is false in characteristic p; there is no upper bound on the exponent of p for the period of a periodic point. 2.23. Let rjJ E K (z) be a rational map with good reduction and let P E Per( rjJ) be a periodic point. Prove tha~ P is attracting if and only if the orbit of its reduction O;p(P) contains a critical point of rjJ.
Exercises
79
2.24. Let c E R and consider the quadratic map ¢(z) = z2 + c. Suppose that there is an integer m ~ 1 such that the limit a = limn~oo ¢mn(O) exists. Prove that a is an attracting periodic point if and only if 6 E Per( ¢), i.e., if and only if 6 is a purely periodic point of ¢.
Section 2.7. Periodic Points and Dynamical Units 2.25. Theorem 2.33 gives a construction of units from periodic points of polynomial maps. Prove the following generalization. Let ¢ E K (z) be a rational map of degree d ~ 2. Assume that ¢ has good reduction and that 00 is a critical fixed point of ¢. (That is, assume that ¢( 00) = 00 and that e oo (¢) ~ 2.) Let [a, 1] E Per~*(¢, K) be a point of exact period n ~ 2. Leti,j E;;Zbe integers satisfying gcd(i - j, n) = 1. Prove that "ia-"ja ,,-'1'_ _"-'1'_
E R*.
¢a-a 2.26. Prove the following version of Theorem 2.35, in which one of the periods divides the other period. Let R be a local ring with residue characteristic p and let q = #k be the order of the residue field. Let ¢ E K (z) be a rational map of degree d ~ 2 with good reduction. Let s be a positive integer with p t sand s t q - 1. Finally, let PI, P2 E pI (K) be periodic points of exact periods nand sn respectively, and write Pi = [Xi, Yi] in normalized form. Prove that
2.27. Let PI, P2 , P3 , P4 E pI, and let the cross-ratio satisfies
where E( 11') mutation.
11' be
a permutation of the set {I, 2, 3, 4}. Prove that
= (_1)si g n (rr) equals 1 (respectively
-1) if 11' is an even (respectively odd) per-
2.28. Let ¢(z) = z2 + 1, let "( be a root of z2 + Z Z6
+ 2, and let a
be a root of
+ Z5 + 4z + 3z 3 + 7z 2 + 4z + 5. 4
In Example 2.37 we explained that there are units Ui,j
= ¢ib) -
¢j (a) E
Rb, o]"
for 0
::::: 1 and 0 ::::: j
< 2,
and that the unit group Rb, a]* has rank at most 5. Find generators and relations for the group of units generated by the six units Ui,j.
Chapter 3
Dynamics over Global Fields Just as algebraic number theory and Diophantine geometry can be studied over local fields and over global fields, so too does arithmetic dynamics have a local theory and a global theory. In this chapter we study some of the fundamental questions in arithmetic dynamics over global fields. Many of these construct ions, theorems , and conjectures have direct analogues in the theory of Diophantine equations , especially the arithmetic theory of elliptic curves. Among the topics covered in this chapter are dynamical analogues of the theory of canonical heights, finiteness of integral points, uniformity of rational torsion points, and cyclotomic and elliptic units. We have attempted to keep this chapter self-contained, but the reader desiring further motivation and background might consult standard textbooks on arithmetic and Diophantine geometry, such as the following: • • • • • •
E. Bombieri and W. Gubler [76], Heights in Diophantine Geomet ry. M. Hindry and J. Silverman [205], Diophantine Geometry. S. Lang [256], Fundamentals ofDiophantine Geometry. 1.-P. Serre [397], Lectures on the Morde//-Weil Theorem. 1. Silverman [410], The Arithmetic ofElliptic Curves. J. Silverman [412], Advanced Topics in the Arithmetic ofEllipt ic Curves.
3.1 Height Functions In order to study the arithmetic properties ofpoints in projective space, it is important to have a method of measuring the size of a point. This "size" should reflect the complexity of the point in an arithmetic sense. In particular, it is good if there are only finitely many points of bounded size. For example, suppose that we naively define the size of a rational number a to be its absolute value [o]. This reflects the size of a as a real number, but there are infinitely many rational numbers with bounded absolute value, so it does not correctly measure a's size from an arithmetic perspective. If we write a = alb as a fraction in lowest terms, then the integers a and b should each contribute to the 81
82
3. Dynamicsover GlobalFields
complexity of a, so we might define the size of a to be the larger of lal and Ibl. With this definition, it is obvious that there are only finitely many rational numbers of bounded size, since there are only finitely many integers of bounded absolute value. We can easily generalize this idea to rational points in projective space.
Example 3.1. Let P E IIDN (iQ) and write P using homogeneous coordinates as
P = [xo, Xl, .. ·, XN]. Since the coordinates are homogeneous, we can multiply through by an integer to clear the denominators, and we can also cancel any common factors, so we may assume that the homogeneous coordinates have been chosen to satisfy Xo, ... ,XN E
Z
and
gcd(xo, ... , XN)
= 1.
Having done this, we define the height ofP to be the quantity
H(P) = max{lxol, .. ·,
IXNI}.
It is clear that for any constant B, the set
{P
E IID
N
(iQ) : H(P) ::; B}
(3.1)
is finite. Indeed, this set clearly has fewer than (2B + 1)N+l elements, since each coordinate Xi of P is an integer satisfying IXil ::; B, so has at most 2B + 1 possible values. (See Exercise 3.2 for an asymptotic estimate for the size of the set (3.1).) When trying to generalize Example 3.1 to an arbitrary number field K, we run into the problem that the ring of integers of K may not be a principal ideal domain, so there is no uniform way to normalize the homogeneous coordinates of a point in IID N (K). For this reason we take a different approach based on the theory of absolute values. We now describe the absolute values on iQ, and then move on to general number fields. For further information about absolute values and completions of number fields, see any standard textbook on algebraic number theory or Diophantine geometry, for example [258, Section 2.1], [256, Chapters I, II], or [205, Section B.I].
Definition. The set ofstandard absolute values on iQ, denoted by
Mlf:b
consists of
the following absolute values:
• MQ contains one archimedean absolute value that is associated to its embedding iQ C 1Ft into the real numbers,
Ixloo
= usual absolute value on 1Ft = maxi X, -x} .
• Let p be a prime, and for any (nonzero) integer a, let
ord, (a) = exponent of highest power of p dividing a. In other words, pordp(a) is the highest power of p dividing a. The set MQ contains nonarchimedean (p-adic) absolute values, one for each prime p, defined by
83
3.1. Height Functions
Now let K /Q be a number field. The set ofstandard absolute values on K is denoted by M K and consists of all absolute values on K whose restriction to Q is one of the absolute values in M!Q. We write Mf( for the (archimedean) absolute values on K lying above I . 100, and M~ for the (nonarchimedean) absolute values of K lying above the p-adic absolute values of Q. The archimedean absolute values of K correspond to embeddings of K into lR. or C, while the nonarchimedean absolute values correspond to prime ideals of the ring of integers of K. Indeed, the ring of integers of K, denoted by RK, may be characterized as the set
R K = {a E K : lal v
:::;
1 for all
v E M~}.
More generally, if S c MK is any (finite) set of absolute values containing Mf(, then the ring ofS-integers ofK is the set
Rs =
{a E K: lal v
:::;
1 for all v ~ S}.
Definition. For any absolute value v E M K on K, let K; denote the completion of Kat v. The local degree ofv, denoted by n v , is the quantity
For example, if v is archimedean, then Qv = lR. and ti; is 1 or 2 depending on whether v corresponds to a real or complex embedding of K, respectively. Similarly, if v is nonarchimedean, then Qv = Qp is the p-adic rational numbers for some prime p, the absolute value v corresponds to some prime ideal p of K lying over p, and using standard notation, nv = e(p)f(p) is the product of the ramification index of p and the degree of the residue field modulo p. The next two results, which we quote without proof, will be used to define and study height functions. Proposition 3.2. (Extension Formula) Let L / K /Q be a tower ofnumberfields, and let v E M K be an absolute value on K. Then 1
[L : K]
L
nw
= n.;
wEML
wlv
Here the notation wlv means that the restriction ofw to K is equal to v, so the sum is over all absolute values on L that extend the absolute value v on K. Proposition 3.3. (Product Formula) Let K /Q be a numberfield. Then for all a E K*.
84
3. Dynamics over Global Fields
For proofs of these two formulas, see for example [258, Section ILl and Section V. I]. The astute reader will have observed that if we take K = Q and v nonarchimedean, then the extension formula becomes the well-known formula
L e(p) f (p) =
[L : Q] .
pip
We now have all of the tools needed to define the height of algebraic points in projective space. Definition. Let K /Q be a number field, and let P E p N (K) be a point with homogeneous coordinates
XO, . .. , XN E K . The height ofP (relative to K) is the quantity
HK(P)=
II
max{lxolv, .. ·,lxNlvt
v •
v EM K
Notice how the height measures the size of the coordinates of P with respect to all of the absolute values on K . We begin by checking that the height is well-defined and proving two elementary properties. Proposition 3.4. Let K /Q be a number fi eld and P E p N (K) a point. (a) The height HK (P ) is independent of the choice of homogeneous coordinates forP. (b) HK (P ) ;::: 1. (c) Let L / K be a finite extension. Then
H L(P)
= HK (P )[L:KI.
Proof (a) Any other choice of homogeneous coordinates for P = [xo , . .. , XN ] has the form P = [axo, . . . , aXN] for some a E K *. Then the product formula (Proposition 3.3) yields
II
max{laxil v}nv = ,
v E MK
II
la l~v max{x ilv}nv = ,
v E MK
II
v max{lxilv}n • z
vE MK
(b) Choose an index j such that X j i= 0 and divide the homogeneous coordinates of P by Xj ' Then one of the homogeneous coordinates is equal to 1, so every factor in the product defining HK(P ) is at least 1. (c) We choose homogeneous coordinates for P that are in K and use the extension formula (Proposition 3.2) to compute
H L(P) =
II w EM L
w= max{lxilv}n ,
II II
m~{lxilv}nw ,
v E M K wE M L wl v
This completes the proof of Proposition 3.4.
o
3.1. Height Functions
85
Remark 3.5. Now that we know that the height is well-defined , we should check that it agrees with the naive definition of the height on IPN (Q) given in Example 3.1. Thus let P = [xQ,"" X N 1 E IP N (Q) with homogeneous coordinates satisfying Xi E Z and gCd(Xi) = 1. Then every nonarchimedean absolute value v E gives IXilv :::; 1 for all indices i and IXi lv = 1 for at least one index i. Hence in the product defining HQ(P ), only the term corresponding to the archimedean absolute value contributes, so
M8
We note again that the set
{P E IPN (Q): H(P):::; B} is finite for any fixed bound B. Later in this section we prove an analogous result for arbitrary number fields.
It is sometimes easier to work with a height function that does not depend on choosing a particular number field.
Definition. Let P E lP'N (Q) be a point whose coordinates are algebraic numbers . The (absolut e) height of P, denoted by H (P ), is defined by choosing any number field K such that P E lP'N (K) and setting
The transformation formula in Proposition 3.4(c) tells us that H (P ) is well-defined, independent of the choice of the field K. In the following , K denotes an algebraic closure of a number field K .
Theorem 3.6. Let KIQ bea numb erfi eld, let P
E lP'N (K) , and let a E Gal (K I K
).
Then
H (CJ(P)) = H (P ). In other words, the height is invariant under the action ofthe Galois group. Proof Let LIK be a finite Galois extension such that P E IP'N (L ). For any absolute value v E M L and any a E Gal(LIK), we can define a new absolute value a(v) on L by the formula
lalcr(v) = la( a)lv' Using the fact that a : L ---. L is a field automorphism, it is easy to verify that a(v) is an absolute value on L , and indeed the map v f---7 a(v) is simply a permutation of the set of absolute values NfL . Further, the automorphism a : K ---. K induces an isomorphism of the completions a : K; ---. Kcr(v), since the effect of a on K is to transform the absolute value v into the absolute value a(v ). In particular, the local degrees and
86
3. Dynamicsover GlobalFields
are equal. We now write P
= [xo, ... , XN]
E jp'N (L) and compute
II max{IC1(xo)lv, ... , 1C1(XN)lv}nv vEM = II max{lxol a(v),· .. ,lxNla(v)r<7(v) vEM = II max{lxol v, ... , IXNlv}
HL{C1(P)) =
L
L
vEM L =HL{P).
Taking [L : Q]th roots completes the proof that H(C1(P)) = H(P).
o
Definition. Let P = [xo, ... , XN] E jp'N (K) be a point with algebraic coordinates. The smallest field over which P can be defined is called the field ofdefinition of P over K and is denoted by K(P). It can be constructed by choosing a nonzero coordinate Xi and setting Xl xo, K(P)=K ( , ... ,XN) - . Xi Xi Xi
It is easy to see that this field is independent of the choice of the index i. Our purpose in defining the height is to have a method of measuring the arithmetic size or complexity of points in projective space, analogous to the way in which the size of a rational number is measured by taking the larger of its numerator and its denominator. The following finiteness theorem is of fundamental importance.
Theorem 3.7. Let K/Q be a number field, and let B be any constant. Then the set ofpoints isfinite. More generally, for any constants Band D, the set ofpoints
{P E jp'N (Q) : H(P) :S Band [Q(P): Q] :S D} is finite. In other words, there are only finitely many points in jp'N (Q) of bounded height and bounded degree. Proof It clearly suffices to prove the second statement, i.e., that there are only finitely many points
P
= [xo, ... , XN]
E jp'N (Q)
satisfying
H(P):S Band [Q(P): Q] = D.
Dividing the homogeneous coordinates of P by some nonzero coordinate, we may assume that some coordinate equals 1. Then for each 0 :S i :S N,
87
3.1. HeightFunctions
SO [lQl(Xi) : lQl] ~ D. Further, if we write K = lQl(P) and d = [K : lQl], then we can estimate the heights of the individual coordinates by
B
~ H(P ) =
II
(
max{ lxolv , ... , IX Nlv}nv ) lid
vEMQ(p )
~ II vEM
=
((max{ lxolv,l}nv ... max{ IXNlv,l}nV)lINY /d
K
(IT( II max{IXil v, ir- Y ldY IN <=0
v E l.! K
= (H(xo)H( xt} . . . H(XN ))l IN
~ H(Xi)l IN
for every 0 ~ i ~ N.
Thus each coordinate of P lies in a field of degree at most D and has height at most BN . Replacing B by B N and taking numbersof exact degree d for each 1 ~ d ~ D, it suffices to prove that the set
{a E Q : H (a ) ~ Band [lQl(a) : lQll = d}
(3.2)
is finite. Let a be in the set (3.2) and write the minimalpolynomial of a as
Fa(X) = X d
+ alX d - l + ... + ad E lQl[X].
Also factor Fa(X) over Cas
The number ai , . . . , ad are the conjugates of a , so Theorem3.6 tells us that
H (a t} = . . . = H(ad) = H( a) . Further, the coefficients al, . .. , ad of Fa(X) are the elementary symmetric polynomials of the roots (up to ±l). For example,
al
= -(al + a2 + .. .+ ad)
More generally, for any 1 ~ k
~
ak = (_ l) k
and ad = (-1) da la 2 · · · ad.
d we have
L
a il ai2 ' " aik'
(3.3)
l :5i l
Note that there are (~) terms in the sum (3.3). Wecan use (3.3) and the triangle inequality to estimatethe absolutevaluesof the coefficients ai , .. . , ad in terms of the absolute values of the roots ai , . . . , a d. Thus for any v E M Q(a ) we compute
3. Dynamics over Global Fields
88
<
(~) max{lallv, 1} max{la2Iv, 1}'"
max{ladlv, 1}.
Further, if v is nonarchimedean, then we may discard the factor of (~). Taking the maximum over k and using the fact that (~) :::; 2d for all k, we find that max{1, lall v, la2Iv,"" ladlv}
:::; 2d max{lallv, 1} max{la2Iv, 1}'" max{ladlv, 1}, where again the 2d is needed only if v is archimedean. Now raise to the n., power, multiply over all v, and take the cfh root to obtain
H([1, aI, a2, ... , ad]) :::; 2dH(al)H(a2)'" H(ad). Theorem 3.6 tells us that H(ai) assumption, so we conclude that
= H(a)
H([1,al,a2,
for every i, and we have H(a) :::; B by
,ad]):::; (2B)d.
In other words, the height of[1, aI, a2, , ad] E JP'd(Q) is bounded by (2B)d. However, we already know that there are only finitely many Q rational points in projective space (see Example 3.1 and Remark 3.5). Hence there are only finitely many possibilities for the minimal polynomial Fa(X) of a, and since each Fa(X) has only d roots, there are only finitely many possibilities for a. This proves that the set (3.2) 0 is finite, which completes the proof of the theorem. We can use Theorem 3.7 to give a very brief proof of a famous theorem of Kronecker. It says that an algebraic integer whose conjugates all lie on the unit circle must be a root of unity.
Theorem 3.8. Let a E Q be a nonzero algebraic number. Then
= 1 if and only if a is a root ofunity. is a root of unity, then lal v = 1 for every absolute value v, so we clearly H (a)
Proof If a have H (a) = 1. Now suppose that H (a) = 1. Directly from the definition of the height we see that for all (3 E Thus H(a n)
= H(a)n = 1 for all n
~
Q and all n
~ 1.
1, which implies in particular that the set
{a, a 2, a 3 , ... } is a set ofbounded height. This set is also clearly contained in the number field Q( a), so Theorem 3.7 tells us that it is a finite set. Hence there are integers i > j > 0 such that a i = a j , which shows that a is a root of unity. 0
89
3.2. HeightFunctions and Geometry
3.2 Height Functions and Geometry The height ofa point P measures the arithmetic complexity of P. We now investigate how the height of P changes when we map it to some other projective space. This will allow us to relate geometric properties of maps to the arithmetic information encapsulated by the height function. Remark 3.9. For ease of exposition we restrict attention to heights on projective space, but the reader should be aware that there is a general theory of height functions on algebraic varieties due to Weil. Height functions provide a powerful tool for converting algebro-geometric relationships into number-theoretic relationships. Theorem 3.11 below provides an example; it converts the geometric information that a map ¢ has degree d into the arithmetic information that ¢( P) has height that is more-or-less the rfh power of the height of P. A summary of the general theory of heights on varieties is given in Section 7.3; see [205, Part B] or [256, Chapters 3-5] for further details.
Definition. A rational map ofdegree d between projective spaces is a map ¢ : jp'N
¢(P)
=
----7
jp'M
[fo(P), ... , fM(P)],
where fo, ... , fM E K[Xo, ... , X N] are homogeneous polynomials of degree d with no common factors. (The polynomial ring K[Xo, ... ,XN ] is a unique factorization domain, so it makes sense to talk about common factors.) The rational map ¢ is defined at P ifat least one of the values fo(P), ... ,fM(P) is nonzero. The rational map ¢ is called a morphism ifit is defined at every point ofjp'N (K), or equivalently, if the only solution to the simultaneous equations
is the trivial solution X o = ... = X N = O. If the polynomials fo, ... ,fN have coefficients in K, we say that ¢ is defined over K. Our goal is to relate the height of a point P to the height of its image ¢(P). To do this in general, we use an important theorem from algebraic geometry called the Nullstellensatz. Here Null
= zero,
stellen = places,
satz
= theorem,
so the Nullstellensatz is a theorem that relates a function to the points at which it vanishes. We give a brief overview of some basic concepts from algebra and algebraic geometry that are needed to understand the statement of the Nullstellensatz. However, we note that for rational maps ¢(z) E K(z) in one variable (i.e., maps ¢ : jp'l --+ jp'l), the Nullstellensatz may be replaced by a simple argument based on the fact that the ring K[z] is a principal ideal domain; see Exercise 3.8. An ideal I in K[Xo, ... , X N ] is homogeneous ifit is generated by homogeneous polynomials. The radical ofan ideal I is the ideal
90
3. Dynamics over Global Fields
Vi = {J E K [Xo, ... , X N] : rEI for some n ~ I} .
r
(N.B. In this definition , the quantity is the n th power of f , not its nth iterate.) The algebraic set attached to a homogeneous ideal I is the set
V(I)
= {P
E JIl>N (K )
: f(P)
= 0 for all f
E
I}.
The Hilbert basis theorem [259, Chapter VI, Theorem 2.1] implies that I is finitely generated, so V( I) is the set of simultaneous zeros ofa finite collection of polynomials. If V c JIl>N (K) is an algebraic set, the ideal attached to V is the ideal
I (V ) =
ideal generated by all hOmOgeneous) polynomials f E K[Xo, . .. , X N] . ( such that f (P ) = 0 for all P E V
It is clear that if V = W , then I(V) = I(W). The converse is not quite true, since the algebraic set attached to the radical .Ji is the same as the algebraic set attached to I . Theorem 3.10. (Hilbert's Nullstellensatz) Let I and J be homogeneous ideals properly contained in K [Xo , . . . , X N]. Then
V (I) = V( J)
if and only if
Vi = /J.
Proof Suppose that .Ji = /J. Let P E V (I) and f E J. Then rEI for some n ~ 1, so r(p) = 0, so f(P) = O. This is true for every f E J, so P E V(J). This proves that V(I) c V(J), and the opposite inclusion follows by interchanging the roles and I and J . Hence V(I ) = V(J ). This proves the trivial direction of the theorem. For a proof of the nontrivial converse , see [198, I.1.3A] or [259, Section X.2]. 0 We are now ready to prove an important result that says that up to a scalar factor, a morphism of degree d causes the height to be raised to the rf't power. Theorem 3.11. Let ¢ : JIl>N (K) -7 JIl> M(K) be a morphism ofdegree d. Then there are constants C1 , C2 > 0, depending on ¢, such that
for all P E JIl>N (K ). (Infact, the upper bound H(¢(P )) ~ C 2 H (P )d is validfor rational maps provided we restrict attention to po ints P at which ¢ is defined.) Proof We begin with some notation. For any point P = [xo , . . . ,x N] with coordinates in K and any absolute value v E M K , we write
(This assumes that we have fixed particular homogeneous coordinates for P.) Similarly, we define the absolute value of a polynomial
3.2. Height Functions and Geometry
f(X 0,
... ,
91 iN
~ aio ...iN x: 0'" X N '"
X) N =
io,o.. ,iN
to be
If Iv = . max laio ...iN lv, 'l,o,···,'l,N
and if ¢ = [fo, ... , f M] is a collection of polynomials, we let
1¢lv = max IfJlv. J Notice that the height of a point P E lP'N (K) may now be written in the compact form
H(P) =
II
(
l/[K:
IPI~v )
vEMK
We define the height of a polynomial
H(J) =
(
II
f
or a collection of polynomials ¢ similarly,
l/[K:
Ifl~v )
H(¢) =
and
vEMK
(
II
l/[K.
1¢I~v )
(3.4)
vEMK
We set one final piece of notation that will make our computations easier. For any absolute value v E M K and any number m, we set
8 (m) = {m if v E M'j( (i.e., if v is archimedean), v 1 if v E M~ (i.e., if v is nonarchimedean). With this notation, we can write a uniform version of the triangle inequality as
Now letP = [xo, ... , XN] E jp'N (K) be a rational point and f E K[Xo, ... , X N ] a homogeneous polynomial of degree d. Then for any v E M K we can estimate
If(P)lv =
I
L
iNX~o ... x~ I
aio ...
i«, .. "iN2:0
v
io+···+iN=d ::::; 8v (# of terms) . max laio ...iNX~o ... x~ Iv' 'lo,··,,'l,N
The number of terms in the sum is equal to at most the number of monomials of degree d in N + 1 variables, which is given by the combinatorial symbol (N r). For our purposes it is enough to know that it depends only on Nand d. Continuing with the computation, we find that
92
3. Dynamics over Global Fields
We apply this inequality with over k to obtain
1=
lk for k
=
0, . .. , N and take the maximum (3.6)
Finally, we raise (3.6) to the n v power, multiply over all v E MK, and take the [K : iQ] root to obtain
Note that in deriving this formula , we have used the fact that
II vEMK
6v (at
v
=
II
an v
and
vEM'j;'
L
nv =
[K : iQ],
VEM'j;'
where the latter is a special case of Theorem 3.2. This completes the proof of the upper bound with the explicit constant
And as indicated in the statement of the theorem, we never assumed that ¢ is a morphism, so the upper bound holds for rational maps . In order to prove the opposite inequality, it is necessary to use the assumption that ¢ is a morphism, or equivalentl y, that ¢ = [fo, ... , 1M] for homogeneous polynomials 10, ,1M having no common zero in lP'N (K) . This condition tells us that , 1M) and (X o, · · · , X N ) in K [X o, . . . , X N ] define the same algethe ideals (fo, braic set in lP'N (K), namel y the empty set. The Nullstellensatz (Theorem 3.10) then implies that these two ideals have the same radical. In part icular,
Hence we can find an integer e
~
0 such that
(The Nullstellensatz says that there is an exponent ei for each X i , and then we take e to be the largest of the ei') Writing this out explicitly, we find homogeneous polynomials g i j E K[Xo, . .. , X N ] such that
X : = 9iOI0 + 9 i l h
+ ... + 9 i /l,[ 1M
for each 1 :::; i :::; N .
We observe that each g i j is homogeneous of degree e - d, since each Ii has degree d. We evaluate these polynomial identities at P = [xo, ... ,XN] E lP'N (K) and estimate v-adic absolute values ,
3.2. Height Functionsand Geometry
93
= max IgiO(P)fo(P) + gil (P)!I(P) + ... + giM(P)fM(P) I
v
O~i~N
~ 6v (M
+ 1) O
So if we let Iglv = maxi,j 1%lv and aM,N,d,e = both sides by IPI~-d yields the estimate
(M
+ l)(N:~dd), then
dividing
(3.7) Now raise this inequality to the [K : Q] root to obtain
H(p)d
ti;
power, multiply over all v E M K , and take the
< aM,N,d,eH(g)H(¢(P)),
where H (g) is the height determined by the coefficients of all of the % polynomials. Its precise value is not important for our purposes; it is enough to note that it does not depend on the point P. We have thus proven that H(p)d ~ CH(¢(P)) for a constant C that does not depend on P, which completes the proof of Theorem 3.11.
o Theorem 3.11 says that the height of cP(P) is approximately equal to the rfh power of P. This means that H is a multiplicative kind of function, similar in some ways to an absolute value. It is often convenient notationally to work instead with an additive function, which prompts the following definition.
Definition. The logarithmic height (relative to K) is the function
The absolute logarithmic height is the function
h(P)
= log
H(P).
Using this logarithmic notation, Theorem 3.11 says that h(¢(P)) and dh(P) differ by a bounded amount. The notion of functions that differ by a bounded amount appears so frequently in mathematics that it has its own notation.
Definition. Let S be a set and let f, g : S formula
--t
IR be real-valued functions on S. The
94
3. Dynamics overGlobal Fields
l=g+O(l) means that the quantity
II(p) - g(p) 1 is bounded as P ranges over S. More generally, if h : S the formula 1 = 9 + O(h)
--t
lR is another function,
means that there exists a constant C such that
II(P ) - g(P)1 ~ Clh(P)1
for all PES.
Using the theory of heights as developed in Theorems 3.7 and 3.11, it is easy to prove that a morphism ¢; : l[DN --t l[DN of degree at least 2 has only finitely many preperiodic points defined over a given number field. This result was first discovered by Northcott [343] and then independently rediscovered many times in varying degrees of generality. It is instructive to compare this global proof with the local proof of finiteness of periodic points given in Corollary 2.26.
Theorem 3.12. (Northcott [343]) Let ¢; : l[DN --t l[DN be a morphism of degree d 2 2 defined over a number field K. Then the set of preperiodic points PrePer( ¢;) C l[DN (K) is a set ofbounded height. In particular, PrePer( ¢;, l[DN (K)) = PrePer( ¢;) n l[DN (K) is a finite set, and more generally, for any D 2 1 the set
U
PrePer( ¢;, l[DN (L))
[ L : Kl~ D
is finite .
Proof Theorem 3.11 tells us that there is a constant C = C (¢;) such that
h(¢;(Q)) 2 dh(Q) - C Applying this with
for all Q E l[DN (K) .
Q = R, ¢;( R), ¢;2 (R), ... , ¢;n-l (R) yields
h(¢;n(R)) 2 dnh(R) - C(l
+ d + d2 + ... + dn- 1 ) 2 dn(h(R) - C).
Now suppose that P is preperiodic, say ¢;m+n(p) = ¢;m(p) with n Setting R = ¢;m (P) and using the preperiodicity yields
(3.8)
2 1 and m 2
o.
and hence (3.9)
95
3.3. The Uniform Boundedness Conjecture
For the last inequality we are using the fact that d 2 2 and n 2 1. Next we use inequality (3.8) with n = m and R = P, (3.10) Finally, combining (3.9) and (3.10) yields
h(P)
1
1
< dm h(qr(p)) + C < dm 2C + C ::; 3C,
which completes the proof that the height of P is bounded by a constant depending only on the map ¢. The remaining assertions of the theorem are immediate consequences of Theorem 3.7, which says that sets of bounded height have only finitely many points of bounded degree. D
Remark 3.13. One may also consider preperiodic points defined over infinite extensions. For example, we might fix an infinite extension L/Q and ask which morphisms ¢ have infinitely many periodic or preperiodic points in jp'N (L). Very little is known about this question in general. We consider a special case. Let K/Q be a number field and let KCYcl denote the maximal cyclotomic extension of K, i.e., the field obtained by adjoining to K all roots of unity. Now let ¢(z) E K[z] be a polynomial of degree d 22 and suppose that the set PrePer (¢, jp'1 (Kcycl)) has infinitely many points. Then Dvornicich and Zannier [146] prove that there is a Mobius transformation f E PGL 2 (k ) such that ¢f(z) has the form ±zd or Td(±z), where Td(z) is a Chebyshev polynomial (see Section 1.6.2). Notice that an alternative dynamical definition of KCYcl is the field obtained by adjoining to K all 'of the points in PrePer(zd) for some d 2 2. This suggests a natural question. Question 3.14. Let¢,'l/J: jp'N ---., jp' Nbemorphismsofdegreeatleast2definedovera number field K. Suppose that ¢ has infinitely many preperiodic points defined over the field K (PrePer( 'l/J)). What can one deduce about the relationship between ¢ and e?
3.3
The Uniform Boundedness Conjecture
Northcott's Theorem 3.12 says that a morphism ¢ : jp'N ---., jp'N has only finitely many K -rational preperiodic points. It is even effective in the sense that we can, in principle, find an explicit constant C( ¢) in terms of the coefficients of ¢ such that every point P E PrePer(¢) satisfies h(P) ::; C(¢). This also allows us to compute an upper bound for #PrePer(¢,jp'N(K)), but the bound grows extremely rapidly as the coefficients of ¢ become large. A better bound, at least for periodic points, may be derived from the local estimates in Chapter 2 as described in Corollary 2.26. However, even that estimate depends on the coefficients of ¢, since it is in terms of the two smallest primes for which ¢ has good reduction. The following uniformity
96
3. Dynamics over Global Fields
conjecture says that there should be a bound for the size of PrePer (¢ , pN (K}) that depends in only a minimal fashion on ¢ and K. Conjecture 3.15. (Morton-Silverman [312]) Fix integers d 2: 2, N 2: 1, and D 2: 1. There is a constant C (d , N, D } such thatforall numberfields K / Q ofdegree at most D and all finite morphisms ¢ : pN ---+ pN ofdegree d defined over K ,
# PreP er (¢ , pN (K })
:::; C(d,N, D} .
Remark 3.16. There are many results in the literature giving explicit bounds for the size of the sets PrePer(¢ , pN (K}) or P er(¢ , p N (K}) in terms of ¢, especially in the case N = 1. Some of these results use global methods, while others use a small prime of good (or at least not too bad) reduction for ¢ . For example, we used local methods in Corollary 2.26 to give a weak bound for # P er (¢, p I (K )). For further results, see [52, 87, 90, 91, 92,100, 101,137,162,171,191, 192, 193, 194, 195, 227,331,265,312,325,326,328,329,330,332,353,355,358,359,361,454]. Remark 3.17. Very little is known about Conjecture 3.15. Indeed, it is not known even in the simplest case (d, N , D ) = (2, 1, 1), that is, for Q-rational points and degree-2 maps on pl . Specializing further, if we let ¢c : p I ---+ pI denote the quadratic map ¢c(z) = z2 + c, then the conjecture implies that sup # P er( ¢c, pI (Q) ) < 00, cEQ
but the best knoWIi upper bounds for # Per ( ¢c, p I (Q) ) depend on c. There are one-parameter families of c-values for which cPc(z) has a Q-rational periodic point of exact period 1, 2, or 3; see Exercise 3.9 and Example 4.9. And one can show that ¢c cannot have Q- rational periodic points of exact period 4 or 5; see [171, 309]. Poonen has conjectured that ¢c cannot have any Q- rational periodic points of period greater than 3. Assuming this conjecture, he gives a complete description of all possible rational preperiodic structures for ¢c; see [361]. Remark 3.18. Another interesting collection of rational maps is the family
¢a,b(Z ) = az
b
+ ;.
These maps have the symmetry property ¢a,b( -z) = - ¢a,b(Z), i.e., conjugation by the map f( z) = -z leaves them invariant. It is known that there are one-parameter families of these maps with a Q-rational periodic point of exact period 1 (in addition to the obvious fixed point at 00), 2, or 4, and that none of the maps ¢a,b(Z) has a Q rational periodic point of exact period 3. See [286] for details, and Examples 4.69 and 4.71 and Exercises 4.1, 4.40, and 4.41 for additional properties of these maps. Remark 3.19. Conjecture 3.15 is an extremely strong uniformity conjecture. For example, if we consider only maps ¢ : pI ---+ p I of degree 4 defined over Q, then the assertion that # PrePer( cP, pI (Q) ) :::; C for an absolute constant C immediately implies Mazur's theorem [292] that the torsion subgroup of an elliptic curve E / Q
3.4. Canonical Heights and Dynamical Systems
97
is bounded independently of E . To see this, we observe that Proposition 0.3 tells us that E tors = PrePer ([2],E) , and hence the associated Lattes map I/JE,2 described in Section 1.6.3 satisfies
x (E tors ) = PrePer (I/JE,2, jp>I ). Note that I/JE,2 has degree 4. In a similar manner, Conjecture 3.15 for maps of degree 4 on jp>1 over number fields implies Merel's theorem [297] that the size of the torsion subgroup of an elliptic curve over a number field is bounded solely in terms of the degree of the number field. Turning this argument around, Merel's theorem implies the uniform boundedness conjecture for Lanes maps , i.e., for rational maps associated to elliptic curves; see Theorem 6.65. Lattes maps are the only nontrivial family of rational maps for which the uniform boundedness conjecture is currently known. In higher dimension, Fakhruddin [162] has shown that Conjecture 3.15 implies that there is a constant C (N, D) such that if K is a number field of degree at most D and if AIK is an abelian variety of dimension N, then
#A (K )tors
:s C(N, D).
He also shows that if Conjecture 3.15 is true over Q, then it is true for all number fields.
3.4
Canonical Heights and Dynamical Systems
It is obvious from the definition of the height that for all
0:'
E
Q.
(3. 11)
Notice that Theorem 3.11 applied to the particular map I/J(z) = zd gives the lessprecise statement (3.12) h(I/J(P )) = dh(P ) + 0(1 ). Clearly the exact formula (3.11) is more attractive than the approximation (3.12). It would be nice if we could modify the height h in some way so that the general formula (3.12) from Theorem 3.11 is true without the 0 (1). It turns out that this can be done for each morphism I/J. To create these special heights , we follow a construction due originally to Tate. Theorem 3.20. Let S be a set, let d
I/J : S -+ S
> 1 be a real number, and let and
h : S-+ IR
be fun ctions satisfying
h(I/J(P )) = dh(P ) + 0 (1)
foral! P E S .
3. Dynamics over Global Fields
98
Then the limit (3.13)
exists and satisfi es:
= h(P) + 0(1) . h(¢(P)) = dh(P).
(a) h(P) (b)
Thefunction
h:S
--; IR is uniquely determined by the properties (a) and (b).
Proof We prove that the limit (3.13) exists by proving that the sequence is Cauch y. Let n > m ~ a be integers. We are given that there is a constant C such that for all Q E S.
Ih(¢(Q)) - dh(Q)1 ::; C We apply inequality (3.14) with Q
(3.14)
= ¢i - l (P ) to the telescoping sum
I: nh(¢n(p )) - d~ h(¢m(p ))j =
t
~i (h (¢i (P)) -
t
~i Ih(¢i (p )) -
dh(¢i-l (P) ))
i=m+ l
<
dh(¢i-l (p ))1
i =m+ l
The inequality (3.15) clearly implies that as m , n --;
00,
so the sequence d- nh(¢n(p )) is Cauchy and the limit (3.13) exists. In order to prove (a), we take m = a in (3.15), which yields
Next we let n --;
00
to obtain
Jh¢(P) - h(p )1::; d ~ i ' which is (a) with an explicit value for the 0 (1) constant. The proof of (b) is a simple computation using the definition of 11"
Finally, to prove uniqueness, suppose that h' and (b). Then the difference 9 = 11, - 11,' satisfies
:S
--; IR also has properties (a)
99
3.4. Canonical Heights and Dynamical Systems
g(P ) = 0 (1)
g(¢(P)) = dg(P) .
and
These formulas hold for all elements P ES , so
d"g(P ) = g(¢n(p )) = 0 (1)
for all n ~
o.
In other words, the quantity dng(P ) is bounded as n --+ 00 , which can happen only if g(P) = O. This proves that h(P) = h'(P ), so 11, is unique. D Definition. Let ¢ : lP'N --+ lP'N be a morphism of degree d ~ 2. The canonical height fun ction (associated to ¢) is the unique function
satisfying
h¢(P) = h(P) + 0(1)
and
The existence and uniqueness of h¢ follow from Theorem 3.20 applied to the maps and since Theorem 3.11 tells us that ¢ and h satisfy
h(¢(P )) = dh(P ) + 0 (1)
for all P E lP'N (Q).
Remark 3.21. The definition hef> (P ) = lim n ---+oo d-nh(¢n(p )) is not practical for accurate numerical calculations. Thus even for P E lP'l(Q), one would need to compute the exact value of ¢n(p), whose coordinates have O(dn) digits. A practical method for the numerical computation of h¢(P) to high accuracy uses the decomposition of kef> as a sum of local heights or Green functions. This decomposition is described in Sections 3.5 and 5.9. See in particular Exercise 5.29 for a detailed description of the algorithm. The canonical height provides a useful arithmetic characterization of the preperiodic points of ¢. Theorem 3.22. Let ¢ : lP'N and let P E lP'N (Q). Then
--+
lP'N be a morphism ofdegree d ~ 2 defined over Q
P E PrePer(¢)
if and only if hq, (P)
= O.
Proof If P is preperiodic, then the quantity h(¢n(P)) takes on only finitely many values, so it is clear that h(¢n(P)) --+ 0 as n --+ 00. Now suppose that hef>(P) = O. Let K be a number field containing the coordinates of P and the coefficients of ¢, i.e., P E lP'N (K) and ¢ is defined over K. Theorem 3.20 and the assumption hef>(P) = 0 imply that
a:»
100
3. Dynamics over Global Fields
Thus the orbit
Gq, (P ) = {P, ¢(p ), ql (p ),q} (P ), . .. } C jpN(K) is a set of bounded height, so it is finite from Theorem 3.7. Therefore P is a preperiD odic point for ¢. Remark 3.23. Further material on canonical heights in dynamics may be found in Sections 3.5, 5.9, and 7.4, as well as [16, 20, 23, 36, 38, 39, 87, 88, 89, 147, 159, 231,228,230,232,234,406,409,446,453].
Theorem 3.22 is a generalization of Kronecker's theorem (Theorem 3.8), which says that h(0:) = 0 if and only if 0: is a root of unity. Kronecker's theorem follows by applying Theorem 3.22 to the lfh-power map ¢(z) = zd, whose canonical height is the ordinary height h. The fact that only roots of unity have height 0 leads naturally to the question of how small a nonzero height can be. If we take the relation h(ad) = dh(0:) and substitute in a = 21 / d , we find that
h(2 1/ d ) =
1
h(2) = 10:2 ,
so the height can become arbitrarily small. However, this is poss ible only by taking numbers lying in fields of higher and higher degree . For any algebraic number a, let
D(a ) = [Q(o:) : QJ denote the degree of its minimal polynomial over Q. Conjecture 3.24. (Lehmer's Conjecture [264]) There is an absolute constant K such that
>0
h(o: ) 2: K/D(o:) for every nonzero algebraic number 0: that is not a root ofunity. There has been a great deal of work on Lehmer's conjecture by many mathematicians ; see for example [7, 8, 75, 94, 264, 366, 421, 422, 424, 445]. (The survey [422] contains an extensive bibliography.) The best result currently known, which is due to Dobrowolski [138], says that
h(o:) > _K_ (10glOgD(0:)) 3 10gD(0:) - D(a) The smallest known nonzero value of D(o:)h(a) is
D((3o)h((3o) = 0.1623576 . .. , where (30
= 1.17628 . .. is a real root of x lO
+ x9 _
x7
_
x6
_
x5
-
x4
-
x3
+ X + 1.
Theorem 3.22 tells us that hq,(P) = 0 if and only if P is a preperiodic point for ¢. This suggests a natural generalization of Lehmer's conjecture to the dynamical setting. (See [317] for an early version of this conjecture in a special case .)
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3.4. Canonical Heights and Dynamical Systems
Conjecture 3.25. (Dynamical Lehmer Conjecture) Let ¢ : jp'N ----+ jp'N be a morphism defined over a number field K, and for any point P E jp'N (K), let D(P) = [K(P) : K]. Then there is a constant r: = Ii(K, ¢) > 0 such that for all P E
jp'N (K)
with P ~ PrePer(¢).
There has been considerable work on this conjecture for maps ¢ : jp'I ----+ jp'I that are associated to groups as described in Section 1.6. For example, in the case that ¢ is attached to an elliptic curve E, it is known that in general [291],
D(P)3log2 D(P) Ii D(P)2
.r.. (log log D(P) ) D(P)
log D(P)
if j(E) is nonintegral [203], 3
if E has complex multiplication [263].
Aside from maps associated to groups, there does not appear to be a single example for which it is known that hq,(P) is always greater than a constant over a fixed power of D(P). Using trivial estimates based on the number of points of bounded height in projective space, it is easy to prove a lower bound that decreases faster than exponentially in D(P); see Exercise 3.17. Remark 3.26. The Lehmer conjecture involves a single map ¢ and points from number fields of increasing size. Another natural question to ask about lower bounds for the canonical height involves fixing the field K and letting the map ¢ vary. For example, consider quadratic polynomials ¢c(z) = Z2 + c as c varies over Q. Is it true that hq,Ja) is uniformly bounded away from 0 for all cEQ and all nonpreperiodic a E Q? In other words, does there exist a constant Ii > 0 such that
hq,Ja) 2':
r:
for all c, a E Q with a ~ PrePer(¢c)?
We might even ask that the lower bound grow as c becomes larger (in an arithmetic sense). Thus is there a constant Ii > 0 such that
hq,Ja) 2':lih(c) forallc,a E Qwitha ~ PrePer(¢c)? This is a dynamical analogue of a conjecture for elliptic curves that is due to Serge Lang; see [202], [254, page 92], or [410, VIII.9.9]. For the quadratic map z2 + c, the height of the parameter c provides a natural measure of its size, but the situation for general rational maps ¢(z) E K(z) is more complicated. We cannot simply use the height of the coefficients of ¢, because the canonical height is invariant under conjugation (see Exercise 3.11), while the height of the coefficients is not conjugation-invariant. We return to this question in Section 4.11 after we have developed a way to measure the size of the conjugacy class of a rational map.
102
3. Dynamics over Global Fields
3.5 Local Canonical Heights The canonical height hq;, attached to a rational map ¢ is a useful tool in studying the arithmetic dynamics of ¢. For more refined analyses, it is helpful to decompose the canonical height as a sum of local canonical heights, one for each absolute value on K. In this section we briefly summarize the basic properties of local canonical heights, but we defer the proofs until Section 5.9. The reader wishing to proceed more rapidly to the main arithmetic results of this chapter may safely omit this section on first reading, since the material covered is not used elsewhere in this book. The construction of the canonical height relies on the fact that the ordinary height satisfies h(¢(P)) = dh(P) + 0(1), so it is "almost canonical." The ordinary height of a point P = [a, 1] is equal to the sum
h(P) = h(a) =
L
n vlogmax{lal v , l },
vEMK
so for each v E MK it is natural to define a local height function
We can understand Av geometrically by observing that for v E M~,
where Pv is the nonarchimedean chordal metric defined in Section 2.1. One says that Av(a) is the logarithmic distance from a to 00. Unfortunately, the function Av does not transform canonically, since Av (¢( a)) is not equal to dAv(a) + 0(1). To see why, note that Av(¢(a)) is large if a is close to a pole of ¢, while Av (a) is large if a is close to the point 00 E pl. (Here the word "close" means in terms of the v-adic chordal metric Pv.) Thus we can hope to find a canonical local height only if we allow an appropriate correction term, as in the following theorem.
Theorem 3.27. Let ¢ : pI -+ pI be a rational function of degree d ~ 2 defined over K, write ¢( z) = F (z) / G (z) using polynomials F, G E K [z] having no common factors, and let v be an absolute value on K. Then there is a unique function
with the following two properties: (a) For all a E pI (Kv) with a -=I- 00 and ¢(a) -=I- 00, the function ),.q;" v satisfies (3.16) (b) The function
a
f----+
),.q;"v(a) -logmax{lal v , I}
extends to a bounded continuous function on all of pI (Kv ) .
103
3.5. Local Canonical Heights Thefunction ~,v is called a local canonical height (associated) to ¢.
D
Proof We defer the proof until Section 5.9; see Theorem 5.60.
Remark 3.28. The local canonical height function ~,v constructed in Theorem 3.27 depends on the choice of a decomposition of ¢ as ¢ = F / G. Ifwe use instead cF/ eG for some c E K*, so G is replaced by eG, then it is easy to see that the new function differs from the old one by a constant, A
1
A
A,v,cc(a)
= A,v,c(a) + d _ 1 log [c],
In the sequel, when we refer to ~,v without further specification, we assume that some particular G has been fixed. However, we note that there are situations in which it may be convenient to normalize ~,v differently for different v; see Exercise 3.29. The utility of local canonical heights is that on the one hand, they are defined on jp'l (Kv) and have various nice metrical and analytic properties, while on the other hand, they fit together to give the global canonical height, as described in the next theorem.
Theorem 3.29. Let K be a numberfield, andfor each v E MK, let ~,v be the local canonical heightfunction constructed in Theorem 3.27. Then
h(a)
=
L
nv~,v(O:)
for all a E jp'l(K)
<,
{oo}.
(3.17)
vEMK
Proof We defer the proof until Section 5.9; see Theorem 5.61.
D
Remark 3.30. If ¢(z) E K[z] is a polynomial, then the local height may be computed as the limit (Exercise 3.24) A
A,v(a)
1
= nl~~ dn
logmax{l¢n(a)lv, I}.
If v is a nonarchimedean absolute value and if ¢ has good reduction at v, or more precisely, if ¢ = F / G with IRes( F, G) Iv = 1, then the local height is given by the simple formula (Exercise 3.30)
~,v(a) = log max{ lal v, I}. In general there is no simple limit formula to compute ~, v (a). However, there is an algorithm that leads to a rapidly convergent series for ~,v(a). Roughly, the Nth partial sum of the series approximates ~,v(a) to within O(d- N ) . For details see Exercise 5.29, which describes an efficient algorithm to compute the Green function 9, and then use Theorem 5.60, which says that the local canonical height and the Green function are related by the formula
104
3. Dynamics over Global Fields
Remark 3.31. Baker and Rumely [26, Chapter 7] give an interesting interpretation of the local canonical height function. They construct an invariant measure on Berkovich space and prove that the measure is the negative Laplacian of a suitable extension of the local canonical height function ).1>, ti- See Section 5.10 for a brief introduction to the geometry/topology of Berkovich space. For the more elaborate machinery required to do harmonic and functional analysis on Berkovich space, including the construction of the invariant measure, we refer the reader to [26, 29] and to the other references listed in Remark 5.77. Remark 3.32. The theory oflocal canonical heights began with Neron's construction on abelian varieties [333], or see [256, Chapter 11]. The general theory of global and local canonical heights associated to morphisms on varieties with eigendivisor classes is described in [88].
3.6 Diophantine Approximation The theory of Diophantine approximation seeks to answer the question of how closely one can approximate irrational numbers by rational numbers. The subject now includes a large and well-developed body of knowledge, while at the same time there is considerable ongoing research on many deep questions. In this section we state a famous theorem on Diophantine approximation and show how it is applied to deduce finiteness results for certain Diophantine equations. In Sections 3.7 and 3.8 we apply the theory of Diophantine approximation in a similar fashion to deduce integrality properties of points in orbits 01> (a). It is clear that any irrational number a E ~ can be approximated arbitrarily closely by rational numbers, since Ql is dense in R The following elementary result quantifies this observation by relating the closeness of x / y and a to the arithmetic complexity of the rational number x/yo We leave the proof as an exercise (Exercise 3.31), or see any elementary number theory text that discusses Diophantine approximation. Proposition 3.33. (Dirichlet) Let a E ~ <, Ql be an irrational number. Then there are infinitely many rational numbers x / y satisfying
Dirichlet's theorem says that every irrational number can be fairly well approximated by rational numbers. Some irrational numbers can be much better approximated, but it turns out that this is not true for algebraic numbers. A succession of mathematicians (Liouville, Thue, Siegel, Gel'fond, Dyson) derived ever better estimates for the approximability of algebraic numbers by rational numbers, culminating in the following deep theorem of Roth, for which he received the Fields Medal in 1958. Theorem 3.34. (Roth) Fix E > 0 and let a E Q be an algebraic number with a ~ Ql. Then there exists a constant", = "'(E, a) > 0 such that
3.6. DiophantineApproximation
105
x
for all- E Q. Y Proof The proof of Roth's theorem is beyond the scope of this book. A nice exposition may be found in [393]. For more general versions, see [205, Part D] and [256, 0 Chapter 7]. We apply Roth's theorem to prove an important result of Thue on the representability of integers by binary forms.
Theorem 3.35. (Thue) Let G (X , Y) E Z [X, Y] be a homogeneous polynomial of degree d and let B E Z. Assume that G(X , Y) has at least three distinct roots in pl (C). Then the equation (3.18) G( X , Y) = B has only finitely many integer solutions (X , Y) E Z2. Proof We begin by factoring G(X , Y) into irreducible factors in Q[X, Y], say
Replacing G (X , Y) by cG (X , Y) and B by cB for a sufficiently large integer c, we may assume without loss of generality that G l , . . . , G; have integer coefficients.' If r 2 2, that is, if G (X , Y) has more than one irreducible factor, then any solution (a, b) to (3.18) has the property that G 1 (a, b) and G 2 (a, b) divide B. Since B has only finitely many distinct factors, we are reduced in this case to showing that there are only finitely many solutions to the pair of simultaneous equations
Gl (X, Y) = B 1
and
This is clear, since the plane curves G 1 = B 1 and G 2 = B 2 have no common components , so they intersect in only finitely many points. Next suppose that r = 1, so G (X , Y) = G l (X , y )e1 • If B is not equal to B~ l for some integer Bi , then G(X , Y) = B has no solutions . Otherwise , we can take roots of both sides and reduce to the case that G (X , Y ) is irreducible in Q[X, Y]. Then G (X , Y ) has only simple roots, and by assumption there are at least three such roots. We factor G(X, Y) in qx, Y], say
with distinct algebraic numbers al , ... , a d. Now divide the equation G (X , Y ) = b by y d to obtain
lOr we can invoke Gauss 's lemma, which implies that the factorization of G(X, Y) in iQ[X, Y] can already be achieved in Z[X, Y].
106
3. Dynamics over Global Fields
We observe that if (x, y) E 'l} is a large solution to G(X, Y) = b, then the righthand side of (3.19) is small, so at least one of the factors on the lefthand side must also be small. However, since a1, ... , ad are distinct, the rational number xly can be close to at most one of aI, ... , ad. Hence we can find a constant C = C(G, b) such that min
l~i~d
I~Y - a·1 < -.£ Iyld t
for all (x, y) E '1. 2 satisfying G(x, y)
= b.
(3.20)
-
In the other direction, Roth's Theorem 3.34 applied to each of a1, ... , ad tells us that for any f > 0 there is a constant", = "'( f, aI, ... ,ad) > 0 such that
. Ix-Y -
mm
l~i~d
a· t
x
-'" I >lyl2+E
forall - E Q.
(3.21)
Y
-
Combining (3.20) and (3.21), we find that
lyld-2-E
< CI n,
By assumption, d 2:: 3, so this shows that there are only finitely many possible values for y. Finally, we observe that for each y, the equation G(x, y) = b implies that there are at most d possible values for x. Therefore the equation G(x, y) = b has only 0 finitely many integer solutions Siegel observed that Thue's theorem can be reformulated in terms of integral values of rational functions. Theorem 3.36. (Siegel) Let ¢(z) E Q(z) be a rational/unction with at least three distinct poles in lP'1 (C). Then
{o: E Q : ¢(o:) E'1.} is a finite set. Remark 3.37. Note that Theorem 3.36 need not be true if the rational function ¢ has fewer than three poles. A simple example with two poles is the function
F(z)
¢(z) = (Z2 _
ov:
where D > 1 is a squarefree integer and F (z) E '1. [z] is a polynomial of degree 2d. If we now take any solution (u, v) E '1. 2 to the Pell equation u 2 - Dv 2 = 1, then ¢( ulv) = v 2d F( ulv) E '1.. The Pell equation has infinitely many solutions, so there are infinitely many rational numbers u I v E Q with ¢( u I v) E '1.. Proof Write
¢ = [F(X, Y), G(X, Y)] using homogeneous polynomials F(X, Y), G(X, Y) E '1. [X, Y] of degree d having no common factors. For any fraction a = alb E Q written in lowest terms, we have
3.6. Diophantine Approximation
107
F (a, b) ¢(a ) = G(a, b)' so ¢(a) E Z if and only ifG (a, b) divides F (a, b). Let R = Res(F, G ) be the resultant of F and G, which is a nonzero integer since F and G have no common factors. Proposition 2.13 says that there are homogeneous polynomials Ii,gl , 12 ,g2 E Z[X, Y] satisfying
h (X, Y )F(X, Y) + gl (X ,Y )G(X , Y) = 12 (X ,Y )F (X , Y) + g2( X ,Y )G(X , Y) =
RX 2d- l , Ry 2d- l .
Substituting (X, Y ) = (a,b) into these equations, we see that if G(a, b) divides F (a,b), then G(a, b) also divides both Ra2d- 1 and Rb2d- 1 . However, a and bare relatively prime, so we have proven that
¢(a/b) E Z
implies that
G(a, b) divides R.
It is important to emphasize that the resultant R depends only on F and G and that it is nonzero. Hence
{~
E Q :¢
( ~)
E
Z}c U {~ E Q : G(a, b) = D} .
(3.22)
DI R
Thue's Theorem 3.35 tells us that each set on the righthand side of (3.22) is finite, which completes the proof of Theorem 3.36. 0
Remark 3.38. Roth 's Theorem 3.34 is not effective, in the sense that it does not provide a method for computing an allowable constant K(to , a) in terms of to and a. Thus our proofs of Thue's and Siegel 's theorems (Theorems 3.35 and 3.36) are also ineffective. Baker's theorem on linear forms in logarithms gives an effective, although weak, version of Roth's theorem , from which one can derive effective versions of Theorems 3.35 and 3.36. This means that our finitenes s result on integral points in orbits (Theorem 3.43) proven in the next section can be made effective, but the stronger integrality statement (Theorem 3.48) cannot, since it relies on the full strength of Roth's theorem. Let C C pN be a smooth projective curve defined over Q and let ¢ : C -+ pI be a nonconstant rational function on C. If C has genus 9 ~ 1, Siegel proved that C (Z) = {P E C(Q) : ¢(P) E Z} is a finite set. For 9 ~ 2, this is superseded by Faltings' theorem that C(Q) is finite, but for elliptic curves (g = 1), the set C(Q) is often infinite. In this case, Siegel greatly strengthened the qualitative statement that C(Z) is finite by showing that the coordinates of the points in C(Q) have numerators and denominators of approximately the same size. The precise theorem, which we generalize to the dynamical setting in Section 3.8, is as follows.
Theorem 3.39. (Siegel [403,404]) Let E / Q be an elliptic curve, let ¢ E Q( E) be a nonconstant rationalfunction on E , andfor each rational point P E E (Q ), write
3. Dynamics over Global Fields
108
¢(P)
= [a(P) ,b(P)] E jp'l(Q)
with a(P) ,b(P) E Z and gcd(a(P),b(P)) = 1.
Then
log la(P) I = l.
lim PEE (Q) h(¢ (P ) )-oo
loglb(P) I
Proof See [410, IX.3.3]. For a general version on curves of arbitrary genus 9 2:: 1,
see for example [205, D.9.4], although as noted above, if 9 2:: 2, then Faltings [164, 0 165] proves that C(Q) is finite. For the convenience of the reader, we conclude this section by stating general versions of Theorems 3.34, 3.35, and 3.36 for number fields. Proofs of these, and even more general, results may be found in [205, Part D] or [256, Chapter 7]. We set the following notation: K a number field; S a finite set of absolute values on K; Rs the ring of S-integers of K. Theorem 3.40. (Roth) Let E > O. For each v E S, extend v to tc in some fashion, and choose an algebraic number a v E tc. Then there is a constant /'l, > 0, depending on K. S. E, and {a v } v ES, such that
v
IImin{lz-a vlv,1}n 2:: H (: )2+f vES
forall zEK.
K
Theorem 3.41. (Thue-Mahler) Let G(X, Y ) E K [X , Y ] be homogeneous of degree d with at least three distinct roots in pi (iC), and let B E K . Then there are only finitely many (X, Y) E R~ satisfying G(X, Y ) = B. Theorem 3.42. (Siegel) Let ¢( z) E K (z ) be a rational function with at least three distinct poles in f<. Then there are only finitely many a E K satisfying 4>( a ) E R s.
3.7
Integral Points in Orbits
In this section we prove that the orbit of a rational point by a rational map contains only finitely many integers, except in those cases in which that statement is clearly false. For ease of exposition, we work with Q and Z, but the result holds quite generally for rings of S-integers in number fields; see [411] or Exercise 3.38. We begin with a definition. Definition. A wandering point for a rational map ¢ : jp'i -+ pi is a point P E jp'i whose forward orbit G¢(P) is an infiniteset. Thus every point is either a wandering point or a preperiodic point? 2The reader should be aware that in topological dynamics , especially with respect to invertible maps, the standard definition says that a point is wandering if it has a wandering neighborhood. In other words , a point P is topologically wandering if there is a neighborhood V of P and an integer n o such that rj>i(V ) n cf>J (V) = 0 for all i > j ~ no. Our definition coincide s with the topological definition if we use the discrete topology.
3.7. Integral Points in Orbits
109
Theorem 3.43. Let ¢( z ) E Q(z ) be a rational map ofdegree d ~ 2 with the property that ¢2(z) tf: Q[z]. Let Q E Q be a wandering pointfor ¢. Then the orbit
contains only finitely many integer points.
Theorem 3.43 is an immediate consequence of the following elementary geometric result combined with Siegel's Theorem 3.36 concerning integral values ofrational functions . Proposition 3.44. Let ¢ E q z ) be a rational fun ction of degree d ~ 2 satisfying ¢2(z) tf: C[z]. so Theorem 1.7 impli es that no iterate of ¢ is a polynomial map. Then
Ifd ~ 3. then #¢-3(00) ~ 3. Proof. We give a pictorial proof using the Riemann-Hurwitz formula . Suppose that ¢ is a rational map with # ¢ - 3(00) ::; 2. There are four possible pictures for the backward orbit of 00, as illustrated in Figure 3.1.
• - - Q - - p - - 00
• --........ •
.>
·-··---
Q - - p - - oo
(A)
(B)
Q--........
Q/~
p - - 00
(C)
Q - - P--........
• - - Q/--
P/~
00
(D)
Figure 3.1: Backward orbits containing few points The weak form of the Riemann-Hurwitz formula (Corollary 1.3) says that
2d - 2 =
L
(d - # ¢- l (p )).
(3.23)
PE p !
We apply (3.23) to each of the four pictures in Figure 3.1. Before computing, we need to determine to what extent the points in the four pictures are distinct. For (A)
3. Dynamics over Global Fields
110
and (B), the three points P , Q, and 00 must be distinct, since P = 00 would mean that
2d - 2 2 (d - #
1) + (d - 1) + (d - 1)
= 3d -
3
in case (A),
1) + (d - 1) + (d - 2) = 3d - 4
in case (B),
l)+(d - 2) +(d - l) = 3d - 4 incase(C), 2) + (d - 1) + (d - 1) = 3d - 4 in case (D).
Thus case (A) yields d :S 1, a contradiction, while the other three cases give d :S 2. This proves that if d 2 3, then #
Proof of Theorem 3.43. Proposition 3.44 tells us that #
could even take
It clearly suffices to prove that N is finite. If n E N with n Q
Q
2 4, then
so we see that
Example 3.45. In order to create a rational function whose orbit contains a large number of integral points, we simply take a finite (reasonably random) sequence of integers Zo, Z2 , ... ,Zk-l and treat the equations
3.7. Integral Points in Orbits -I,n+ 1 (
'f'
Zn
)
111
= Zn+1
for n = 0, 1, .. . , k - 1
as a system of homogeneous linear equations for the 2d + 2 coefficients of ¢. If the degree of ¢ satisfies 2d + 1 2: k , then there should be a nontrivial rational solution. We carry out this procedure with d = 2 to find the rational function
2 ¢(Z) = 899x - 2002x 33x 2
+ 275
584x + 275
-
such that the orbit of 0 contains quite a few integer points:
o~
1 ~ 3 ~ -2
..«: 5 ~
-7
Of course, as Theorem 3.43 predicts, the list of integer points must end, and indeed the next few iterates make it seem likely that there are no further integral points in the orbit ofO. Thus -
7
¢
---t
2917 ¢ 299 - - - t
296398306 10198813
17011876043966969359 938619769242091763
Example 3.45 (see also Exercise 3.41) shows how to construct orbits with many integer points by using rational maps of large degree. The following trick, adapted from work ofChowla [103] and Mahler [285] on elliptic curves (see also [405]), says that 0 ¢ (Q) n Z may be arbitrarily large even for rational maps of a fixed degree. Proposition 3.46. For all integers N 2: 0 and d 2: 2 there exists a rational map ¢(z) E Q( z) with the follo wing properties:
• ¢2(Z) tt iC[z]. • 0 is a wandering pointfor ¢.
• 0, ¢(O),¢ 2(0), ¢ 3(0) , . . . , ¢ N (0) E Z. Proof Let 'l/J( z) E Q( z ) be any rational map of degree d for which 0 is not a pre-
periodic point. For each 0
~
n
~
N, write
as a fraction in lowest terms, and let B = bob1 ... bN. Consider the rational function ¢(z ) = B'l/J(z/ B). Clearly ¢n(z) = B'l/Jn(z/ B), so for 0 ~ n ~ N we have
Hence ¢n(o) E Z for all 0
~ n ~
N.
o
Although Proposition 3.46 shows how to make 0 ¢ (Q) nZ arbitrarily large, it is in some sense a cheat. What is really happening is that we are clearing the denominators of rational points in an orbit. We can use the following notion of minimal resultant to rule out this behavior.
112
3. Dynamics over Global Fields
Definition. For a rational map ¢(z) E Ql(z), write ¢(z) as ¢(z) = F¢(z)/G¢(z), where F¢,G¢ E Z[z] have integer coefficients and the greatest common divisor of all of their coefficients is 1. Then F¢ and G¢ are uniquely determined by ¢ up to multiplication by ±l, and we define the resultant of ¢ to be the quantity
Res(¢) = IRes(F¢,G¢)I. (See Section 2.4 for the definition and basic properties of the resultant of two polynomials. See also Section 4.11 for a general discussion of minimal resultants of rational maps over number fields.) We can use the resultant to rule out the denominator-clearing trick of Proposition 3.46, and having done so, we conjecture that the number of integer points in an orbit is bounded solely in terms of the degree of the map. (See Theorem 6.70 for a special case.) This is a dynamical analogue of a conjecture of Lang [254, page 140]; see also [202, 407]. Conjecture 3.47. Let ¢(z) E Ql(z) be a rational map ofdegree d 2: 2 with ¢2(z) ~ Ql[z], and let a E lP'1 (Ql) be a wandering point for ¢. Assume further that ¢ is affine minimal in the sense that
Res(¢) =
min fEPGL 2 (iQI) f(z)=az+b
Res(¢f).
(In other words, we cannot make the resultant smaller by conjugating by an affine linear transformation az + b.) Then there is a constant C = C (d) depending only on the degree of ¢ such that
#(O¢(a) n Z) 5:
c.
3.8 Integrality Estimates for Points in Orbits Theorem 3.43 says that, except for the obvious counterexamples, an orbit G¢(a) contains only finitely many integer points. In this section we prove a dynamical analogue of Siegel's Theorem 3.39, which asserts that the numerators and denominators of certain rational numbers have approximately the same number of digits. Siegel's proof of Theorem 3.39 uses the existence of the multiplication-by-m maps [m] : E --t E on an elliptic curve. These finite unramified maps have the effect of significantly increasing the height of points while leaving distances relatively unchanged. We adapt Siegel's argument to the dynamical setting, but since our maps are on lP'1, they are always ramified. This causes some additional complications, since distances shrink significantly near ramification points. Theorem 3.48. (Silverman [411]) Let ¢(z) E Ql( z) be a rational map with the property that ¢2(Z) rt Ql[z] and Ij¢2(1/z) ~ Ql[z]. Let a E Ql be a wandering point for ¢, and write
3.8. Integrality Estimates for Points in Orbits
113
as a fraction in lowest terms. Then
lim log lanl = 1.
(3.25)
n--->oo log Ibnl
Remark 3.49. It is clear why we must assume that ¢2(z) ~ Q[z] in the statement of Theorem 3.48, but it may be less clear why we also require that 1j¢2(1/z) ~ Q[z]. Letting j(z) = 1/ z, we observe that
1j¢k(1/z) =
U- 1 0 ¢k 0
f)(z) =
U- 1 0 ¢ 0
f)k(z) = (¢f)k(z).
So if 1j¢2(1/z) E Z[z], then (¢f)2n(z) E Z[z] for all orbit of a = l/b for an integer b. We have
ti
2': 1. Now consider the
The quantity (¢f)2n(b) is an integer, so a2n = 1 and b2n = (¢f)2n(b) E Z. Hence the limit in (3.25) for even values of n is o. Thus the assertion of Theorem 3.48 is not true for rational maps ¢( z) such that 1/ ¢2 (1/ z) is a polynomial. Example 3.50. To illustrate Theorem 3.48, we take ¢(z) list the first few values of ¢n(1) = an/bn in Table 3.1. n
an
= z + l/z and a =
bn
1 1 0 1 2 1 2 5 2 3 29 10 4 941 290 5 969581 272890 1014556267661 264588959090 6 7 1099331737522548368039021 268440386798659418988490 Table 3.1: Orbit 0 1 (¢) for ¢(z)
1 and
log(a n ) log(bn ) -
0.69315 1.60943 1.46239 1.20759 1.10128 1.05110 1.02613
= z + 1/ z, writing ¢n(l) = an/bn.
Notice how both the numerator and denominator of ¢n(1) grow extremely rapidly. Of course, the elementary height estimate (Theorem 3.11) tells us that the maximum of lanl and Ibnl grows this rapidly, but the fact that they both grow at approximately an equal rate lies much deeper and is the content of Theorem 3.48. And to illustrate the speed with which the fractions grow, even for a very simple map of degree 2 such as ¢(z) = z + liz, here is the exact value of ¢9(1), which is the last value that fits on one line using very small (5 point) type:
114
3. Dynamics over GlobalFields
1726999038066943724857508638586386504281539279376091 034086485112150 121338989977841573308941492781 37790870974605039248107160960958052 743612256926142413111204802346 7330784 739529329885668846964890 •
This ninth iterate has logarithmic ratio log(a n )
log (bn )
=
10g(17269990380 ... 492781) ( ) ~ 1.00690. log 37790870974 ... 964890
Proof The idea underlying the proof of Theorem 3.48 is fairly simple. Choose some f > 0, and suppose that
lanl ;::: Ibnl H
€
for infinitely many n ;::: O.
(3.26)
This means that ¢n(a) = an/b n is very large, so ¢n(a) is close to 00. It follows that a is quite close to one of the points in the inverse image ¢ -n ( 00), say a is close to (3 E ¢ -n (00). But a is in Q, so one can hope that if n is sufficiently large, then a and (3 are so close to one another that they contradict Roth's Theorem 3.34. Unfortunately, this naive approach does not work, because the point (3 depends on n, so the constant in Roth's theorem changes with each new value of n. A more sophisticated idea is to use a fixed (large) integer m and apply Roth's theorem to the rational point ¢n-m (a) and a nearby point in ¢ -m (00). Note that
so the height of ¢n-m (a) is much smaller than the height of ¢n(a). We fix an integer m satisfying dm > 6/f (we will see later why this is a good choice for m) and we let (3 be the point in ¢ -m ( (0) that is closest to ¢n-m (a). How close is (3 to a? It turns out that this depends on the ramification index of ¢ at various points. Ifwe make the (incorrect) assumption that ¢ is everywhere unramified, then ¢ preserves distances up to a scaling factor, which allows us to make the following estimates, where the constants C l , C 2 , ... may depend on ¢, a, and m, but they do not depend on n: n l ~ ; : : I b 1= l¢n(a)llanl an
ClP(¢n(a), (0), definition of P, since I¢n(a) I > 1, = ClP(¢n(a), ¢m((3)), since ¢m ((3) = 00, ~ C2P(¢n-m(a),(3), assuming ¢ is unramified, ~ C31¢n-m(a) - (31, definition of p, ~
>
C4
- H(¢n-m(a»3'
Cs
~ H(¢n(a))3/d""
Cs
Roth's Theorem 3.34 (with exponent 3), property of heights (Theorem 3.11),
3.8. Integrality Estimates for Points in Orbits
115
since m satisfies d"
> 6/E.
Takinglogarithms,we have proventhat log lanl
::;
2 -log(C5 )
for all n satisfying(3.26), i.e., lanl 2:
E
Ibnll+€.
We reiterate that the constant C 5 depends only on ¢, a, and m; it does not depend on n. Hence if n satisfies (3.26), then an, and also b«, are bounded. It follows that the ¢-orbit of a contains only finitely many points satisfying(3.26), and therefore . log lanl hmsuP I Ib I ::;l+E. n--->oo og n
(3.27)
Repeating the argument with the rational map 1j¢( 1/ z) and the initial point 1/00 yields . log Ibnl lim sup I I I::; 1 + E. (3.28) n--->oo og an Since (3.27) and (3.28) hold for all E > 0, we conclude that the limit in (3.25) is 1. This would complete the proof of Theorem 3.48 except for the unfortunate fact that rational maps 1P'1 ---. 1P'1 are always ramified. Further, our proof sketch made no mention of the assumption that ¢2 (z) is not a polynomial, and the theorem is false for polynomials! In order to fix the proof, we begin by studying how ramification affects the distance between points.
Lemma 3.51. Let ¢ : 1P'1(C) ---. 1P'1(C) be a rational map oj degree d 2: 2, let p : 1P'1 (C) X 1P'1 (C) ---. IR be the chordal metric as defined in Section 1.1, and for Q E 1P'1 (C), let (3.29) eQ (¢) = max eQ' ( ¢) Q'E-l(Q)
be the maximum ofthe ramification indices ojthe points in the inverse image ojQ. Then there is a constant C = C( ¢, Q), depending on ¢ and Q, such that
min
Q'E-l(Q)
p(P, Q')e Q (
::;
Cp(¢(P), Q)
Jorall P E 1P'1(C).
(3.30)
In other words, if ¢(P) is close to Q, then there is a point in the inverse image oJQ that is close to P, but ramification affects how close.
Proof We dehomogenize using a parameter z such that Q -I 00 and 00 ~ ¢-l(Q). This means that we can write Q = (3 and P = a and that we are looking for the (3' E ¢-1((3) that is closest to oo. Writing¢(z) = F(z)/G(z) as a ratio of polynomials, the set ¢-1((3) is precisely the set of roots of the polynomial F (z) - (3G (z). If we factorthis polynomialover C as
3. Dynamics over Global Fields
116
with distinct 131, ... ,13r, then 13i E ¢ -1 (13) has ramification index ei. For notational convenience, we write e = eQ(¢) = maxei.
We may assume that ¢(P) is quite close to Q, i.e., that p(¢(P), Q) is small, since otherwise, the fact that the chordal metric satisfies p ~ 1 lets us choose a C for which the inequality (3.30) is true. In particular, P =I 00, and writing a = z(P), we may assume that a is quite close to at least one of the points 13k in ¢-1(13). For example, we may require that a satisfy and This implies in particular that for all i
=I k.
Hence
IF(a) - 13G(a) I = Iblla - 1311 e 11a - 132I e2 .. ·Ia - 13rl er =
Iblla - 13kl e k II la- 13il e i i#
: : : Iblla - 13kl e k II ~113k - 13il e i if'k
=
C 1 1a - 13kl
ek
,
where the constant C 1 is positive and depends only on ¢ and 13. Further, the exponent satisfies ek ~ e, so we obtain the estimate
The fact that ¢(a) = F (a) / G(a) is close to 13 and the assumption that G(13) implies that G(a) is bounded away from 0, so dividing by G(a) yields
=I 0
This in turn implies the same estimate for the chordal metric, since we can estimate 1¢(a)1 using 1131 and we can estimate lal using l13kl. Therefore
o Next we show that if we stay away from totally ramified periodic points, then iteration of ¢ tends to spread out the ramification.
3.8. Integrality Estimates for Points in Orbits
117
Lemma 3.52. Let 1J : p I - pI be a rationalmap ofdegree at least 2 and let Q E pI be a point such that Q is not a totally ramifiedfixed point of 1J2. Then
lim eQ(1Jm) = O. (deg 1J) m
m -+ oo
(See Lemma 3.51for the definition of eQ(1J).) Proof The proof of this geometric result uses the Riemann-Hurwitz Theorem 1.1 in a manner similar to the way that we have used it in the past. Let d = deg(1J), fix an integer m, and let P E 1J - m ( Q). We use the fact that the ramification index is multiplicative,
To ease notation, we write We consider several cases. Case I. The points P, cP(P), cP2(P) , ... , cP=-l(p) are distinct. Note that this covers the case that Q is a wandering point. The Riemann-Hurwitz formula allows us to estimate
ep(1Jm) = eOeIe2 ' " em- I
'" < _ ( eo + el +m
+ em_ I ) m
= C eo - 1) + (el :'S
(2d~ 2 + 1) m
-
arithmetic-geometric inequality
1 ~+' " + (em- I - 1) + 1) m Riemann-Hurwitz formula.
(3.31)
This estimate is clearly much stronger than the stated result, since the righthand side of (3.31) has a finite limit as m - 00. Indeed, it is an elementary exercise to verify that (1 + t/m)m :'S e t is valid for all t 2: 0, so
ep(1Jm) :'S e2d -
2
for all m 2: 1 provided P,1J(P), ... ,1Jm - I (P) are distinct.
(N.B. ee is a ramification index, while the e in e2d -
2
is the number 2.71828 . .. !)
Case II. P is purely periodic of period k ~ 3, and m ~ k. The assumption that 1Jk(p ) = P implies that e, = e",i(p)(1J) depends only on i modulo k. Writing m = qk + r with 0 :'S T < k, we use the multiplicativity of ramification, applied to 1Jk, to estimate
118
3. Dynamics over Global Fields
Note that P, ¢(P), ... , ¢k-l(P) are distinct, so we can apply Case I to each of the ramification indices on the righthand side of (3.32). This yields
q ep(¢m)::; (2d;2 +l)k (2d;2
+lr
2d - 2 )m-r 2d- 2 ::; ( - k - + 1 e
applying (3.31) to ¢k and ¢r, using (1 + t/ry
::; e',
)m e2d- 2.
2d - 2 ::; ( - k - + 1 Finally, we observe that
~ (2d; 2+ 1) = 1- (1 - ~)
(1 _~) : ; ~
for all d
~
2 and k
~
3.
This proves that e p (¢m) ::; e2d- 2 (~d) m.
Case III. P is purely periodic of period k ~ 2, and m ~ k: We use the assumption that Q is not a totally ramified fixed point of ¢2 to deduce that at least one of e p (¢) and e¢(P) ( ¢) is strictly smaller than d - 1, so
ep(¢2) = ep(¢)e¢(p)(J) = eOel ::; d2 - d. Now using the fact that ¢2(p) = P, we see that em is even and equal to el if m is odd, so
ep(¢m) = eOele2··· em-l
(3.33)
= e¢",(p)(¢) is equal to eo ifm
multiplicativity of ramification index,
ifm is even, (e oe 1 )m/ 2 - { (eoel)(m-l)/2 eo ifm is odd,
(d2 - d)m/2 ifm is even, 2 ::; { (d - d)(m-l)/2d ifm is odd, =
(1-~) Lm/2J d":
Case IV. P is preperiodic. If P is periodic, we are already done from earlier cases. Suppose that P is not periodic and let j ~ 1 be the smallest integer such that ¢1 (P) is periodic. Let R = ¢1 (P) and let k be the exact period of R. Then P, ¢(P), ... , ¢1-1(p) are distinct and R = ¢1 (P) is periodic, so from Case I we have
ep(¢m) = ep(¢J) . e¢j(p) (¢m- j )
::;
e2d- 2eR(¢m- j )
::;
e2d- 2eR(¢m).
We also note that R is not a totally ramified fixed point of ¢2, since ¢-l (R) contains at least the two distinct points ¢k-l(R) and ¢1-1(p). (This is where we use the assumption that P is not itself periodic.) Hence we can apply Case II or III to eR(¢m), which gives the desired estimate for ep(¢m).
3.8. Integrality Estimates for Pointsin Orbits
119
In all four cases we have proven estimates for e p (qr) that imply that
This completes the proof of Lemma 3.52. More precisely, we proved that there are constants CI and C2, depending only on d and with C2 < 1, so that for all m 21. And if P is a wandering point, we have shown that e p ( qr) is bounded by a constant 0 depending only on the degree of ¢. We next prove an elementary lemma that relates the chordal metric p(x, y) to the Euclidean distance [z - yl.
Lemma3.53. Let p be the chordal metric on pI (C). Thenfor all x, y E C C pI (C), 1
p(x,y):S "2 p(y, oo)
p(x,y)
Proof Note that p(z, 00) = 1/ Jlzl2 have
p(x, y) =
2Ix-yl·
p(y 00)2
+ 1, so directly from the
Ix -yl Jlxl2+ IJlyl2 + 1 = Ix -
'2
definition of p we
Ylp(x, oo)p(y, 00).
Then the triangle inequality for p in the form p(x, 00) + p(x, y) 2 p(y, 00) (see (1.2) in Section 1.1) yields
p(x, y) 2
Ix -
yl{p(y, 00) - p(x, y) }p(y, 00).
Making the further assumption that p(x, y) equality.
:S
~p(y, 00), we obtain the desired in0
ProofofTheorem 3.48. We now have the tools needed to complete the proof of Theorem 3.48. Writing ¢n(a) = an/b n as usual, our initial goal is to prove that the set
is finite. We observe that for n E N(¢, a, f), the point ¢n(a) is close to 00, since
By assumption, ¢2(z) ~ qz], so 00 is not a totally ramified fixed point of ¢2. This allows us to apply Lemma 3.52, which gives us an integer mo = mo(d, f) such that eoo(¢m) :S dm for all m 2 mo. (3.35)
i
P
Having fixed an m 2 mo, we apply Lemma 3.51 to the map ¢m and the points ¢n-m(a) and Q = 00 to obtain a point f3n E ¢-m(oo) satisfying
=
120
3. Dynamics over Global Fields (3.36)
where C 1 = C1 (qr
, 00) depends
only on m and ¢. Hence if we define
then N( ¢, a, E) is the union of N( ¢, a, E, (3) for (3 E ¢-m( 00), so it suffices to prove that the set N( ¢, a, E, (3) is finite for each (3 E ¢-m (00). Note that if N( ¢, a, E, (3) is infinite, then (3.34) and (3.36) imply that lim
nEN(,p,a,E,(3)
p(¢n-m(a), (3) = O.
(3.37)
n-oo
In other words, ¢n-m(a) approaches (3 in jp'1(C) as n --t 00. We consider first the case that (3 i- 00. The limit (3.37) tells us that
p(¢n-m(a), (3) :S
~p((3, 00)
for all but finitely many n E N(¢, a, E, (3). (3.38)
Note that the inequality (3.38) allows us to apply Lemma 3.53 with x = ¢n-m(a) and y = (3, which yields
where as usual C 2 > 0 depends only on ¢ and m. We are now ready to repeat the calculation from page 114, but this time done rigorously to account for the fact that ¢ has ramification. As usual, all constants may depend on ¢ and m, but are independent of n:
1
lanl
E
from (3.34),
2: p(¢n(a),oo) n
2: C3P(¢n-m(a),(3t",W 2: C4 1¢n- m(a ) _ (3l
e oow n
> C5 - H(¢n-m(a))3e oo(,pm)
>
C6 - H( ¢n(a)) 3e oo(,pm)ld m
C6
>
- H(¢n(a))'/2
)
from (3.36),
)
from (3.39), Roth's Theorem 3.34 with exponent 3, from Theorem 3.11, which says that
h(P)
=
d-mh(¢m(p))
+ 0(1),
from (3.35),
C6 lanlE/2
ci
/ E is bounded for n E N(¢, a, E, (3), and the same is true Hence lanl :S of Ibnl since Ibnl H E :S lanl, so ¢n(a) takes on only finitely many values for
121
3.8. Integrality Estimates for Pointsin Orbits
n E N(¢, a, E, (3). But by assumption, a is a wandering point for ¢, so we conclude that N( ¢, a, E, (3) is finite in the case that (3 ¥- 00. Suppose now that (3 = 00. Then a similar, but elementary, argument not requiring Roth's theorem yields
1
from (3.34),
lanl' 2: p(¢n(a), 00)
2: C7P(¢n-m(a), (3t'x(
C ( =
7
)
m
from (3.36),
eoo(
Ja;_m + b;_m
definition of p,
- H(¢n-m(a)roo(
definition of height, where note that bn - m ¥- 0, since a is wandering and 00 is periodic,
>
from Theorem 3.11,
Cs
>
Cg
- H (¢n(a) roo (
m
Cg
>----;-;::-
from (3.35),
- H(¢n(a))'/6
Cg Ian \,/6 As above, we conclude that an and bn are bounded, and hence that N( ¢, a, E, 00) is finite. We have now proven that N(¢, a, E, (3) is finite for all (3 E ¢-m(oo), and hence that N( ¢, a, E) is finite. The finiteness of N( ¢, a, E) is equivalent to the statement that
log lanl .,-----,-,-----,< 1+E
for all but finitely many n
log Ibnl -
2:
o.
(3.40)
We now apply the same argument to the rational map 'ljJ(z) = 1j¢(1/z) and the point l/a. It is easily verified that 'ljJn(l/a) = bn/a n, so we obtain the complementary inequality for all but finitely many n Since (3.40) and (3.41) hold for all
E
> 0, it follows
2: 0.
(3.41 )
that
lim log lanl = 1 n--->oologlbnl ' which completes the proof of Theorem 3.48.
o
122
3. Dynamics over Global Fields
3.9 Periodic Points and Galois Groups In this section we study the Galois groups of the field extensions generated by periodic points of a rational map. Much of the theory is valid for rational maps defined over an arbitrary perfect field and even for nonperfect fields, as long as ¢ is separable and one replaces the algebraic closure tc of K with the separable closure KseP. For further material on this topic and the more general Galois theory of iterates, see [3, 179,220,222,310,311 ,345,346,347,425]. Let ¢ (z) E K (z) be a rational function of degree d 2 2 with coefficients in a perfect field K. The periodic points of ¢ have coordinates in the algebraic closure k of K , since they are solutions to equations of the form
More precisely, write ¢ = [F, G] using homogeneous polynomials F, G E K[X, Y]. Then ¢n = [Fn ,Gn] for homogeneous polynomials Fn , Gn E K[X, Y] of degree d" , and the periodic points of ¢ of period n are the solutions in pI (k) to the equation
o.
Y Fn(X, Y) - XG n(X ,Y ) =
Thus counted with multiplicity, there are exactly d" + 1 such points . In this section we study the field extensions of K generated by the coordinates of the periodic points. Recall that if P = [ao, . . . , aN] E pN (k ), then the field of definition of P over K is the field K (P ) obtained by choosing a nonzero a i and setting
K (P ) = K
(ao,a l
ai
a i
, .. . ,
aN ) . a i
It is easy to see that this field is independent of the choice of the index i. The Galois group Gal (k/ K ) acts on the points ofpN (k) in the natural way,
u(P ) = [u(ao), dad, ... , u(aN)], and it is not hard to verify (Exercise 3.47) that
K(P)
= Fixed field of {u
E Gal(k/ K)
: u(P)
= P}.
Recall that the set of n-periodic points of ¢ is the set
Some of the points in Per n (¢ ) may have period that is smaller than n . For example, Per n (¢ ) contains all of the fixed points of ¢ . This suggests that we look at a smaller set consisting of the primitive ti-periodic points,
We begin by verifying that both Per n (¢ ) and Per~* ( ¢) are Galois-invariant.
123
3.9. Periodic Pointsand Galois Groups
Proposition 3.54. Let ¢( z) E K (z) be a rational function of degree d. The set of n-periodic points Per., (¢) and the set ofprimitive ti-periodic points Per~* (¢) are Galois-invariant sets. Proof Let (J E Gal(K/ K). Then
¢((J(P)) = (J(¢(P))
for all points P E JlD 1 (K ),
since ¢ is a rational function with coefficients in K. Let PEPer n ( ¢ ). Then
so (J(P) E Pern(¢). This proves that Pern(¢) is Galois-invariant. Similarly, we see that
which proves that P is a primitive n-periodic point if and only if eJ( P) is a primitive 0 n-periodic point. Hence Per~* (¢) is also Galois-invariant. Proposition 3.54 tells us that the set of primitive n-periodic points generates a Galois extension of K. We denote this extension by
Kn,<jJ = K(P: P E Per~*(¢)). These fields are analogues of the fields generated by roots of unity (cyclotomic fields), so we call Kn,<jJ the nth dynatomicfieldfor ¢. We denote the Galois group of the nth dynatomic field by
Example 3.55. Consider the quadratic polynomial ¢>(z) = z2 + 1. The sets of primitive 2nd , 3rd , 4th , and 6th periodic points are given, respectively, by the roots of the polynomials * ¢2(Z) - Z 2 ¢>2(Z) = ¢(Z) _ Z = Z
~(~! ~
¢>;(z)
=
¢*(z) 4
= ¢4(Z) -
¢*(z)
=
6
: Z
¢2(Z) - Z
+ Z + 2,
= z6 + z5 + 4z4 + 3z3 + 7z2 + 4z + 5,
= Z12 + 6zlO + z9 + 18z8 + 4z7 + 33z6 + 8z5 + 40z4 + 9z3 + 30z2 + 6z + 13,
(¢6(z) - z)(¢(z) - Z)
(¢3(Z) - Z)(¢2(Z) - Z)
= Z54
_ Z53
+ 27z52 -
25z51 + ...
-
13750z
+ 45833.
As this simple example clearly shows, it is infeasible to study dynatomic fields via explicit equations except for very small values of n.
124
3. Dynamics over Global Fields
The Galois group Gn,
a (¢i (p )) = ¢i (a (P ))
for all a E Gn ,
Thus the action of a on points in the orbit of P is determined by its action on P. This suggests that we decompose the action of G n,t/> on Per~· (¢) into its action on the set of orbits and its action within each orbit. A nice way to visualize the action of G n,t/> on the points ofPer~· (¢) is to consider the following picture:
Pr - 1 ¢(Pr-d ¢2(Pr _d
An element of Gn,
Notice that 1r u is a permutation of the set {I , 2, ... ,r }, so we can think of it as an element of the permutation group Sr . This gives a map 1r : G n,t/> -+ Sr . Similarly, if we set i u = (iu (1) , i u (2) , . . . , iu(r )), then we obtain a map
We perform a computation to see how i and
(ar)(Pj
)
1r
interact with one another:
= a(r (Pj ) ) = a(¢iT(j) (P7rT(j))) = ¢iT(j )(a (P7rT(j)) ) = ¢iT(j ) (¢iU ( 7r T (j ) ) p 7rO(7r (j)) ) T
=
qi
T
(j )+i (7r (j)) (p(7r o
T
07r T
)(j) ) .
From this equation we obtain the two formulas and Thus the map
1r :
G n,
(3.42)
3.9. Periodic Points and Galois Groups
125
is not a homomorphism, since it is twisted by the permutation action of S; on (7Lln7Lr. This is an example of the following general construction.
Definition. Let Sand H be groups, and let A be an index set on which S acts. For simplicity, we assume that H is an abelian group and we write its group law additively. The group structure on H makes the collection of maps Map(A, H) into a group. Thus ifi 1 , i z E Map(A, H), then
There is a natural action of S on the group Map(A, H) via its action on A,
7r: Map(A, H)
-+
Map(A, H),
7r(i)(a) = i(7r(a)).
The wreath product of Hand S (relative to A) is the twisted product of the groups Map(A, H) and S for this action. In other words, as a set the wreath product consists of the collection of ordered pairs Wreath(H, S)
= Map(A, H) x S,
and its group law is defined by twisting the natural group law on the product, (i 1 ,7rd
* (i z, 7rz) =
(7rz(id +i z, 7r l 7r Z) '
Theorem 3.56. Let ¢ E K (z) be a rational function ofdegree d. Let 0 1 , ... , Or be the distinct ¢ orbits in Per~* (¢) and choose a point Pj E OJ in each orbit. For each a E G n , and each I :::; j :::; r, define integers 0 :::; ia(j) < n and I :::; 7r a (j) :::; r by the formula a(Pj ) = ¢irr(j) (P1r rr (j ) ' Let Wreath(7Lln7L, Sr) be the wreath product ofthe symmetric group S; and 7Lln7L relative to the natural action ofS; on the set {I, 2, ... , r}. Then the map
W: G n ,
-+
Wreath(7Lln7L,Sr),
is an injective homomorphism. Proof In order to show that W is a homomorphism, we need to verify that
Writing this out in terms of the (twisted) definition of the group law on the wreath product, we need to prove that
This is precisely the formula (3.42) proven above, which shows that W is a homomorphism.
126
3. Dynamics over Global Fields
Now suppose that W (a) = 1. This means that i a (j) 1 :::; j :::; r, so a(Pj) = Pj. Hence
= 0 and 7ra (j) = j for all
so a fixes every point in Per~* (¢). Therefore a fixes Kn,r/>, so a that W is injective.
= 1, which proves D
The above discussion shows that the Galois group Gn,r/> of the dynatomic extension Kn,r/>/K roughly splits up into two pieces, a permutation piece determined by a permutation action on the orbits, and a cyclic piece determined by the action within each orbit. The permutation piece can be quite complicated, but one might hope that the dynamics of ¢ can be used to study the cyclic piece. We now try to make these vague remarks more precise. Let
G~,r/>
= Ker(7r: Gn,r/> ---. Sr) = {a
E Gn,r/>: a(Oj)
= OJ for all 1 :::; j:::; r},
and let
K~,r/> = fixed field of G~,r/>' Then
More precisely, if we fix a particular primitive n-periodic point P E Per~*(¢), then the extension K~,q'> (P) / K~,q'> is a cyclic extension of order dividing n whose Galois group is determined by the homomorphism
i : Gal(K~,r/>(P)/ K~,q'»
'-t
Z/nZ,
satisfying
a(P) = ¢i a (P).
(3.43)
This formula gives us some control over the Galois group of the relative abelian extension K~,r/> (P) / K~,r/>'
Remark 3.57. If L = K(a) is any Galois extension and if a E Gal(L/K) is any automorphism, then a(a) can always be expressed as a polynomial in a. Thus there is nothing special, a priori, in the fact that the action of Galois in (3.43) is given by a polynomial. However, ifwe fix ¢(z) of degree d and take n very large compared to d, then the extension K~,r/> (P) / K~,r/> is interesting because there is an automorphism a described by a polynomial (or rational function) ¢ of small degree compared to the degree of the extension.
3.10
Equidistribution and Preperiodic Points
There are many theorems and conjectures concerning the distribution of torsion points and points of small height on elliptic curves and abelian varieties. In this section we describe, without proof, some dynamical analogues. For further details, see Zhang's survey article [453] and the papers listed in its references.
3.10. Equidistribution and Preperiodic Points
127
We begin with the Manin-Mumford conjecture, which asserts that if X is an irreducible subvariety of an abelian variety A such that X n A tors is Zariski dense in X, then X is a translate by a torsion point of an abelian subvariety of A. The original Manin-Mumford conjecture was proven by Raynaud [367, 368]. Replacing torsion points by preperiodic points leads to a dynamical conjecture, where the dynamical analogue of a translate of an abelian subvariety of A is a preperiodic subvariety oflP'N as in the following definition.
Definition. Let ¢ : IP'N
----* IP'N be a morphism. A subvariety X C IP'N is a periodic variety for ¢ if there is an integer n ::::: 1 such that ¢n(x) = X. The subvariety is a preperiodic variety for ¢ if ¢m(x) is periodic for ¢ for some m ::::: O.
Conjecture 3.58. (Dynamical Manin-Mumford Conjecture) Let ¢ : IP'N a morphism defined over C and let X C IP'N be a subvariety. Then X
is Zariski dense in X
----*
IP'N be
n PrePer( ¢)
if and only if the subvariety X
is preperiodic for ¢.
The set of torsion points on an elliptic curve E, or more generally an abelian variety, is equidistributed with respect to the natural (Haar) measure on E(C). More precisely, we identify E(C) = C/ L as described in Section 1.6.3, and then for any open set U C C lying in a fundamental domain for E(C) we have
. #(E[n] n U) }~ #E[n]
=
Area(U).
A deeper equidistribution result of an arithmetic nature says that the Galois conjugates of torsion points are equidistributed. In order to state a dynamical analogue for morphisms ¢ : IP'N ----* IP'N, we need a 4>- invariant measure on IP'N (C) as described in the following proposition.
Proposition 3.59. Let ¢ : IP'N
----* IP'N be a morphism of degree d defined over C. There is a unique probability measure f-L¢ on IP'N (C) satisfying
and We call f-L¢ the canonical ¢-invariant probability measure on IP'N (C). Proof For a general construction that covers both archimedean and nonarchimedean base fields, see [453]. We also mention a standard result in dynamics (the KrylovBogolubov theorem [226, Theorem 4.1.1]), which says that any continuous map ¢ : X ----* X on a metrizable compact topological space X admits a Borel probability measure f-L¢ satisfying ¢* (f-L¢) = f-L¢, i.e., f-L¢ ( ¢ -1 (A)) = f-L¢ (A)
for every Borel-measurable subset A of X.
D
128
3. Dynamics over Global Fields
For the remainder of this section we fix an algebraic closure Q ofQ and an embedding Q '---+ Co So when we speak of a number field K and its algebraic closure K, we assume that they come with compatible embeddings into Co Definition. Let KjQ be a number field. For any algebraic point P E lP'N(K), let C(PjK) denote the set of Galois conjugates of P, i.e.,
C(PjK) = {a(P) E lP'N(K): a E Gal(KjK)}, and let
op denote the Dirac measure on lP'N (C) supported at P, s
(U)={l
°
p
ifPEU, if P ~ U.
We associate to P E lP'N (K) the discrete probability measure
1 Itp
= #C(PjK)
I:
oQ
QEC(PjK)
supported on the Galois conjugates of P. Definition. Let KjQ be a number field and let PI, P2 , P3 , ... E lP'N (K) be a sequence of points with algebraic coordinates. Fix a probability measure It on lP'N (C). We say that the sequence {Pd c- I is Galois equidistributed with respect to It if the sequence of measures J-tPi converges weakly' to J-t. We are now ready to state a dynamical equidistribution conjecture for Galois orbits of preperiodic points on lP'N, and more generally for points of small height. Conjecture 3.60. (Dynamical Galois Equidistribution Conjecture) Let KjQ be a numberfield, let ¢ : lP'N ---. lP'N be a morphism ofdegree d ~ 2 defined over K, and let PI, P2 , P3 , ... E lP'N (K) be a sequence of distinct points such that no infinite subsequence lies entirely within a preperiodic subvariety of lP'N. (a) If H, P2 , P3 , ... E PrePer( ¢), then the sequence {Pdi:2:1 is Galois equidistributed in lP'N (C) with respect to the canonical d-invariant probability measure J-t¢.
(b)
If lim.,., h¢(Pi) =
00 0, then the sequence {Pdi:2:1 is Galois equidistributed in lP'N (C) with respect to the canonical ¢-invariant probability measure It¢.
It is clear that Conjecture 3.60(b) implies Conjecture 3.60(a), since preperiodic points have canonical height equal to O. A version of the conjecture is known provided that the sequence of points satisfies a somewhat stronger Zariski density condition as in the following theorem. 3Recall that a sequence of measures J.Li on a compact space X converges weakly to J.L if for every Borel-measurable set U, the sequence of values J.Li(U) converges to J.L(U) as i -+ 00.
3.11. Ramification and Units in Dynatomic Fields
129
Theorem 3.61. (Yuan [450]) Let ¢ : JIDN ----+ JIDN be a morphism of degree d 2: 2 defined over K and let P1 , P2 , P3 , . .. E JIDN (K) be a sequence ofpoints satisfying the following two conditions: (a) Every infinite subsequence of {Pi }i>l is Zariski dense in JIDN. (b) hq,(Pi ) ----+ 0 as i ----+ 00. (In the terminology of [453], sequences with property (a) are called generic and sequences with property (b) are called small.) Then the sequence {Pdi>l is Galois equidistributed with respect to the canonical o-invariant probability measure !Lq, onJIDN(C).
Proof The proof is beyond the scope of this book. See Yuan [450] for a general version over archimedean and nonarchimedean base fields and algebraic dynamical systems on arbitrary projective varieties. Earlier results and generalizations are given by Autissier [15, 16], Baker-Ih [24], Baker-Rumely [28], ChambertLoir [98], Chambert-Loir-Thuillier [99], Favre-Rivera-Letelier [169] and SzpiroUllmo-Zhang [432]. 0 The classical Bogomolov conjecture, which says that sets of points of small height on abelian varieties lie on translates of abelian subvarieties, was proven by Ullmo [435] and Zhang [452]. We state a dynamical analogue. Conjecture 3.62. (Dynamical Bogomolov Conjecture) Let ¢ : JIDN ----+ JIDN be a morphism of degree d 2: 2 defined over a number field K and let X C JIDN be an irreducible subvariety that is not preperiodic. Then there is an f > 0 such that the set
{p
E
X(K): hq,(P) < f}
is not Zariski dense in X. Notice that Conjecture 3.62 implies Conjecture 3.58, since the set of points with h¢(P) < E includes all of the preperiodic points. Finally, in closing this section, we mention that canonical invariant measures have been constructed on Berkovich spaces; see Remark 5.77 and the references listed there.
3.11 Ramification and Units in Dynatomic Fields In Section 3.9 we used periodic points to construct field extensions Kq"n and studied their Galois groups. We now take up the question of the arithmetic properties of these algebraic number fields. In general, the three basic questions that one would like to answer about a given number field are these: Where is it ramified? What is its ideal class group? What are its units? In addition, one wants to know how the Galois group acts on ideal classes and on units. In this section we provide partial answers to the question of ramification and units for dynatomic fields. We first recall the classical case of cyclotomic extensions, which provide a model for the dynatomic theory. Let (is a primitive nth root of unity and a E Gal(Q/Ql). Then
130
3. Dynamics over Global Fields
IT(( ) =
(i(a)
fora unique jfo] E (7L./ n7L.)* .
This defines an isomorphism
j : Gal(Q (( )/ Q )
-----t
(Z/ nZ)*,
IT t------t
j (IT ),
expres sing the action of IT on Q(() as a polynomial action I/>(z) = zi . Now let p be a prime not dividing n , let p be a prime of Q (() lying above p, and let ITp E Gal(Q / Q ) be the corresponding Frobenius element. The definition of Frobenius says that
IT p ( () == C/
(mod p).
(3.44 )
However, the n th roots of unity remain distinct when reduced modulo p, so the congruence (3.44) implies an equality in Q((),
IT p ( () = (P. This exact determination of the action of Frobenius as a polynomial map on certain generating elements ofQ(() is of fundamental importance in the study of cyclotomic fields . To some extent, we can carry over the analysis of Q(() to dynatomic fields , although the final results are not as complete as in the classical case . Let p be a prime ideal of the ring of integers of K g,n' let p be the residue characteristic of p, and let q be the norm of p. We assume throughout that p f n. Choose a prime ideal liJ in K g,n(P) lying above p. Assuming that p does not ramify in Kg,n(P), the associated Frobenius element ITp is determined by the condition ITp(a)
== a q
(m od
liJ)
for all a in the ring of integers of Kg,n(P), On the other hand , by construction the action of ITp on P E Per~* (I/» is given by by formula
The next proposition allows us to characterize the primes at which the extension
K g,n(P)/ K g,n may be ramified. Proposition 3.63. Let K be a number field, let I/> E K (z) be a rational map of degree d 2: 2, let P E Per~* (I/> , K) be a point in pI (K) ofexact p eriod n, and let p be a prime ofK satisfying the following three conditions:
I/> has good reduction at p. p does not divide n . If A
P mod p has exact p eriod n . In particular, the set {p : P mo d p has p eriod strictly less than n }
is a finite set ofprimes of K.
(3.45) (3.46 ) (3.47)
131
3.11. Ramification and Units in Dynatomic Fields
Proof Let p be a prime of K satisfying (3.45), (3.46), and (3.47). Let m be the exact period of F and let r be the order of AJ,(F) in lF~. Theorem 2.21 tells us that either n = m or n = mr, since (3.46) rules out powers of p appearing in n. If Aq,(P) = 1, then also AJ,(F) = 1, so r = 1 and n = m. On the other hand, if Aq,(P) i= 1, then (3.47) tells us that
But AJ,(Ft
= i
by definition of r, so nlm cannot equal r. Hence n
=
m
III
all cases, which shows that F mod p has exact period n for all primes p satisfying (3.45), (3.46), and (3.47). This proves the first part of the proposition, and since each of the three conditions is satisfied for all but finitely many primes, the second
D
~~~R
Corollary 3.64. Let K be a number field, let c/>( z) E K (z) be a rational map of degree d 2: 2, and let Kn,q, be the nth dynatomic field for c/>. Let S be the set of primes p of K such that either c/> has bad reduction at p or p divides n or p divides the quantity
II
(Aq,(P) -1).
(3.48)
PEPer~*(q,)
Aq,(P)#l
Then Kn,q,1 K is unramified outside ofS. Proof The field extension Kn,q,1 K is generated by the points of Per~* (c/». Proposition 3.63 tells us that if p ~ S, then those points remain distinct when reduced modulo primes lying over p. Hence the extension Kn,q,1 K is unramified at p. D Example 3.65. We continue studying the quadratic polynomial c/>(z) = z2 + 1 from Example 3.55 (see also Example 2.37). The polynomial c/>(z) has everywhere good reduction, so if we assume for the moment that none of its periodic points have multiplier equal to 1, then the quantity (3.48) in Corollary 3.64 is equal to
II
(Aq,(OO) -1)
q,~(a)=O
=
II
((c/>n),(oo) - 1)
= Res(c/>~(z),
(c/>n),(z) - 1).
q,~(a)=O
Denoting this resultant by D. n (c/> ), it is not hard to compute the values of D. n (c/» for small values of n. We obtain
D.2(c/» = 72 , D.3(c/» = (33 . 11)3, D.4(c/» = (32'11 . 13·41)4, D.5(c/»
= (33.7.83.331.140869)5,
D.6(c/» = (34.5.7.23.73.223.2251. 347495839)6.
132
3. Dynamics over Global Fields
The field K rjJ,2 is generated by the roots of ¢2 (z) = z2 + z + 2, so we have explicitly KrjJ,2 = K( A). For higher values of n, we can check the primes dividing I::i. n (¢ ) by computing directly the discriminant of the polynomial ¢~(z), whose roots generate KrjJ,n/K. Thus for example, Disc(¢j) = _3 6 . 113 and Disc(¢4') = 34 .11 4 .13 3.41 4 . The local theory of units described in Section 2.7 allows us to construct units (or at least 8-units) in dynatomic fields Kn,rjJ and their composita. For convenience, we make two definitions.
Definition. Let 8 be a finite set of places of K and let L / K be a finite extension. We say that u E L is an 8-unit ifit is an 8 L -unit, where 8 L is the set of places of L lying over 8.
Definition. Let ¢(z) E K(z) be a rational map. We write 8rjJ for the set 8rjJ =
M'K U {primes at which ¢ has bad reduction}.
Thus ¢ has good reduction at all primes in the localized ring Rs. In particular, if
¢(z) =aozd+,.·+ad is a polynomial, then the finite primes in 8rjJ are the primes where some a; is nonintegral together with any primes for which ao is not a unit. Having set this notation, we now state globalized versions of the dynamical unit theorems from Section 2.7.
Theorem 3.66 (Global version of Theorem 2.33). Let ¢(z) E K[z] be a polynomial of degree d 2': 2, let 0: E Per~* (¢) with n 2': 2, and fix integers i and j satisfying gcd(i - j, n) = 1. Then ¢i(o:) - ¢J(0:) ¢(o:) - 0:
is an 8rjJ-unit in KrjJ,n' For the next result, we recall that the cross-ratio of four points Pi, P2 , P3 , P4 in lP'1 is the quantity
Theorem 3.67 (Global version of Theorem 2.34). Let ¢( z) E K (z) be a rational function ofdegree d 2': 2, let P E Per~* (¢), andfix integers i and j satisfying gcd(j, n)
Then is an SrjJ-unit in KrjJ,n'
= gcd(i - 1, n) = gcd(i - i, n) =
1.
133
3.11. Ramification and Units in Dynatomic Fields
Theorem 3.68 (Global version of Theorem 2.35). Let ¢(z ) E K (z) be a rational function ofdegree d ;::: 2, let m and n be integers with m 1n and n 1m, and let
P
= [x, y] E Per;; (¢)
Q = [x', y'] E Per~* ( ¢ ) .
and
Denote by S p the set ofplaces Sp
=
{v E M~ : v(x)
> 0 and v (y ) > o}
U {v E M~ : v (x )
< 0 or v (y ) < o} ,
and similarly for S Q. Then x y' - x' y is an (8 U Sp U SQ) -unit in the compositum K ,mK ,n.
Theorems 3.66, 3.67, and 3.68 allow us to construct many S-units in dynatomic fields K ,n and their composita K ,mK ,n. Further, since Gal(K,n / K ) frequently contains an element a whose action on points P E Per~* (¢) is characterized by the equation a(p) = ¢(P), we obtain a partial description of the action of the Galois group on the dynamical units. For example, the units in Theorem 3.67 transform via
We compare this to the construction of cyclotom ic units in cyclotomic fields. Let q be a prime power and let ( be a primitive qlh root of unity. Then the cyclotomic field Q( () contains the units for 2
< i ::; q; 1 with gcd(i , q).
The Galois group Gal (Q(() /Q) is the set of elements a t characterized by
at (()=(t
withO
The action of the Galois group on the cyclotomic units is given by the explicit formula
at
( _ 1 -- ~ (t - 1 . ( ~)
Further, the cyclotomic units generate a subgroup of finite index in the full group of units Z [(]*, and the index of this subgroup is related to the class number ofQ(() . The situation for dynatomic fields is not nearly as complete. One problem is that the dynatomic fields K ,n tend to have very large degree over K , so the dynamical unit theorems cannot produce enough units to give a subgroup of finite index in the full unit group . Further, the Galois group Gal( K ,n/ K ) is usuall y huge, and we have an explicit description only of the subgroup generated by the element a . However, since for general number fields there is no known way to systematically produce any units with any explicit Galois action, the dynatomic construction might be said to fall under the heading of "half a loaf is better than none."
134
3. Dynamics over Global Fields
Example 3.69. Consider the rational map
¢(z) = Z2 - 4. After some algebra, we find that
¢( Z) - Z = Z2 - Z - 4,
¢2(Z) - Z 2 ¢(Z)-Z =Z +z-3, ¢3(Z) - z = (Z3 - Z2 - 6z + 7)(Z3 + 2z2 - 3z - 5). ¢(z) - z
-'--:--:-'~-
It is not hard to check that ¢( z) - z divides ¢n(z) - z for all n 2: 1 (Exercise 1.19(a», but the further factorization of ¢3 (z) - z into a product of cubics is less common; see Exercise 3.49. Ifwe let a, (3 E C satisfy
a 3 - a 2 - 6a + 7 =
a
and
then we have Peri*(¢)
= {I ± 2vTI } ,
Peri*(¢)
Per;*(¢)
= { -1 ~
vTI},
= {a,¢(a),¢2(a),(3,¢((3),¢2((3)},
where we recall that Per~* (¢) denotes the set of points of exact period n for ¢. Let K = iQ(a). The polynomial z3 - z2 - 6z + 7 is irreducible over iQ, but it factors completely in K since its roots are a, ¢(a), and ¢2(a). Hence K is Galois over iQ with Galois group generated by the map (Y determined by (Y(a) = ¢(a). Notice that the discriminant of Z3 - z2 - 6z + 7 is 192 , again confirming that its roots generate a cyclic cubic Galois extension. Further, Z[a] must be the full ring of integers RK of K, since
(Note that K
=I- iQ, so Disc(KjiQ) > 1.) Then the fact that 19 is prime forces
R K =Z[aJ. We apply Theorem 3.66 with i = 2 and j = 1 to obtain the unit
It is easy to check that N K / IQ (uI) = 1, confirming that i = 2 and j = agives the unit
Ul
is a unit. Similarly, taking
Exercises
135
In this case, N K / Q (U2 ) = -1. The field K is totally real of degree 3, so its unit group R'K has rank 2. It is not hard to see that the two units Ul and U2 are independent, so they at least generate a subgroup of finite index. (In fact, {-I, Ul, U2} generates the full unit group R'K, but we leave the verificationof this fact to the reader.) We can compute the action of the Galois group on Ul and U2,
a(ud = a(0:2 + a(u2) = a(0:2 +
0: 0: -
4) = ¢(0:)2 + ¢(o:) - 4 = 3) = ¢(0:)2 + ¢(o:) - 3 =
0: 0:
4 4
-
70: 2 + 8 = 70: 2 + 9 =
-0:
+ 1,
-0:
+ 2.
These new units are related to the original units by and Using points of period 2 and 3 for ¢, we can create units in larger fields. Thus let L = K( v'13). Then L contains both 0: and the points in Per~* (¢), so Theorem 3.68 with nl = 2 and n2 = 3 says that U3
with
0:
-1 + v'13 ,= ---2
is a unit in the ring of integers of L. If we take the norm from L down to K, we find that N L / K (U3) = U2 is one of the units that we already discovered. Similarly, NL/K('y - ¢(o:)) = -0: + 2. If instead we compute the norm of U3 from L down to Q('y), we find using + , _ 3 = 0 that
,2
3
2
NL/Qhl('y-o:)=, - , -6,+7=-,+1=
3-v'13 2 .
This unit and -1 generate the unit group of the ring of integers of Q( V13). For additional information about cyclic cubic extensions generated by periodic points of polynomials, see [306], and for a general analysis of units generated by 3-periodic points of quadratic polynomials, see [313, Section 8].
Exercises Section 3.1. Height Functions
3.1. Show that theconstant C(d, N, D) inConjecture 3.15 must depend oneach ofthe quan-
tities d, N, and D bygiving a counterexample if any one of them is dropped. 3.2. Let
v(B) = #{P E lP'N (Q) : H(P)
:s: B}.
(a) Find positive constants ci and C2 such that c1B N + 1 < v(B) < c2B N + 1
for all B
~
1.
136
Exercises
(b) For N
= 1, prove that lim v(B) B~oo
B2
=
12. 71'2
(c) More generally, prove that the limit limB~oo v(B)/ B N + I exists and express it in terms of a value ofthe Riemann (-function. 3.3. Prove that
#{ P E jp'N (Q) : H(P)
< Band D(P) < D} ::; (12D)N 2ND B N D(D+I). 2
Aside from the constant, to what extent can you improve this estimate? In particular, can the exponent of B be improved? 3.4. Let F(X) = aoX d + alX d- 1 factor F(X) as
F(X)
+ ... + ad E
= ao(X
Q[X] be a polynomial of degree d, and
- QI)(X - (2) ... (X - Qd)
over the complex numbers. Prove that
T dH(QI)'" H(Qd) ::; H([ao, al, ... , ad]) ::; 2dH(QI)'" H(Qd)' (Hint. Mimic the proof of Theorem 3.7 for the upper bound. To prove the lower bound, for each v pull out the root with largest IQilv and use induction on the degree of F.) Can you increase the 2- d and/or decrease the 2d ? 3.5. Prove that the number of (N io, ... .i»
+ l l-tuples (io, ... ,iN) E ZN+I of integers satisfying and io + ... + iN = d :2: 0
is given by the combinatorial symbol (N;td). Note that this is equal to the number of monomials of degree d in the N + 1 variables Xo, ... , XN. Section 3.2. Height Functions and Geometry
3.6. Let ¢ : jp'2 --; jp'2 be the rational map ¢(X, Y, Z) = [X 2 , y 2 , X Z]. Although ¢ is not defined at [0,0,1], we can define Per(¢, jp'2) to be the set of points satisfying ¢n(p) = P for some n :2: 0 and ¢i (P) i- [0, 0, 1] for all a ::; i < n. Prove that Theorem 3.12 is false by showing that Per(¢, jp'2 (Q)) is infinite. What goes wrong with the proof? Try to find a "large" subset S ofjp'2 such that Per(¢, S(K)) is finite for every number field K. 3.7. Let K/Q be a number field, let P = [xo, . . . ,XN] E jp'N (K), and let b be the fractional ideal generated by Xo, ... , XN . Prove that
3.8. Let K/Q be a number field and let ¢(z) E K(z) be a rational map of degree d Recall that the height H (¢) is defined by writing ¢
:2:
2.
= [F(X, Y), G(X, Y)] = [aoXd + alX d- 1 + ... + adyd, boX d + ... + bdyd]
and setting H(¢) = H([ao, ... ,ad,bo, ... ,bdl). (See (3.4) on page 91.) Prove that there are positive constants CI(d) and C2 (d) such that forallP E jp'1(K). Find expressions for CI and C2 in terms of d. This gives an explicit version of Theorem 3.11 for jp'l.
Exercises
137
3.9. Let
=
3.10. Let d 2: 2 and let cPd(Z) Zd. Prove that there is an absolute con stant c such that for all number fields K / Q of degre e n we have
#PrePer(cPd, jp' I(K )) :::; e[K: Q ]log log ([K : Q J) . Prove that aside from the constant, this upper bound cannot be improved. (Hint. You will need the fact that the Euler totient function <.p satisfies <.p( m) :::; em j log log m ; see for example [11, Theorem 13.14].) In particular, the uniform boundedness conjecture (Conjecture 3.15) is true for cPd. This is one of the few maps for which the conj ecture is known . The other s are the Chebyshev polynomials and Lattes maps; see Theorem 6.65.
Section 3.4. Canonical Heights and Dynamical Systems 3.11. Let cP : jp'N ---> jp'N be a morphism defined over K , let automorphism oflP'N, and let cP f = I-I 0 cP 0 f. Prove that
I
E P GLN+l (K) be an
N for all P E lP' (K) .
Thus the canonical height is conjugation-invariant , i.e., it is independent of change of coordinates.
3.12. Let cP,7j; : IP'N ---> IP'N be morph isms of degree at least two that commute with one another, i.e., cP (7j; (P )) = 7j; (
for all P E IP' (Q) .
3.13. Let cP : ]P'N ---> ]P'N be a morphi sm of degree at least two defined over a number field K and let P E IP'N (k) have the property that
(')<jJ(U(P)) n (')<jJ(p):I= 0 for all rr E Gal(k jK). Prove that either P is preperiodic for cP or else there is some point in the orbit of P satisfying cPn (P) E IP'N (K). Is this result true if instead KjQp is a p-adic field? 3.14. In the setting of Theorem 3.20, suppose that the set S is a topological space and that the maps cP : S ---> Sand h : S ---> ~ arc cont inuous maps. Prove that h : S ---> ~ is also a continuous map . (Hint. Show that the functio~s h", (P) = d: " h (cP m(P)) for m = 1,2, 3, .. . are continuous and converge uniformly to h.) 3.15. Let h be the canonical height for cP(z) = Z2 + c. For any given e E Q , there is a minimum nonzero value for h<jJ (z) as z ranges over nonpreperiodic points in Q . Find that minimum value for: (a) e = O.
Exercises
138
(b) c = -2. (c)c=-1. (d) ** arbitrary c E Z. (e) ** arbitrary cEQ. (Hint. Try to do (a), (b), and (c) directly, but we note that it is easier to do them using the theory oflocal canonical heights for polynomials; see Exercises 3.24 and 3.28. For (d) and (e) the goal is to describe the minimum value of h¢ in tenus of c.) 3.16. Let ¢(z) = Z2 - 1 . Analyze the canonical height for points in the following sets. (a) 8¢( 00), the attracting basin of 00, where we recall (Exercise 2.1) that the attracting basin of 00 for a polynomial rjJ is the set
8¢(00) =
{a E C:
lim rjJn(a) = n~oo
oo} U [oo}.
(b) F¢" 8¢ (00). (Note that all ofthe points in this set are eventually attracted to the attracting 2-cycle 0
..!.-..
-1.)
(c) .:T¢. (See also Exercise 3.27.) 3.17. Let rjJ : IP N ---+ IP N be a morphism defined over a number field K, and for any point P E IP N (K), let D(P) = [K(P) : Q]. Prove that there is a constant C = C(¢) > 0 such that 11,
¢
(P)
> dC D1( P ) 2
for all P E IPN(K) with P
-
d 'F
PrePer(A..). 'f/
(Hint. Use the estimate for the number of points of bounded degree and height given in Exercise 3.3.) This is a very weak version of the dynamical Lehmer Conjecture 3.25. 3.18. ** Let cEQ with c # 0, -2 and let rjJ(z) = z2+ c. Prove that there exist K = K(C) and N = N(c) > 0 such that K
A
h¢(a) 2: [Q(a) : Q1N
>0
for all a E Q with a tf- PrePer( rjJ).
(This is not currently known for any value of c other than c = 0 and c = - 2.)
3.19. Let a E Q* and let
f(x)
= aox d + a1x d- 1 + ... + ad E Z[x]
be the minimal polynomial of a, normalized so that ao Factor f(x) over Cas f(x) = ao TI(x - ai). Prove that
H(a)d = ao
>
0 and gcd(ao, ... , ad)
1.
II max{ 1, lail} = 11 10glf(e27rio)IdB. d
i=l
0
The quantity H(a)d, or equivalently HQ(a)(a), is also known as the Mahler measure ofa and denoted by M(a). Lehmer's conjecture (Conjecture 3.24) can be stated in tenus of Mahler measure: There is an absolute constant K > 1 such that if a E Q* is not a root of unity, then
M(a) >
K.
Exercises
139
3.20. Let ¢(z) E Q(z). Write a program to estimate h¢(a) directly from the definition. Use your program to compute the following heights to a few decimal places. (See also Exercises 3.30 and 5.31.) (a) ¢(z)=z2-1anda=~. (b) ¢(z) = Z2 + 1 and a = ~. (c) ¢(z) = 3z 2 - 4anda = 1. 1 (d) ¢(z)=z+-anda=1.
(e) ¢(z) =
3z 2
z
Z2 _
3.21. Let ¢(z) =
1 1 and 00= 1.
Z2 -
Z
+ 1. The ¢-orbit of the point 2 is called Sylvester's sequence [428],
CJ¢(2) = {2,3, 7,43,1807,3263443,10650056950807, 113423713055421844361000443, ... }. (a) Prove that Sylvester's sequence satisfies ¢n+l(2) = 1 + ¢O(2)¢1(2)¢2(2)¢3(2) ... ¢n(2). (b) A rough approximation gives h¢(2) ~ 0.468696, so ¢n(2) is approximately equal to 2. e°.468696n Prove a more accurate statement by showing that
n = 0,1,2, ... , is positive, strictly increasing, and converges to ~. (Hint. First conjugate ¢(z) to put it into the form Z2 + c.) 2
(c) Deduce that there is a real number H such that ¢n(2) is the closest integer to H" for all n 2 O. Note that this is far stronger than the general height estimate
3.22. Let 8 > 0, let d 2 2 be an integer, and let (xnk:>o be a sequence of positive real numbers with the property that Xo
> 1+8
(a) Prove that the sequence (b) Prove that the limit
and Xn
is strictly increasing and that H
=
lim
x;jd
X n ---. 00
as n ---.
00.
n
n~oo
exists. (Hint. Take logs and use a telescoping sum to show that the sequence is Cauchy.) (c) Prove that n2 I 8 IH - X n ::; d _ 1 for all n 2 O. 3.23. Let d
2 2 and let ¢(z) E Z[z] be a monic polynomial of degree d, say ¢(z)
=
zd + az d- 1
+ ...
E
Z[z].
Prove that for every E > 0 there is a constant C = C(¢, E) such that for all a E Z satisfying 1001 2 C we have for all n 2 O.
Exercises
140 Section 3.5. Local Canonical Heights
The general theory oflocal canonical heights is developed in Section 5.9. However, the theory becomes much simpler if 1> is a polynomial, because the local canonical height then has a simple limit definition similar to the limit used to define the global canonical height. Exercises 3.24-3.30 ask you to develop some of the theory oflocal canonical heights for polynomials. 3.24. Let 1>(z) E K[z] be a polynomial. Prove that the limit (3.49) exists and that the resulting function has the following two properties: (a) For all a E
tc.,
(b) The function
a
1---+
>-¢,v(a) -logmax{lalv, I}
is continuous on tc, and has a finite limit as lal v -+ 00. Hence the function defined by (3.49) is a local canonical height as described in Theorem 3.27. Prove that the (global) canonical height is equal to the sum of the local heights,
h¢(a) =
L
nv>-¢,v(a).
vEM K
3.25. Let 1>(z) E K[z] be a polynomial and let v be an absolute value on K. Prove that the local height >-¢,v as defined by (3.49) has the following properties. (a) >-¢,v(a)~OforallaEKv. (b) >-¢,v (a) = a if and only if I¢n (a) Iv is bounded, equivalently, if and only if a is in the v-adic filled Julia set K; (1)) of 1> (see Exercise 2.1). 3.26. Let 1>(z) E qz] be a polynomial with complex coefficients and let B¢(oo) be the attracting basin of 00 for 1>. (See Exercises 2.1 and 3.16 for the definition of B¢ (00).) Prove that the local height >-¢ : ([ -+ IR as defined by (3.49) has the following properties: (a) >-¢ is a real analytic function on B¢( 00) -, {oo}. (b) The function >-¢ is harmonic on B¢ (00) -, {00}. In other words, writing z function >-¢ is a solution to the differential equation
= x + iy, the
on the open set B¢ (00) -, {oo}. It satisfies the boundary conditions that >-¢ vanishes on the boundary of B¢ (00) and has a logarithmic singularity at z = 00, i.e.,
>-¢(z) -log Izi
is bounded as z
-+
00.
(c) >-¢ is the unique function that has the properties described in (b). In classical terminology, the function >-¢ is the Green function for the filled Julia set lC(¢)
jp'1(iC) -, B¢(oo).
=
Exercises
141
3.27. Let 4>(z) = Z 2 - 1. Analyze the local canonical height '\ ",.v for points in: (a) the attracting basin B1> (00) of 00. (b) F1> " B1' (oo). (c) :1",. (See also Exercise 3.16 .)
3.28. Let 4>(z ) E Z[z] be a monic polynomial of degree d ~ 2 and let a rational number written in lowest terms. Prove that
= alb
E Q be a
with equality if and only if a is in the filled Julia set K:(4» of 4>. 3.29. Let K IQ be a number field and let 4>(z) E K (z ) be a ration al map of degree d ~ 2. For each finite place v E M~ , write 4>(z) = Fv(z)/Gv(z) as a ratio of polynomials that are normalized for v, i.e., the coefficients of F; and G» are v-adic integers and at least one coefficient is a v-adic unit. Let ,\~~': be the local canonical height, as described in Theorem 3.27, normalized using G» in (3.16). (a) Prove that ,\~~': is well-defined, independent of the decomposition 4>(z ) as a ratio of vnormalized polynomials. For nonarchimedean places v this serves to pin down a specific function ,\¢,': that depend s only on 4> and v (cf. Remark 3.28). (b) Give an example to show that it may not be possible to write 4>(z ) F (z )/G(z ) such that F and G are simultaneously normalized for all finite places v E M~ . (c) More generally, prove that every 4> E K (z ) can be written as F(z )/G (z ) with F and G simultaneously normalized for all finite places v E M~ if and only if K has class number 1.
=
3.30. Suppose that function
4>( z) E K (z) has good reduction at a finite place v E M K. Prove that the '\¢,v(a ) = log max{ lal v , 1}
for a E K;
is a local canonical height by showing that it has the required propertie s, Section 3.6, Diophantine Approximation 3.31. Prove Dirichlet's Theorem 3.33, which says that for every a E lR" Q there are infinitely many x] y E Q satisfying
I~ -al ~
y\'
(Hint. Look at the numbers ya - LyaJ for 0 ~ y ~ A. They all lie in the interval [0,1]. Divide the interval into A equal pieces and use the pigeonhole principle.) 3.32. Let B be a nonzero integer. (a) Prove that every solution (x , y) in integers to the equation (3.50)
<
satisfies max l jz ], Iyl } J4B13. (Hint. The polynomial X 3 +Y3 factors in Q[X, Y]. ) (b) Try to find an explicit upper bound for max{ lx l, Iyl } for the similar-looking equation X 3 + 2y 3 = B. The difficulty you face illustrates the fact, seen during the proof of Theorem 3.35, that the equation G(X , Y ) = B is relatively easy to solve if G(X , Y ) has distinct factors in Q [X, Y ], but very difficult if it does not.
Exercises
142
(c) Find all solutions in integers x ~ y to the equation (3.50) for the following values of B : (i) B = 2. (ii) B = 91. (iii) B = 728. (iv) B = 1729.
Section 3.7. Integral Points in Orbits 3.33. Let ¢( z ) = z + 1/ z and write ¢ n (1) = an I bn in lowest terms as in Example 3.50. (a) Prove by a direct computation that there are constants c, c' > 0 such that C
log an
I
< -- < c - logb -
for all n
n
> 2. -
(b) Try to prove directl y that log an f log bn ---> 1 as n ---> 00. (c) Prove that ¢n( l) ---> 00 as n ---> 00. (Notice that 00 is a rationally indifferent fixed point, since ¢' (00) = 1.) 3.34. Let c E Z be a squarefree integer and let ¢e (z) = z + c] z . Further, let a E Q be a wandering point for ¢e. (a) If 1 - 4c is not a perfect square, prove that # (0(0') n Z) ~ 2. (b) Ifl -4c = d2 is a perfect square, prove that # (O(O')nZ) ~ 2 unless a in which case there are three integer points in the orbit.
= (-I±d)/2,
3.35. Let ¢(z) E Q(z) be a rational function of degree d ~ 2 such that ¢2 is not a polynomial. Suppose that ¢ has everywhere good reduction and that ¢( (0) = 00. Let a E Q be a wandering point for ¢. Prove that
# (O
= 2. Try to improve the bound for larger values of d.
3.36. Let ¢(z) E Q(z) be a rational function of degree d ~ 2, let a E Q be a wandering point for ¢, and write ¢ n (O' ) = an/ bn E Q as a fraction in lowest terms. (a) Suppose that there is a nonzero integer B such that bnlB for infinitely many n. Prove that ¢2( Z) E Q[z]. (Hint . Make a change of variables and apply Theorem 3.43.) (b) Suppose that there are infinitely many n such that bn Ibn + 1 . Prove that either 00 is a fixed point of ¢ or else ¢2(Z) E Q[z]. 3.37. Let ¢(z) E iC( z) be a rational function of degree d. This exercise gives a quantitative strengthening of Proposition 3.44. (a) Let P be a nonperiodic point for ¢. Prove that for all n
~
O.
(b) Let P be a fixed point of ¢ that is not totally ramified. Prove that
#¢-n(p) ~ d n -
2
+2
for all n
~
2.
(c) Generalize (b) to the case that P is a periodic point for ¢ of exact period m under the assumption that ¢'"' is not totally ramified at P . Use (a), (b), and (c) to deduce that if ¢ is not a polynomial map, then
#¢- n( p)
~
3
if either
n ~ 4 { n ~ 3
and
d = 2,
and
d
~
3.
Exercises
143
3.38. Let K/Q be a number field, let S C MK be a finite set of absolute values on K, and let Rs be the ring of S-integersof K. Let ¢( z) E K (z) be a rationalfunction of degree d ~ 2 with ¢2(Z) .;. K[z], and let 0: E K be a wanderingpoint for ¢. Prove that 01>(0:) n Rs is a finite set. (This exercise generalizes Theorem 3.43.)
3.39. Let K be a number field, let S C M K be a finite set of absolute values on K that includes all the archimedean absolute values, and let Rs be the ring of S-integers of K. Let ¢(z) E K(z) be a rational function of degree d ~ 2 satisfying ¢2(Z) .;. K[z]. (a) Let n = 4 if d = 2 and let n = 3 if d ~ 3. Prove that the set
{o
E
K: ¢n(o:) E Rs}
is finite. (b) Give an examplewith (n, d) = (3,2) and oo nonperiodicfor ¢ such that the set in (a) is infinite. (c) Same question as in (b) with n = 2 and d arbitrary. (d) Repeat (b) and (c) with oo a fixedpoint of ¢.
3.40. Let ¢l, ... ,¢r E Q( z) be rational functions of degree at least 2, and let be the collection of rational functions obtained by composing an arbitrary finite number of ¢l, ... , ¢r. Note that each ¢i may be used many times. For example,if r = 1, then is simply the collection of iterates {¢l}. Also note that in general, compositionof functions is not commutative, so if r ~ 2, then is likely to be a very large set. (a) A rational map ¢ E K(z) is said to be of polynomial type if it has a totally ramified fixed point. Prove that for such a map, there is a linear fractional transformation f E PGL2(K) such that ¢f E K[z]. Further, if ¢ is not conjugate to Zd, prove that one can take f to be in PGL2(K). (b) Assume that contains no maps of polynomial type. Prove that there are finite subsets 1 and 2 of satisfying and (Here ¢ denotes the compositionof ¢ with every map in .) (c) Let 0: E Q. The -orbit of 0: is the set 0<1>(0:) = {¢(o:) : ¢ E }. Continuing with the assumptionthat contains no maps of polynomialtype, prove that the -orbit of 0: contains only finitely many integers, i.e., prove that 0<1> (0:) n Z is a finite set.
3.41. Definethe (logarithmic) height ofa rational map ¢(z) E Q(z) by writing
¢(z)
= F(z) = ao + alZ + G(z) bo + blz +
+ adzd + bdzd
with F(z), G(z) E Q[z] and setting
h(¢)
= h([ao, al, ... , ad, bo, bl, ... , bdl).
Prove the followingquantitative version of Proposition3.46. For all d ~ 2 there is a constant 0 = Oed) such that for all integers N rational map ¢( z) E Q( z) of degree d with the followingproperties:
~
0 there exists a
Exercises
144 • rji (z ) i C[z]. • 0 is a wandering point for . • 0, (0), 2(0), 3(0) , . . . , ¢ N (0) E Z. N
• h( ¢) S; C . d . Try to do quantitatively better than this, for example, replace the height estimate with one of the form h( ¢) S; C · fJ N for some s < d.
3.42. Recall that a rational map ¢ (z ) E Q(z) is affine minimal ifits resultant Res( is affine minimal and ¢>2 is not a polynomial, we have conjectured (Conjecture 3.47) that the size of O(a) n Z is bounded solely in terms of the degree of ¢. For each integer d ~ 2, let
C (d)
= sup { # (O (a) n Z) :
(z) E Q(z), ¢2(Z) i Q[z], } ¢ is affine minimal, and . a E ]P'N (Q) " P rePe r(¢)
Thus Conjecture 3.47 says that C(d) is finite. (a) Prove that for every integer N ~ 0 there exists an affine minimal rational map ¢(z) E Q(z) such that ¢2 (z) i Q[z], such that 0 is not a preperiodic point for ¢, and such that
are all integers. In particular, C(d) - t 00 as d - t 00 . (b) Prove that the function ¢(z) in (a) can be chosen to have degree L(N - 1)j2J. Hence C(d) ~2d+2 .
(c) Let ¢>(z) be the function ¢ (z ) = 481z
5
-
Z5 -
4
2565z + 9385z 465z 4 + 2185z 3
3 -
-
2
14955z + 12094z - 3720 . 6975z 2 + 8254z - 3720
Trace the orbit of O. How many integers do you find? (d) Find a rational function as in (a) of degree 2 and with the property that ¢i (0) is an integer for all 0 S; i S; 6. Hence C (2) ~ 7, which improves the bound from (b). Can you find infinitely many such functions? Can you prove that C(2) ~ 8? (e) Find a rational function as in (a) of degree 3 and with the property that ¢i(O) is an integer for all 0 S; i S; 8. Hence C(3) ~ 9, which improves the bound from (b). Can you find infinitely many such functions? Can you prove that C(3) ~ 1O? Section 3.8. Integrality Estimates for Points in Orbits
3.43. Let K; be a field complete with respect to the absolute value v, let ¢ : ]P'i - t ]P' i be a rational map of degree d ~ 2 defined over K v , and let p v : ]P'l (K v ) x ]P'l(K v ) - t lR be the associated chordal metric. (See Sections 1.1 and 2.1 for the definition of the chordal metric when v is archimedean and nonarchimedean, respectively.) Generalize Lemma 3.51 by showing that for every Q E ]P'i (K v ) there is a constant C« = C v (¢ , Q) such that
Here e Q(¢) is as defined in Lemma 3.51.
145
Exercises
3.44. Let K be a number field and let 1> : pI --4 IP'I be a rational map of degree d 2:: 2 defined over K . Prove that there is a finite set of absolute values S C MK such that for all v 1- Sand all Q E 1P'1 (K), the constant Cv ( 1>, Q) in Exercise 3.43 may be taken equal to l. 3.45. Let 1>(z) E Q(z) be a rational map of degree d 2:: 2 with 1>2(z) 1- Q[z], let a E Q, and write 1>n(a ) = anlb n E Q as a fraction in lowest terms as usual. Prove that
3.46. Let KIQ be a number field, let v E M K be an absolute value on K , and let 1>( z) E K (z) be a rational function of degree d 2:: 2. Suppose that a E 1P'1 (K) is a wandering point for 1> and that 'Y E 1P'1 (K ) is any point that is not a totally ramified fixed point of 1>2. Prove that . -log pv (1)n(a), 'Y) lim =0. n -oo h( 1>n(a)) Takingfirst 'Y = 00 and then 'Y to number fields.
= 0, explain how to use this result to generalize Theorem 3.48
Section 3.9. Periodic Points and Galois Groups 3.47. Let P E IP'N(K) be a point in projective space and let K (P ) be its field of definition. Prove that K (P ) = fixed field of {u E Gal(KIK ) : u{P ):=: Pl. In mathematical terminology, this says that the field of moduli of P is a field of definition for P . Wediscuss fields of moduli and fields of definition for (equivalence classes of) rational maps in Chapter 4.
Section 3.11. Ramification and Units in Dynatomic Fields 3.48. Let 1>{z) = Z2 + c and let
; (z)
= 1>3(Z) -
z 1>(z) - z
be the polynomial whose roots are periodic points of period 3. (a) Prove that 3(z) E Z[c, x] is a polynomial in the variables c and z and that it has integer coefficients. (b) Let a be a root of <1>;; (z) and assume that the field Q(c, a) is an extension of Q( c) of degree 6 (i.e., ;;(z) is irreducible in Q(c) [z]). Theorem 2.33 implies that and are units. Compute these units explicitly as elements of Q[c, o]. (c) Prove that there is a field automorphism a : Q (c, a) --4 Q(c, a) characterized by the fact that it fixesQ (c) and satisfies u(a ) = 1>(a) = a 2 +c. (Note that in general, Q( c, a) will not be a Galois extension of Q (c).) Prove that u 3 is the identity map. (d) Compute the units a (u 1) and a (U2). (e) Compute the units u 2(uI) and U2(U2). (f) Express u(uI), U(U2), u 2(uI), and U2(U2) in the form ±ui u~ .
Exercises
146
3.49. Let ¢>( z ) = Z2 + c and let q>; (z ) be as in the previous exercise. (a) Prove that q>;(z) factors into a product of two cubic polynomials in the ring Q(c)[z] if and only if c has the form c = _(e2 + 7)/4 for some e E Q(c) . (b) Suppose that c = _ (e2 + 7) /4. Show that there is a polynomial ge(z) E Q[e , z] such that the factorization q>3(z) has the form ge(Z)g-e(z). Compute the discriminant of ge(z) and verify that it is a perfect square in Q[e]. Conclude that if ge(Z) is irreducible over Q( e), then its roots generate a cyclic Galois extension K; ofQ(e). (c) Let Q be a root of ge(z). Use the results of the previous exercise to construct units in Q (e) (Q). Analyze these units for some small values of e, say e = 1, 5,7,9. (Note that we investigated the case e = 3 in Example 3.69.) 3.50. ** Let ¢>( z) be a generic monic polynomial of degree d ~ 2, i.e., ¢>( z) is a polynomial of the form Z d + al z d- l + ...+ ad , where al , .. . , ad are indeterminates. Let p be a prime and let Q E Per;*(¢» . Theorem 3.66 says that the elements
U i ,j
=
¢>i(Q) _ qJ(Q) ¢>(Q) _ Q '
0:::; j < i < p,
are units. What is the rank of the group that they generate? Is the answer the same for the polynomial ¢>(z) = Z2 + c? 3.51. ** Let ¢>(z ) be a generic rational function of degree d ~ 2, let p be a prime, and let p E Per; *(¢». Theorem 3.67 describes how to construct units K,(p, ¢>p,¢>ip, qJp). What is the rank of the group that they generate? Is the answer the same for the rational function ¢>(z) = (Z2 + b)/(bz 2 + I) ?
Chapter 4
Families of Dynamical Systems Most of our work in previous chapters has focused on a single rational map ¢(z) and the effect its iterates have on different initial values. We now shift focus and consider the effect of varying the rational map ¢( z). In order to do this, we study the set of all rational maps. This set turns out to have a natural structure as an algebraic variety, as does the set of rational maps modulo the equivalence relation defined by PGL 2 conjugation. There are many threads to this story. The specific topics that we touch upon in this chapter include: 1. 2. 3. 4. 5. 6.
Dynatomic polynomials and fields generated by periodic points. The space of quadratic polynomials (the simplest nontrivial case). Rational maps that are PGL 2-equivalent over k, but not over K (twists). Field of defintion versus field of moduli for rational maps. Minimal models for rational maps. Moduli spaces of rational maps (with marked periodic points).
All of these topics have close analogues in the geometric and arithmetic theory of elliptic curves and abelian varieties. For purposes of comparison, we list the correspondences: 1. 2. 3. 4. 5. 6.
Division polynomials and fields generated by torsion points. The space of elliptic curves (the simplest abelian varieties). Abelian varieties isomorphic over K, but not over K (twists). Field of definition versus field of moduli for an abelian variety. Minimal Weierstrass equations and Neron models. Elliptic modular curves and moduli spaces of abelian varieties.
In this chapter, we let K be a perfect field and fix an algebraic closure k of K, although much of what we do is also valid for nonperfect fields if we use instead a separable closure of K. We write Gal(k/ K) for the absolute Galois group of K. It is convenient to use the profinite group Gal(k/ K), but we observe that it generally suffices to work with finite extensions and finite Galois groups, since the coefficients of any particular rational map ¢(z) E k (z) lie in a finite extension of K. 147
148
4. Families of Dynamical Systems
4.1 Dynatomic Polynomials Let ¢( z) E K [z] be a polynomial. Then the fixed points of ¢ are the roots of ¢( z) - z (plus the point at (0), and more generally, the points of period n for ¢ are the roots of ¢n(z) - z. However, the polynomial ¢n(z) - z may have roots of period smaller than n, since if ¢k(a) = a and kin, then also ¢n(a) = a. It is natural to try to eliminate these points of strictly smaller period and focus on the points of exact period n. Example 4.1. We recall an analogous situation. The roots of the polynomial z" - 1 include all nth roots of unity, not only the primitive ones. The nth cyclotomic polynomial is defined using an inclusion-exclusion product, nth
cyclotomic polynomial
=
II(zk -
l)lL(n/k).
(4.1)
kin It is the polynomial whose roots are the primitive Mobius function J-l defined by J-l(l) = 1 and
ei
nth
roots of unity. Here J-l is the
ec)_{(-lY ifeI=···=er=l, . a If any e; 2: 2.
J-l (PI .. ·P r
(4.2)
See [216, Section 2.2] or [11, Chapter 2] for basic properties of the Mobius function and the Mobius inversion formula. It is easy to check that the product (4.1) is a polynomial, using the fact that the (complex) roots of z" - 1 are distinct and the following basic property of the J-l function (Exercise 4.2):
L J-l (~) = {I ~f n = 1, a
kin
(4.3)
Ifn> 1.
Taking our cue from the example provided by the cyclotomic polynomials, we might define the nth dynatomic polynomial by the formula
n(z)
= II(¢k(z) - zt(n/k). kin
However, it is not clear that n(z) is a polynomial, since ¢n(z) may have multiple roots, as shown by the following example. Example 4.2. Let ¢(z) be the polynomial
¢(z) Then
=z
2
-
3 4.
4.1. Dynatomic Polynomials
149
¢(z) - z = z2 - Z -
~ = ( z - ~)
3 2 4 3 ¢ (z) - z = z - - z2 - z - - = 2
¢2(Z) - Z = ¢(z) - z
16
(
(z+
~) ,
3 z- - ) 2
(
1 z+ - ) 2
3
'
(Z +~) 2 2
Thus ¢2(z) - z vanishes with multiplicity 3 at the point z = - ~ , and although it is true that the ratio ~g/~: is a polynomial, it is somewhat distressing to observe that its root is a fixed point of ¢, not a point of primitive period 2. For simplicity, the preceding discussion dealt with polynomial maps ¢(z). We now tum to general rational maps ¢(z) E K(z) and develop tools that are useful for studying their periodic points. Let ¢( z) be a rational function of degree d and write ¢ = [F(X, Y), G(X, Y)] using homogeneous polynomials F and G. Then the roots of the polynomial
Y F (X ,Y ) - XG (X, Y ) in ]F'l are precisely the fixed points of ¢. If we count each fixed point according to the multiplicity of the root, then ¢ has exactly d + 1 fixed points . More generally, we can apply the same reasoning to an iterate ¢n of ¢ and assign multiplicities to the n- periodic points.
Definition. Let ¢( z) E K (z) be a rational function of degree d, and for any n 2: 0, write
¢n = [Fn( X ,Y ),Gn(X,Y )] with homogeneous polynomials Fn , Gn E K [X,Y ] of degree d" , (See Exercise 4.9 for a formal inductive definition of Fn and G n . ) The ti-period polynomial of ¢ is the polynomial q" n(X, Y ) = YFn(X,Y ) - XGn (X,Y ). Notice that q"n (P ) = 0 if and only if ¢n(p) = P, which justifies the name assigned to the polynomial q"n. The n1hdynatomic polynomial of ¢ is the polynomial'
q"* n(X , Y) = II(YFk (X, Y ) - XG k (X, y))!"(n/k) =
kin
II q" k" (X y)!"(n/k) kin
where JL is the Mobius function . If ¢ is fixed, we write n and ~. If ¢(z) E K [z] is a polynomial, then we generally dehomogenize [X ,Y] = [z,l] and write n (Z) and ~ ( z ) . All of the roots P of; ,n(X , Y ) satisfy ¢n(p ) = P , but we saw in Example 4.2 that ; ,n (X, Y ) may have roots whose periods are strictly smaller than n . Following Milnor [302], we make the following definitions. I See
Theorem 4.5 for the proof that ep; .n is indeed a polynomi al.
150
4. Families of Dynamical Systems
Definition. Let cjJ{ z) E K (z) be a rational map and let P E pI be a periodic point for cjJ .
• P has period n if n(P) =
a.
• P has primitive (or exact) period n if n(P) = m< n.
• P has formal period n if ~ (P) =
a and m{P ) =I a for all
a.
We set the notation Pern{cjJ)
= {p E pI: n {P ) = a},
Per~ { cjJ) = {P E pI : ~ {P ) = Per~*{cjJ)
= {p
E
a},
pI: n{P ) =
a and m{P) =I a for all 1 ~ m < n}.
Thus P er., (cjJ) is the set of points of period n, Per~ (cjJ) is the set of points of formal period n, and Per~*( cjJ) is the set of points of primitive (or exact) period n. Sometimes we treat these as sets of points with assigned multiplicities, e.g., if P E Per~ (cjJ) , the multiplicity of P is the order of vanishing of ~ at P. It is clear that primitive period n
==>
formal period n
==>
period n ,
but neither of the reverse implications is true in general.
Remark 4.3. The polynomial ,n is homogeneous of degree d" + 1, so counted with multiplicity, the map cjJ has exactly d" + 1 points of period n. And if we let IId(n ) = deg(; ,n{X, Y)) =
LJL ( ~) (dk + 1),
(4.4)
ki n
then counted with multiplicity, the map cjJ has exactly lid ( n ) points of formal period n. The number IId(n) grows very rapidly as d or n increases . See Exercise 4.3. The first few period and dynatomic polynomials for cjJ( z ) = z2 + c are listed in Table 4.1 on page 156. Notice how complicated n and ~ are, even for small values of n. For example, 6 has degree 54 as a polynomial in z and degree 27 as a polynomial in c.
Remark 4.4. Rather than using the homogeneous polynomials n and ~, it is sometimes more natural and convenient to consider instead the associated divisors in pI, especially in generalizing the theory to higher-dimensional situations. This is the approach taken in [313], where for any (nondegenerate) morphism cjJ : V --+ V of smooth algebraic varieties, the O-cycle ,n is defined as the pullback of the graph of cjJn by the diagonal map A : V --+ V x V. See Exercise 4.8. We now prove the important fact that the dynatomic polynomial ; ,n(X, Y) is indeed a polynomial.
4.1. Dynatomic Polynomials
151
Theorem 4.5. Let ¢( z) E K (z) be a rationalfunction ofdegree d jpI
~
2. For each P E
(K), let and
or, in terms ofdivisors in Div(lPi ), div(,n) =
L ap(n)(P)
and
div( ; ,n) =
p
L aj,( n)(P). p
(a) ;,n E K[X, Y], or equivalently,
aj,(n) ~ 0 for all n ~ 1 and all P E jpI. (b) Let P be a point ofprimitive period m and let >.( P) = (¢m)' (P) be the multiplier of P. Then P has formal period n. i.e., aj,(n) > 0, if and only if one of the following three conditions is true: (i) n = m. (ii) n = mr and >.( P) is a primitive r 'h root ofunity. (iii) n = mrp", >.(P) is a primitive r 'h root of unity, K has characteristic p, and e ~ 1.
In particular, ofn.
if K
has characteristic 0, then a j,(n) is nonzero for at most two values
Proof By the definition of~, we have the relation
aj,(n) =
L J.l(n/m)ap(m). min
We begin with a lemma that describes the value of a p (n) for fixed points.
Lemma 4.6. Let 'ljJ (z) E K (z) be a rational function of degree d ~ 2, let P E jpI (K) be a fixed point of'ljJ, let>' = >.p( 'ljJ ) = 'ljJ' (P) be the multiplier of'ljJ at P , and let
ap(N) = ordp(1jJ,N(X, Y)) be as in Theorem 4.5. Thenfor any N
>.N # >. # 1 and >.N =
~
2,
1
=}
1
=}
>. = 1 and N # 0 in K >. = 1 and N = 0 in K
=} =}
= ap(l) = 1,
(4.5)
ap(N) > ap(l) = 1, ap(N) = ap(l) ~ 2, ap(N) > ap(l) ~ 2.
(4.6)
ap(N)
(4.7) (4.8)
Proof Making a change of variables, we may assume that P = 0, and then the assumption that P is a fixed point of 'ljJ means that 'ljJ( z) has the form (4.9)
4. Families of Dynamical Systems
152
where 0: -I 0, e 2: 2, and O(ze+l) is shorthand for a function that vanishes to order at least e + 1 at O. Note that
z) =
ap(l) = ord(1P(z) z=o
I { e 2: 2
if),
.
-11,
If A = 1.
Weconsiderfirstthe case that A -11. Using the weaker form 1P(z) = AZ + O(z2) of (4.9) and iterating gives
so
ap(N) = ord(1P N (z) - z) z=o N = ord((A
z=o
-l)z + O(z2))
-{ -1
2: 2
if),N
-11,
if AN = 1.
This completes the proof of the lemma in this case. Next we consider the case A = 1. Then ap(l) = ord z=o(1P(z) iteration of (4.9) yields
z) = e,
and
(4.10) Hence
ap(N) = ord('¢N (z) - z) z=o e = ord(No:z
z=O
+ O(ze+I))
-{ =e=ap(l)
> e = ap(l)
if N -lOin K, if N
= 0 in K. D
This completes the proof of Lemma 4.6
Resuming the proof of Theorem 4.5, we observe that if the primitive period m does not divide n, then ap(n) = 0, and hence also ap(n) = O. We may thus assume that min. Since we will deal with several different maps, we write ap(
ap(
N=~. m
= 0 unless mlk, we find that
L J1 (~) ap(
kiN
=
LJL (~) ap(
153
4.1. Dynatomic Polynomials
We further observe that P is a fixed point of 'ljJ, so Lemma 4.6 applies to 'ljJ and P. We consider several cases, the first two of which may done simultaneously.
Case I(a). >..(p)N =II. Case I(b). >..(P) = 1 and N =I 0 in K. In Case I(a) we have )..(p )k 1= 1 for all kiN, so we can apply (4.5) of Lemma 4.6 to conclude that ap('ljJ , k) = apCljJ , 1) for all kiN. Similarly, in Case I(b) we can apply (4.7) of Lemma 4.6 to conclude that ap('ljJ ,k ) = ap (ib, 1) for all kiN. Hence in both cases we find that
ap('ljJ , N) = LtL
(~) ap('ljJ, k) =
(L tL
kiN
( ~) ) ap('ljJ , 1)
kiN
=
{ap ('ljJ , 1) 2 1 if N = 1, o if N > 1.
Using the equality ap('ljJ, 1) = ap(qr, 1) = ap(c/J,m), this shows in Case I(a,b) that * ('" ) _ { ap(c/J,n ) 2 1 if n = m ,
ap 'f', n -
=
o
if n > m.
=
Case II. >..(P) 1 and N 0 in K. Let p be the characteristic of K and write N = peM with p f M . We observe that if kiM and 0 :::; i:::; e, then k 1= 0 in K and )..p ('ljJpi ) = ).. p('ljJ )pi = 1, so (4.7) of Lemma 4.6 applied to 'ljJpi tells us that
ap('ljJ ,pik) = ap('ljJpi , k) = ap ('ljJpi , 1) = ap('ljJ,pi). This allows us to compute
ap('ljJ,N) = LtL k iN
( ~) ap('ljJ ,k) = L
ttl
kiM i =O
= (LtL kiM
(pei~) ap (1/J,pi k ) P
(~)) (ttL (pe- i)ap(1/J,pi)) 2=0
= {a p(1/J,pe) - ap ('ljJ ,pe- l )
o
2 1 if M
= 1,
if M
> 1.
The fact that the value is positive when M = 1 follows from applying condition (4.8) of Lemma 4.6 to the difference ap (1/J pC- l ,p) - ap('ljJpe- l , 1). Hence in Case II, we have shown that ap(c/J,n) 2 0, and further, ap(c/J, n) 2 1 if and only if n = mp", Since Case II includes the assumption that )..(P ) = 1, this proves Theorem 4.5 in this case.
Case III. >..(P) =11 and >..(p)N = I. Let r be the exact order of )..(P ) in K*, so r lN and r > 1. Then (4.5) of Lemma 4.6
154
4. Families of DynamicalSystems
tells us that if r f k, then ap('ljJ , k) defining ap('ljJ ,N) as
ap('ljJ ,N ) = (L + L)1l kiN kiN rtk rlk =
= ap('ljJ , 1). This
allows us to write the sum
(~) ap('ljJ,k)
(L Il( ~) ap('ljJ ,l)) + (L Il( ~) ap('ljJ ,k)) kiN rtk
= (L Il kiN = L
kiN rlk
( ~) ap('ljJ , 1)) + (LIl ( ~) (ap('ljJ,k) kiN rlk
ap('ljJ ,
1)))
11 ( Nt ) (ap('ljJ , kr) - ap('ljJ, 1))
kl ~ r
=a p( 'ljJr ,N - {a p( 'ljJ, l) lr)
o
If N
ifN=r, if N i= r.
(4.11)
= r, so n = mr, then we have
ap(¢, n) = ap('ljJ, N ) = ap('ljJr , 1) - ap('ljJ, 1) = ap('ljJ,r) - ap('ljJ, 1) 2: 1 from (4.6) of Lemma 4.6. We may thus assume that N
ap(¢,n)
> r, so (4. 11) says that
= ap('ljJ,N) = ap('ljJr ,Nlr) .
We now observe that >"p('ljJr ) = >..p('ljJY = 1. Hence if N i= 0 in K, then we can apply Case I(b) to ap('ljJr ,Nlr), and if N = 0 in K, then we can apply Case II to ap('ljJr ,Nlr) . This compl etes the proof of Theorem 4.5. 0 As an easy application of Theorem 4.5, we now prove that a rational map
¢(z) E K (z) possesses periodic points of infinitely many distinct periods, i.e., the set of primitive-n periodic points Per~* (¢) is nonempty for infinitely many n . Corollary 4.7. Let ¢ (z ) E K( z) be a rational map of degree d 2: 2. Then fo r all prim e numbers f except possibly for d + 2 exceptions, the map ¢ has a point of primi tive period f. Proof We begin by discarding the finitely many primes f satisfying either of the followin g conditions:
• K has characteristic £. • There is some Q E Fix(¢) with >"(Q)
i= 1 and >..(Q)i! = 1.
155
4.2. Quadratic Polynomials and Dynatomic Modular Curves
The set Fix (¢) contains at most d + 1 points, so these condit ions eliminate at most d + 2 primes . take a point P E Pere(¢ ). If P does not For any of the remaining primes have primitive period then it must be a fixed point of ¢, since its period certainly divides e. Then our assumption s on eimply that either )..(P ) = lor )..(p )e i=- 1, and further that ei=- 0 in K. It follows from (4.5) and (4.7) of Lemma 4.6 that in both cases we have ap (e) = ap(I ). Hence
e,
e,
L PEPere(.p)n F ix(.p)
ap( e) =
L
ap( I) ::;
P EPer e(4))nFix(4>)
L
ap (l ) = d + 1.
P EF ix(4))
Thus the total multiplicity of all of the fixed points of ¢ in the e-period polynomial
is at most d + 1. However, the degree of
e-
For many applications , Corollary 4.7 is sufficient, but a more detailed analysis yields a much stronger result. We state the full theorem in characteristic 0 and refer the reader to Pezda 's two papers [354, 357] for the more complicated description required in characteristic p. See also [161, 172, 173, 174] for higher-dimensional results of a similar nature.
Theorem 4.8. (LN. Baker [19]) Let ¢( z ) E K (z ) be a rational map ofdegree d :::: 2 defined over afield K ofcharacteristic O. Suppose that ¢ has no primitive n -periodic points. Then (n, d) is one ofthe pairs (2, 2), (2, 3) , (3, 2), (4, 2).
If ¢ is a polynomial, then only (2,2)
is possible.
Proof For the proof, which is function-theoretic in nature, see [19] or [43, § 6.8].
o 4.2
Quadratic Polynomials and Dynatomic Modular Curves
In this section we expand on the material from the previous section in the special case of the quadratic polynomial
¢(z ) = ¢c(z ) = z2 + C. For further material on the iteration of quadratic polynomials, see for example [17, 18,113,115,171,220,221 ,222,305,309,350,361 ,388].
4. Families of Dynamical Systems
156
+ (4c3 + 4c2)Z2 - Z + (C4 + 2c3 + C2 + c) Z16 + 8CZ 14 + (28c2 + 4C)Z12 + (56c3 + 24c2)ZlO 4 4 + (70c + 60c3 + 6c2 + 2C) Z8 + (56c5 + 80c + 24c3 + 8c2)Z6 2 6 5 4 3 + (28c + 60c + 36c + 16c + 4c )Z4 7 6 4 + (8c + 24c + 24c5 + 16c + 8c3) Z2 4 2 7 6 5 3 8 - Z + (C + 4c + 6c + 6c + 5c + 2c + C + c)
+ Z5 + (3c + 1)Z4 + (2c + 1)Z3 + (3c2 + 3c + 1)z2 2 2 3 + (c + 2c + 1)z + (c + 2c + c + 1) 10
Table 4.1: Period and dynatomic polynomials for ¢(z) =
4.2.1
Z2
+ c.
Dynatomic Curves for Quadratic Polynomials
Note that as long as the field K does not have characteristic 2, then any quadratic polynomial can be put into the form z2 + c by a simple change of variables working entirely within the field K. More precisely, if we let f( z) = (2z - B )/ (2A), then
¢ f( z) =
U- 1 0 ¢o J) (z ) = z2 +
(AC - ~B2 + ~B) .
(4.12)
The roots of the period polynomials n(z) = ¢~ (z ) - z and associated dynatomic polynomials ~ ( z ) are periodic points of the map ¢c(z ) = Z 2 + c. In order to investigate how the periodic points of ¢c(z) vary as a function of c, we observe that ~ (z) is a polynomial in the two variables z and c. Thus in studying
157
4.2. Quadratic Polynomials and Dynatomic Modular Curves
quadratic polynomial maps, it is natural to write ll>~ (c, z) and treat ll> ~ as a polynomial in Z[c, z]. (Table 4.1 lists the first few period and dynatomic polynomials for ¢c(z).) Treating ll>~ ( c, z) as a polynomial in two variables leads to a natural association polynomial ¢( z) E K [z] of } degree 2 with a point { P E K of formal period n
~
{ (c,Q) E K x K : ll> ~ (c,Q) =
O}.
Thus given a polynomial ¢(z) E K[z] of degree 2 and a point P of formal period n, we make a change of variables as in (4.12) so that ¢I (z) = z2 + c and set Q = 1-1(P). Note that this entire procedure takes place within the field K (which we assume has characteristic different from 2). Thus the solutions to the equation ll>~(y, z) = 0 parameterize pairs (¢, P), where ¢ is a conjugacy class of quadraticpolynomials and P is a point offormal period n for ¢. Further, the solution is K -rational if and only if ¢ and Pare K -rational, Definition. The dynatomic modular curve Y l (n) C A2 is the affine curve defined by the equation ll> ~ (y , z )
=0.
The normalization of the projective closure of Yl (n) is denotedby X I (n). Example 4.9. It is easy to see that X l (1) and X l (2) are rationalcurves. Indeed, the projective closures ofYl (l ) and YI (2) are smooth conics,
and
Xl (2) : z2
+ ZW + yw + w 2 = O.
It turns out that Xl (3) is also rational, but this is less clear from the equation in Table 4.1. In order to parameterize X l (3), suppose that ¢(z) = Az 2 + B z + C is any quadratic polynomial with a periodic point of primitive period 3. Conjugating by a linear map z ~ Q Z + /3, we may assume that the given 3-cycle has the form o-> 1 -> t -> 0 for some t. This gives the equations ¢ (1) =A+B+C=t,
¢(O) = C = 1,
Solving for A, B, C in terms oft yields ¢(z) =
t2
t+1 2 2 Z t -t
t3
-
-
+1
t2
-
t-t
2
Z
+ 1.
Now we apply the linear change of variables (4.12) to put ¢ into the form z2 + c. Thus letting I (z) = (2z - B )/(2A ), we find that
I ¢ (z) = z2
+
t6
-
4t 5
+ 9t 4 - 8t3 + 4t 2 -4t 4 + 8t 3 _ 4t 2
2t + 1
Our computation showsthat for every value of t
' ~
{O, I} , the point
4. Families of Dynamical Systems
158 t 6 - 4t 5 (
+ 9t 4 - 8t 3 + 4t 2 -4t 4 + 8t 3 - 4t 2
2t
+
1 '
_ t3 + t2 -2t 2 + 2t
1)
is a solution to the equation ; (y , z) = z6 + Z5
+ (3y + l )z4 + (2y + l )z3 + (3y 2 + 3y + l )z2 + (y2 + 2y + l) z + (y3 + 2y2 + y + 1) = O.
(You may check this directly using a computer algebra system .) We have thus constructed a nonconstant rational map s--«
~g
o ------+
t6 - 4t 5
+ 9t 4
8t 3 + 4t 2 --- 1)2
t 3 --- t 2 + 1 ) . , 2t (t --- 1) (4.13) General principles (Liiroth's theorem [198, IV.2.5.5]) tell us that Xl (3) is birational to pl. More concretely, we can prove that the map (4.13) has degree 1 by constructing its inverse . Thus let (c,b) be a root of ;. We set g(z) = (b 2 + c - b)z + b, so g sends 0 to band 1 to ¢(b). Thus the 3-cycle b ----t ¢(b) ----t ¢2(b) ----t 0 becomes the following 3-cycle for ¢ g:
t
(
1
-
2t
-
+1
-4t 2(t
~g
------+
b2
+ b+ 1 + c
~g
------+
O.
This gives the map (c, b) f----t b2 + b + 1 + c, which is inverse to (4.13) , a fact that can also be checked directly with a computer algebra system.
4.2.2 Dynatomic Curves as Modular Curves The curve YI (n ) and its completion X I (n) are modular curves in the sense that their points are solutions to the moduli problem of describing the isomorphism classes of pairs (¢, 0:), where ¢ is a polynomial of degree 2 and 0: E A I is a point of formal period n for ¢. Here two pairs (¢l , o:d and (¢2,0:2) are PGL 2-isomorphic if there is a linear fractional transformation f E PGL 2 satisfying and In order to state this more carefully, we define
Formal(n)
=
. ¢ E k[z], deg(¢ ) = 2, 0: E k} { (¢,o:) . 0: has formal period n for ¢
.
PGL 2-lsomorphism
.
We have demonstrated that the elements of Formal(n) are in one-to -one correspondence with the points of YI (n). But much more is true: the correspondence is algebraic in an appropriate sense. Before stating this important result, we must define what it means for a family of maps and points to be algebraic.
4.2. Quadratic Polynomials and Dynatomic Modular Curves
159
Definition. Let V be an algebraic variety. An algebraic family ofquadratic polynomials over V with a markedpoint offormal period n consists of a quadratic polynomial
'ijJ(z) = Az2 + Bz
+ C,
A, B, C E K[V],
whose coefficients A, B, C are regular functions on V and such that A does not vanish on V(K), and a morphism A : V ----t p} such that for all P E V(K), the point A(P) is a point offormal period n for the quadratic polynonomial
'ijJp(z) = A(p)z2
+ B(P)z + C(P)
K[z].
E
The family is defined over K if the variety V and morphism A are defined over K and the functions A, B, C are in K[V]. Example 4.10. The pair
A(t)=t-l
and
is an algebraic family of quadratic polynomials over lP'1 -, {O, oo} with a marked point of formal period 2.
Theorem 4.11. Let K be a field ofcharacteristic different from 2. (a) The map
Yl(n)
---+
Formal(n),
(c, CY)
r---+
(z2
+ c, CY),
(4.14)
is a bijection ofsets. (b) Let V be a variety and suppose that the points ofV algebraically parameterize a family of quadratic polynomials 'ijJ together with a marked point A offormal period n. Then there is a unique morphism ofvarieties
with the property that
'f/(P) = ('ijJp(Z),A(P)) E Formal(n)
(c)
for all P E V(K),
(4.15)
where we use (4.14) to identify Formal(n) with Y 1 (n). is defined over the field K, then the morphism 'f/ is also defined over K.
If the family
Proof (a) We have shown this earlier in this section. (b) By definition 'ijJ has the form
'ijJ(z)=Az 2+Bz+C,
A,B,C E K[V],
with A not vanishing on V and A a morphism A : V ----t A l such that for all P E V(K), the point A(P) is a point offormal period n for the quadratic polynonomial
'ijJp(z)
=
A(p)z2
+ B(P)z + C(P)
E
K[z].
4. Families of Dynamical Systems
160
For any point P E V(K), let Ip(z) be the linear fractional transformation
2z - B(P)
fp(z) =
2A(P)
.
Note that Ip is well-defined for every P E V(K), since we have assumed that A is nonvanishing on V. We define a map T/ from V to A,.2 by the formula
T/(P) = (cp,ap)
with
{
cp
=
A(P)C(P) - ~B(p)2
p
=
A(P)A(P) + ~B(P).
Cx
+ ~B(P),
(4.16)
Note that T/ : V --+ A,.2 is a morphism, Le., it is given by everywhere-defined algebraic functions on V. We are now going to verify that the image of T/ is the curve Y1 (n). The computation that we performed in deriving formula (4.12) shows that
'l/;{:(z)
=
z2 + cpo
To ease notation, we let ¢p(z) = z2 + Cp . We also let n,P, ~,p, \lIn,P, and \lI~,p be the period and dynatomic polynomials for ¢p and 'l/;p, respectively. Note that the period polynomials are related by
\lI~~p(z) = Ip1o('l/;p(z)-z)olp = (fp1o'l/;plpt(z)-z = ¢p(z)-z = n,p(z). Hence the dynatomic polynomials also satisfy
(\lI~,p)fp = n,P.
(4.17)
We are given that A(P) E Per~('l/;p), which is equivalent to saying that A(P) is a root of \lI~,p. It follows from (4.17) that
I p 1(A(P)) = A(P)A(P) + ~B(P) is a root of ~ p, and thus is in Per~ (¢ p ). This proves that the image of the map T/ defined by (4.1'6) is contained in Y1 (n), so T/ is a morphism from V to Y1 (n). Further, this map T/ respects the identification of Ydn) with Formal(n) from (a), since it takes P to a pair (cp,ap) that is isomorphic to the pair ('l/;P(Z),A(P)) via the conjugation Ip E PGL2 (K ). Finally, it is clear from the construction that the map T/ is uniquely determined as a map (of sets) from V(K) to Formal(n), so it is the unique morphism V --+ Y1 (n) satisfying (4.15). (c) The definition (4.16) of T/ shows immediately that T/ is defined over K, since all of A, B, C, and A are assumed to be defined over K. D
Remark 4.12. In the language of algebraic geometry, Theorem 4.11 says that Y1 (n) is a coarse moduli space. In fact, the curve Y1 (n) is actually a fine moduli space for all n 2: 1; see Exercise 4.18. The underlying reason is that there are no nontrivial elements ofPGL 2 that fix a quadratic polynomial and its points of formal period n, i.e., the moduli problem has no nontrivial automorphisms.
4.2. Quadratic Polynomials and Dynatomic Modular Curves
4.2.3
161
The Dynatomic Modular Curves X 1(n) and Xo(n)
The curve YI (n), and by extension X I (n), has the interesting property that the rational map ¢ acts on the points of YI (n) via the map
(y, z)
f------7
(y, Z2
+ y) =
(y, ¢y(z)).
(4.18)
By abuse of notation, we will also use ¢ to denote the map (4.18). This map is welldefined, since if ex is a point of formal period n for the polynomial ¢y(z), then ¢y (ex) is also a point offormal period n for ¢Y' Further, the nth iterate ¢n is the identity map on YI (n) and Xl (n), so Aut(YI (n)) and Aut(X I (n)) contain subgroups of order n generated by ¢. In general if V is any algebraic variety and if G c Aut(V) is any finite group of automorphisms of V, then there exist a quotient variety W = VI G and a projection map n : V ----t W with the property that n(y) = n(x) if and only if there is agE G such that y = g(x). (See, for example, [321, II §7].) Further, if V is defined over a field K and ifG is K-invariant,2 then VIG is defined over K. The construction of VI G when V is a nonsingular curve is particularly simple. Each 9 E G induces an automorphism of the function field g* : K(V)
----t
K(V)
fixing K, so we may view G as a subgroup of Aut(K(v)1K). Then one shows that the fixed field K (V) G of G has transcendence degree lover K, so there exists a unique nonsingular projective curve WI K with function field K(W) = K(V)G. The inclusion K(W) c K(V) defines the projection n : V ----t W making W into the quotient of V by G. Finally, if G is K -invariant, one shows that W has a model defined over K. Remark 4.13. We note that taking the quotient of a variety by an infinite group of automorphisms, as we will need to do in Section 4.4, is considerably more difficult than taking the quotient by a finite group of automorphisms. Indeed, in the infinite case it often happens that the quotient does not exist at all in the category of varieties.
Definition. With notation as above, we let Yo (n) be the quotient of YI (n) by the finite subgroup of Aut(YI(n)) generated by ¢. Similarly, Xo(n) is the quotient of XI(n) by the finite subgroup of Aut(XI(n)) generated by ¢. By construction, the points of Yo (n) classify isomorphism classes ofpairs (¢, 0), where ¢ is a quadratic polynomial and 0 is the orbit of a point of formal period n. The moduli-theoretic interpretation of the projection map from YI (n) to Yo (n) is
2The elements in G are morphisms V - t V that are defined over some extension of K. The Galois group Gal(K/ K) acts on G by acting on the coefficients of the polynomials defining the maps in G. We say that G is K -invariant if each element of Gal( K / K) maps G to itself.
162
4. Families of Dynamical Systems
Example 4.14. We have seen in Example 4.9 that X I (2) and X I (3) are rational curves, i.e., they are isomorphic to pI , so their quotient curves X o(2) and X o(3) must also be rational curves. However, it is still of interest to make the quotient maps explicit. We do this for X I (2) and leave X I (3) as Exercise 4.19. The affine curve YI (2) has equation
YI (2) : z2 + Z + Y + 1 = 0, where a point (c, 0: ) E YI (2) corresponds to the quadratic map cPc (z ) = z2 + c and point 0: E P er; (cPc) . The automorphism cP ofYI (2) is given by (y, z) t---; (y, Z2 + y). From general principles we know that cP maps Y1 (2) to itself, or we can see this explicitly by the formula
(z2 + y)2 + (Z2
+ y) + y + 1 = (z2 + Z + Y + 1)(z2 - Z + Y + 1).
The affine coordinate ring of YI (2) is
q z,y] rv C[ ] (z2 + Z + Y + 1) = z, since we can express y in terms of z as y = - z2 - Z - 1. In terms of z alone, the automorphism cP ofYI (2) corresponds to the map on q z] given by
z
t------>
Z2 + Y = Z2 + (- Z2 - Z - 1) = - z - l.
So the affine coordinate ring of the quotient curve Yo(2) is the subring of C ]»] that is fixed by the automorphism z t---; - z - 1. Since the map YI (2) ~ Yo (2) has degree 2, we look for an invariant quadratic polynomial in z. Setting
Az 2 + B z + C = A( - z - 1)2 + B ( -z - 1) + C and equating coefficients, we find that A = B . Hence the invariant subring is
q z2 + z], which is the affine coordinate ring ofYo(2).
Suppose now that (cP, {0:,,8}) is a quadratic polynomial and an orbit consisting of two points . How does this correspond to a point on Yo(2) whose coordinate ring is z2 + z]? First, we use a PGL 2 conjugation as in (4.12) to put cP into the standard form cPc(z) = z2 + c. Then 0: and ,8 are related by
q
,8 = cPc(O:) =
0:
2 +c
Notice that this implies that 2 0: + 0: = (,8 - c)
and
0:
= cPc ((J) = ,82 + c.
+ (,82 + c) =
,82
+ ,8.
Letting c = -0: 2 - 0: - 1 and using the isomorphisms YI (2) ~ Al and Yo(2 ) ~ Al given earlier, we have the following commutative diagram: (c,a).-.a
4.2. Quadratic Polynomials and Dynatomic Modular Curves
163
Remark 4.15. Morton [307] describes an alternative method for finding an equation for the curve Yo(n ) and the map Yi (n) --+ Yo(n ). The roots of ~ ( e, z) = can be grouped into orbits, say 0'1 , .. . , a r are representatives for the different orbits, so
°
The points in each orbit all have the same multiplier, and we define a polynomial r
On(e, x) =
IT (x - A,p(ai ))' i =l
Then one can show that On (c, X) E Z [e, X] and that the equation
gives a (possibly singular) model for Yo(n). Using this model, the natural map from Yt{n ) to Yo(n ) is
(y, z)
t-------;
(y, ( ¢~) /(z) ).
See Morton 's paper [307] and Exercise 4.13.
Remark 4. 16. The dynatomic modular curves Y1 (n) and Yo(n ) defined in this section are analogous to the modular curves that appear in the classical theory of elliptic curves. Briefly, the elliptic modular curve y 1ell(n) classifies isomorphism classes of pairs (E, P), where E is an elliptic curve and PEE is a point of exact order n . Similarly, the elliptic modular curve y~ll ( n ) classifies pairs (E, C ), where E is an elliptic curve and C c E is a cyclic subgroup of order n. The group (Z/nZ)* acts on y 1ell (n ) via m
* (E , P) =
(E , [m]P)
for
m E (Z /nZ)*,
and the quotient ofy{lI (n ) by this action is y~lI (n ) . The reader should be aware that standard terminology is to write
for elliptic modular curves. But since in this book we deal exclusively with dynatomic modular curves, there should be no cause for confusion. In situations in which both kinds of modular curves appear, it might be advisable to use identifying n superscripts such as y t (n) and y 1ell(n) to distinguish them. We showed earlier that X 1 (1), X 1 (2), and Xl (3) are all (irreducible) curves of genus 0, and similar explicit computations show that Xl (4) has genus 2 and Xl (5) has genus 14. See [171, 305, 309] and Exercise 4.20. The geometry of X 1 (n) and Xo(n ) for general n is described in the following theorem, which is an amalgamation of results due to Bousch and Morton.
4. Families of Dynamical Systems
164
Theorem 4.17. (a) The affine curve YI (n) defined by the equation
is nonsingular. (b) The dynatomic modular curves Xl (n) and Xo(n) are irreducible. (c) The projection map
(z, c) - > c, exhibits X I (n) as a Galois cover of p l. The Galois group is maximal in the sense that it is the appropriate wreath product, cf Section 3.9. (d) Let sp denote the Euler totient function (not to be conf used with the rational map 1» , let f..L be the Mobius function, and let K, be the function
(This is essentially half the degree of 4" n" cf Remark 4.3.) Then the genera ofX I (n) and X a(n) are given by the formulas
genus X j ln ) = 1 + n ; 3 K,(n) -
L
~
mK,(m)
m in, m < n
genusXo(n)
= 1 + n 2~ K,(n) 3
~
L
K,(m)
min, m
4~ L
f..L ( : )
m 2 / .
2
m in , m odd n / m even
(Ifn is odd, then the final sum in the formula for the genus of Xo(n) is empty.) Proof The properties of YI (n) and Xl (n) were originally proven by Bousch [83] , with subsequent proofs by Lau and Schleicher [261] using analytic methods and Morton [307] via algebraic arguments. The latter two papers give various generalizations, including results for the dynatomic modular curves associated to maps of the form zd + c. The formula for the genus of Xo(n) is due to Morton [307]. 0 Remark 4.18. The genera of Xl (n) and Xo(n) grow rapidly; see Table 4.2.
n genus Xj In ) genusXo (n)
I 0 0
2 0 0
3 0 0
4 2 0
5 14 2
6 34 4
7 124 16
8 285 32
9 745 79
10 1690 162
Table 4.2: The genera ofthe dynatomic modular curves X I (n) and Xo(n) for z2 + c.
4.2. Quadratic Polynomials andDynatomic Modular Curves
165
Res(i, ;) = 4e + 3 Res(i, ;) = -16e2 - 4e - 7 Res(i, :) = -16e2 + 8e - 5 Res(i, ;) = -256e4 - 64e3 - 16e2 + 36e - 31 Res( i, ~) = -16e2 + 12e - 3 Res(; , :) = -(4e + 5)2 Res(;, ~) = -(16e2 + 36e + 21)2 Res(; , ~) = -(64e3 + 128e2 + 72e + 81)3
Table 4.3: The first few bifurcation polynomials for z2 + e.
4.2.4
Bifurcation, Misiurewicz Points, and the Mandelbrot Set
When does the polynomial ¢( z) = Z2 + e have a point of formal period n whose exact period m is strictly less than n? This will occur if and only if~ (z) and ;'" (z) have a commonroot, so if and only if e is a root of the resultant equation Res(~(z),
:r,(z)) = O.
Note that this is a polynomial equation for the parameter e. We list the first few given in the last section examples in Table 4.3. Thus the polynomial¢(z) = Z2 is the only example with a fixed point of formal period 2. Similarly, the polynomial ¢(z) = z2 - ~ has a point offormal period 4 whose exact period is 2. One pattern that is apparent from even the small list in Table 4.3 is the fact that ifmln, then Res(~(z), :r,(z)) is the m th power ofa polynomial in Z[e]. See Exercises 4.7 and 4.12 for a descriptionof the m th root of Res( ~ (z), ;'" (z)). The roots of Res( ~ (z), :r, (z)) are special points in the Mandelbrot set, which we recall is the subset of the e-plane given by
i
M = {e E C : ¢n(o) is bounded as n
-->
oo}.
Alternatively, the Mandelbrot set M is the set of e for which the Julia set J (¢c) is connected. See Remark 1.34 and the picture of the Mandelbrot set (Figure 1.2) on page 27. The solutions to the equation Res(~(z),
;',,(z)) (e) = 0
4. Families of Dynamical Systems
166
are called bifurcation points.' They connect the components of M 's interior (the bulbs of M). For example,the point e = - ~ connects the main cardioid of M to the disk to its left. It is clear that the bifurcation points are algebraic numbers. Beyond that, little is known about their arithmetic properties, although Morton and Vivaldi conjecture that the bifurcation points of type (m,n) are all Galois conjugate to one another. (See Exercise 4.12.) An elementaryproperty of the Mandelbrotset M , which we now prove, is that it is contained in a disk of radius 2. Note that M is not contained in any smaller disk, since
0 ~-2 ~2 ~2 showsthat - 2 E M . Proposition 4.19. The Mandelbrot set is contained in the disk ofradius 2,
Me Proof. Supposethat
{eE c: lei:::; 2}.
lei > 2 and let z« =
We have IZl l = lei > 2 by assumption, so (4.19) and induction tell us first that IZnl is an increasing sequence, and indeed that
Hence IZnl ->
00 ,
so e is not in M .
o
The set of quadratic maps whose critical points are preperiodic, but not periodic, defines an important subset of the Mandelbrot set. Definition. A point e E C is called a Misiurewicz point if 0 is strictly preperiodic for
= - 2 and c = i. Thus for c = -2 we have 0 -> -2 -+ 2 -> 2, so c = -2 has type (2,1). Similarly for c = i we have 0 -> i -+ i-I -> -i -> i-I, so c = i has type (2,2).
Example 4.20. Examples of Misiurewicz points include c
Remark 4.21. If c is a Misiurewiczpoint, then the map
4.2. Quadratic Polynomials and Dynatomic Modular Curves
167
It is clear that every Misiurewicz point is contained in the Mandelbrot set M , since preperiodicpoints certainlyhave bounded orbit. This sufficesto prove an arithmetic result.
Proposition 4.22. The set ofMisiurewicz points is a set ofbounded (absolute) height in Q. More precisely, the height ofa Misiurewicz point 1 satisfies H (,) :S 2. Hence there are only finitely many Misiurewicz points defined over any given numberfield. Proof A Misiurewicz point C = 1 is the root of a polynomial of the form
for some m ~ 1 and n ~ 1. These are monic polynomials with coefficients in Z, so not only is 1 an algebraic number, it is an algebraic integer. Further, the minimal polynomialof 1 is a factor of Mm,n(c). On the other hand, every root of Mm,n(c) is in the Mandelbrotset M, since if c is a root of Mm,n(c), then 0 is preperiodicfor ¢e, so it certainly has bounded orbit. It follows from Proposition4.19 that every Galois conjugateof 1 has absolute value at most 2. Let K = Q( I) and let 11, ... Lr: be the full set of Galois conjugates of I ' Then i
II
HK(r ) =
r
max{1,lll vr" =
vEM K
II max{l, lril} :S 2
r
.
i= l
o
Takingthe r 1h root yields H (r ) :S 2.
We now describe an analytic characterization of Misiurewicz points. It depends on the following deep result giving an analytic uniformization of the complementof the Mandelbrot set. Theorem4.23. (Douady-Hubbard) There is a conformal isomorphismfrom the exterior of the unit disk to the complement ofthe Mandelbrot set,
e: {w E C : Iwl >
I} ~ C " M .
Proof It is not hard to show that for all sufficiently large z (dependingon c) there is
a consistent way to choose square roots so that the limit
We(z) = lim 2\/¢~(Z) n -> oo
converges and defines a holomorphic function We on some region Izi > R e . (See ehas an analytic continuation Exercise 4.15.) If c ~ M, then one can prove that W to C <, M. The isomorphism in the theorem is the inverseof the map
e
C" M
-t
{w E C : Iwl > I} ,
See [142. 143, 141], [43, §9.1O], or [95, VIII §§3,4] for details.
o
Remark 4.24. An immediateconsequenceof Theorem4.23 is the connectivityof the Mandelbrot set, since the theorem implies that lP'I (C) <, M is simply connected.
168
4. Families of Dynamical Systems
The unifonnization map () from Theorem 4.23 can be used to give an analytic description of the Misiurewicz points.
{w E C : Iwl > I} ~ C "M described in Theorem 4.23. Consider the doubling map x I-t 2x on Q/7l... Let t E Q/7l.. be preperiodic, but not periodic, for the doubling map. Let m and n be the smallest positive integers for which we can write t in the form Theorem 4.25. Let () be the isomorphism
(The fraction need not be in lowest terms. See Exercise 4.16.) Then the limit 21ri t Ct = lim ()(re ) r-tl+
exists and is a Misiurewicz point of type (m, n) in M, although distinct values of t E Q/7l.. may yield the same Misiurewicz point. Remark 4.26. For a given t E Q/7l.., the "spider algorithm" [213] can be used to compute Ct numerically. The spider algorithm is mainly topological and combinatorial in nature, although the limiting process that yields Ct E C is analytic.
4.3
The Space Rat, of Rational Functions
The set of quadratic polynomials {A z 2 + Bz + C} has dimension three, since it may be identified with the set of triples (A, B, C) with A =I- O. In fancier language, the space of quadratic polynomials is equal to the algebraic variety
{(A,B,C) E A3 : A =I-
o].
We have seen in Section 4.2.1 that every quadratic polynomial can be conjugated to a polynomial of the form z2 + C, and that polynomials with different C values are not conjugate to one another. Thus the space of conjugacy classes of quadratic polynomials has dimension 1. It may be identified with the variety A I. In the next few sections we study analogous parameter spaces for more general rational maps and their conjugacy classes. We begin in this section by explaining how the set Ratd of rational maps of degree d has a natural structure as an algebraic variety and how the natural action of the algebraic group PGL 2 on Rat., is an algebraic action. Then in Section 4.4 we discuss (mostly without proof) how to take the quotient of Rat, to construct the moduli space Md of conjugacy classes of rational maps of degree d. We continue in Section 4.5 by describing a natural collection of algebraic functions on Md that are created using symmetric functions of multipliers of periodic points. These functions can be used to map Md into affine space. Finally, in Section 4.6 we use these functions to prove that M 2 is isomorphic to A 2 . A rational map ¢ : pI ~ pI of degree d is specified by two homogeneous polynomials
4.3. The Space Rat, of Rational Functions
169
such that F and G have no common factors, or equivalently from Proposition 2.13 , such that the resultant Res( F, G ) does not vanish. Thu s a rational map of degree d is determined by the 2d + 2 paramet ers aD, a 1, ... , ad, bo, b1, . . . , bd. However, if u is any nonzero number, then [uF,uG] = [F, G], so the 2d + 2 parameters that determine ¢ are really well-defined only up to homogeneity. This allows us to identify in a natural way the set of ration al maps of degree d with a subset of projecti ve space . , ad), let To ease notation, for any (d + l l-tuple a = (aD,
Fa(X , Y) = aoX d + a1Xd - 1y
+
+ ady d
be the associated homogeneous polynomial. Similarly, if a and bare (d + 1)-tuples, we write [a, b] E JP'2d+ l for the point in projective space whose homogeneous coordinates are [aD, ... , ad, bo, · ··, bdl.
Definition. The set of rational functions ¢ : JP'1 ---+ JP'1 of degree d is denoted by R atd. It is naturally identified with an open subset ofJP'2d+l via the map {[a, b ] E JP'2d+l : Res(Fa ,Fh )
-# a} .z;
Ra t d,
[a, b] 1-----' [Fa, Fh]. The collection of rational maps Rat d' which a priori is merely a set, thus has the structure of a quasiprojective variety. In fact, Rat., is an affine variety, since it is the complement of the hypersurface Res(Fa , F h ) = ain the proje ctive space JP'2d+ l .
Proposition 4.27. The variety Rat., is an affine variety defined over Q. The ring of regularfun ctions Q [Ratd] ofRat d is given explicitly by
Equivalently, Q[Ratd l is the ring ofrationalfunctions ofdegree a in the localization ofQ [ao, al, . . . ,ad , bo, bi , .. . , bd]at the multiplicatively closed set consisting ofthe nonnegativepowers ofRes(Fa , Fh ) . Proof We remind the reader that in general, if F E K[Xo, . . . , X r ] is a homogeneous polynomial of degree ti , then the complement of the zero set of F ,
is an affine variety of dimension nal function
7'.
(See [198, Exercise 1.3.5].) Explicitl y, each ratio-
with i o + i 1 + ...+ i r = n is a regular (i.e., everywhere well-d efined) function on V. There are ( r ~n) such functions , and together they define an embedding
170
4. Families of Dynamical Systems
of V into affine space. The affine coordinate ring of V is the ring of polynomials in these lio ... i r ,
K[V] = K [f.. .].'to +.'1,1 +"'+Zr=n . . "0"'1··'''1' In the language of commutative algebra,
K[V] = K[Xo, ... ,Xn , 1/F](O) is the set of rational functions of degree 0 in the localization of K[Xo, ... , X n ] at the multiplicatively closed set (Fik~_o. Applying this general construction to
o
gives the results stated in the proposition.
Remark 4.28. The geometry of Ratd (C), especially near its boundary, presents many interesting problems. See [121, 122,369,370]. Example 4.29. Let
be the resultant of aoX 2 + alXY + a2y2 and bOX 2 + b1XY + b2y 2. Then the collection of 84 functions
gives an embedding of Rat2 into A84. Of course, this is not the smallest affine space into which Rat2 can be embedded. Projecting onto appropriately chosen hyperplanes, there is certainly an affine embedding of the 5-dimensional space Rat2 into A 11; see [198, Exercise IV.3.11]. Example 4.30. The set of rational functions of degree 1 is exactly the set of linear fractional transformations,
Rat,
= PGL 2 = {[aX + (3Y, "IX + 8Y] : a8 -
(3"1
f- o} C
1P'3.
We note that PGL 2 is not merely a variety, it is a group variety, which means that the maps
PGL 2 X PGL 2 -------; PGL 2,
(fI, h)
1-----+
and
fIh
defining the group structure are morphisms.
PGL 2
-------;
PGL 2 ,
1
1-----+
1-1,
4.3. The Space Rat., of Rational Functions
171
Each point of Rat., determines a rational map jp'l ---> jp'l. Further, as we vary the chosen point in Ratd, the rational maps "vary algebraically." We can make this vague statement precise by saying that the natural map
¢ : jp'l X Ratd ([X,Y],[a,bJ)
Ratd,
------t
jp'l X
f-----*
([Fa(X,Y),Fb(X,Y)],[a,bJ),
(4.20)
is a morphism of varieties. The following definition is useful for describing families that vary algebraically.
Definition. Let V be an algebraic variety. The projective line over V is the product
A morphism 'l/J : jp'~ ---> jp'~ over V is a morphism that respects the projection to V, i.e., the following diagram commutes, where the diagonal arrows are projection onto the second factor:
./
'\. V Then 'l/J can be written in the form
'l/J = [F(X, Y), G(X, Y)], where F, G E K(V)[X, Y] are homogeneous polynomials with coefficients that are rational functions on V. The degree of'l/J is the degree of the homogeneous polynomials F and G.4 Any morphism .\ : V ---> W of varieties induces a natural morphism jp'~ ---> jp'tv, which, by abuse of notation, we also denote by .\. Thus
.\(P, t)
for (P, t) E jp'~ .
= (p, .\(t))
With this notation, the map (4.20) says that there is a natural morphism
over Ratd. The next proposition says that this ¢ is a universal family ofrational maps of degree d.
Proposition 4.31. Let V be an algebraic variety and let 'l/J : jp'~
------t
jp'~
be a morphism over V ofdegree d. Then there is a unique morphism 4More generally, if'lj! : 1I'~ is the degree of 'Ij!.
-+
1I'~ is a morphism of S-schemes, then 'Ij!*OpN (1) ~ OpN (d) and d
s
s
172
4. Families of Dynamical Systems
such that the inducedmap ,\ : p~
-->
Phatdfits into the commutative diagram ,p
pI
-------t
V
pI
V
lA
lA Phatd
<J>
-------t
(4.21 )
pI
Ra t .,
Remark 4.32. The commutative diagram (4.21) in Proposition 4.31 says that any algebraic family of degree d rational maps W : JIl'~ --> p~ factors through the Ratd family. Thus ¢ : Phatd --> Phatd is a universal family of rational maps of degree d. It is an example of afine moduli space. See [322, 323, 335] for further information about moduli spaces. Proof Let U c V be an affine open subset and write K[U] for its affine coordinate ring. The fact that 'IjJ is a morphism over V implies that it restricts to give a morphism e : JIl'h --> ph over U. This restriction of 'IjJ to ph has the form 'IjJ = [o:oX d + O:IX d- Iy
+ ... + O:d y d, f3oX d + f3 IX d- I y + with 0:0 , .. . , O:d, f30,
+ f3dy d], ,f3d E K[U].
In other words, the coefficients of the polynomials defining 'IjJ are regular functions on the affineopen set U . In particular, if there is a map ,\ : V --> Ratd making(4.21) commute, then it must be given by ,\ = [0:0"" , O:d, {3o, . . . , (3d] at any points of V at which the O:i and f3i are defined and the homogeneous polynomials in (4.21) have no common roots. Now given any point in V , we can find a neighborhood U of that point and O:i , f3i E K[U] as above such that 'l/J : ph --> ph is given by
'l/J ([x , y], t)
=
[o:o(t )x d + ...+ O:d(t)yd, f3o( t)x d + ...+ f3d(t )yd] for
([x, y], t) E ph. (4.22)
Using a natural notation, we abbreviate this by writing
W(P, t ) =
[Fo: (t ) (P), F,6(t ) (P)]
.
In order to make the diagram(4.21) commuteon p h c P~ , we are forced to define,\ on Uby
'\(t ) = [0:0(t),0:1(t), .. " O:d(t ),f30(t ), f3I(t )' ... ,f3d(t )] E p 2d + l .
(4.23)
Further, this ,\ will have the desired properties provided that its image lies in Rat d. However, the fact that 'IjJ is given by (4.22) for every point in Ph implies that for every t E U, the homogeneous polynomials Fo: (t ) and F,6(t ) have no common root in pl. It follows from Proposition 2.13(a) that their resultant
4.3. The Space Ratd of Rational Functions
173
Res(Fn , F(3 ) E K[U] is nonzero at every point in U. Hence the image of the map>. = [a, b] defined by (4.23) is in Ratd . We have now proven that every point in V has a neighborhood U for which there is a map>. : U ----t Ratd making (4.21) commute. Further, the maps on different U must agree on the intersection by the uniqueness discussed earlier. Fitting them together, we find that there is a unique map>. : V ----t Ratd making (4.21) commute . 0 An ongoing theme of this text has been the observation that a pair of rational maps ¢ l , ¢ 2 E Rat., determine (arithmetically) equivalent dynamical systems if they are PGL 2-conjugate, i.e., if there is a linear fractional transformation f E PGL 2 such that ¢2 = ¢{ = f -l ¢f. The conjugation action ofPGL 2 on Rat., is algebraic in the following sense.
Proposition 4.33. The map (4.24) is an algebraic group action ofPGL 2 on Rat., and is defined over Q. This means that the map (4.24) is both a morphism defined over Q and a group action.
Proof The proof is mostly a matter of unsorting the definitions. Let
f = [aX + {3Y" X + bY]
E PGL 2 C
jp'3
and
¢
= [Fa, Fb] E Ratd C
jp'2d+l .
Then
+ (3Y, I'X + oY) - {3Fb(a X + (3Y, I'X + OY), ,Fa(aX + (3Y ,I'X + oY ) + a Fb(aX + (3Y, I'X + OY)] .
¢ I ([X , YJ) = [oFa (aX -
(4.25) The homogeneity of Fa and Fb shows that (4.24) at least gives a well-defined rational map jp'3 X
jp'2d+ 1
---t
jp'2 d+l ,
(J,¢) ~ ¢/ .
Further, an elementary resultant calculation (Exercise 2.7) shows that the resultant of the two polynomials appearing in the righthand side of (4.25) is equal to
It follows that if f E PGL 2 and ¢ E R atd, then ¢I is a well-defined point in Ratd. This proves that the map (4.24) is a morphism. The fact that it is a group action is then a straightforward, albeit tedious, calculation. 0
174
4. Families of DynamicalSystems
Example 4.34. Inprinciple it is possible to explicitly write down the action ofPGL 2 on Ratd, but in practice the expressions become hopelessly unwieldy for even moderate values of d. As illustration, we describe the action for d = 2. Let j
= [aX + (3Y, 'YX + oY]
and
and set
Multiplying out both sides of (4.25) and equating the coefficients of X 2, XY, and y 2 yields the following formulas for the coefficients of Fal and Fbi:
a~ = a 20ao
+ a'Y0al + loa2 - a 2{3bo - a{3'Ybl - {3'Y2b2, a~ = 2a{3oao + (ao + {3'Y)oal + 2'Y02a2 - 2a{32bo - (ao + {3'Y){3b l - 2{3'Yob2, a~ = {32 0ao + {302 al + 03a2 - {33b o - {32 0bl - {302b 2, b~ = -a2'Yao - a'Y2al - 'Y3a2 + a 3bo + a2'Ybl + a'Y2b2, b~ = - 2a{3'Yao - ({3'Y + aohal - 2'Y20a2 + 2a 2{3b o + (ao + {3'Y )ab l + 2a'Yob2, b~ = _{32'YaO - {3'Y0al - 'Y02a2 + a{32b o + a{3ob l + a0 2b2.
4.4
The Moduli Space M d of Dynamical Systems
The intrinsic properties of the dynamical system associated to a rational map ¢ depend only on the PGL 2-conjugacy class of ¢, so it is natural to take the quotient of the space Rat., by the conjugation action ofPGL 2 .
Definition. The moduli space of rational maps of degree d on pI is the quotient space
M d = Ratd / PGL 2 , where PGL 2 acts on Rat., via conjugation ¢f = j-I¢j as described in Proposition 4.33. For the moment, the quotient space Md is merely a set, in the sense that for any algebraically closed field R,
We denote the natural map from Rat., to Md by
In order to endow Md with the structure of a variety, we observe that the action ofPGL 2 on Rat., induces in the usual wayan action ofPGL 2 on the ring of regular functions Q[RatdJ on Ratd. Thus an element R E Q[Ratd] is a function whose domain is the set {¢ : pI ----+ pI} of rational functions on pI, and for j E PGL 2 we define Rf E Q[RatdJ by the formula Rf (¢)
= R(¢f).
4.4. The Moduli Space
Md of Dynamical Systems
175
In this way we obtain a map
so it makes sense to talk about functions in Q[RatdJ that are invariant under the action ofPGL 2 . We write
for the ring ofPGL 2-invariant functions on Ratd. Remark 4.35. For technical reasons, it is sometimes advantageous to replace PGL 2 by a slightly different group. The special linear group is the subgroup of GL 2 defined by
8L 2 = {
(~ ~)
:
aJ -
~1 =
1} ,
and the projective special linear group P8L 2 is the image of 8L 2 in PGL 2 . The groups 8L 2 and P8L 2 act on Rat., via conjugation in the usual way. Geometrically, the actions ofPSL 2 and PGL 2 on Rat., are identical, since it is easy to see that the natural inclusion 8L 2 C GL 2 induces an isomorphism
However, if K is not an algebraically closed field, then the map
need not be an isomorphism; see Exercise 4.22. The following theorem provides the abstract quotient Ratd / PSL 2 with the structure of an algebraic variety. Theorem 4.36. There is an algebraic variety Md defined over Q and a morphism
(4.26) defined over Q with the following properties: (a) The map (4.26) is P8L 2-invariant, i.e., the following diagram is commutative:
P8L 2 x Ratd
fit
Ratd
(f,e/> )--->1/
) Ratd
l(i
C)
Md
In terms ofelements, (
f
E PGL 2 .
176
4. Families of Dynamical Systems
(b) The map on complex points
is surjective and each fiber is the full PSL 2 (C) -orbit ofa single rational map. Thus there is a bijection ofsets
(c) The variety M d is a connected, integral (i.e., reduced and irreducible), affine variety ofdimension 2d - 2 whose ring ofregularfunctions is the ring ofPSL 2 invariant functions on R at d'
(d) Let VIC be a variety and let T : Rat., - t V be a morphism with the property that T(tfyf) = T(tfy) for all tfy E Ratd(C) and all f E PGL 2(C). Then there is a unique morphism t : M d - t V satisfying t( (tfy) ) = T( tfy ). ProofSketch. A full proof of Theorem 4.36 (see [416]) uses the machinery of geometric invariant theory [322] and is thus unfortunately beyond the scope ofthis book. Geometric invariant theory tells us that there is a certain subset OfJP2d+ 1 , called the stable locus , on which the conjugation action of PSL 2 is well behaved. The main part of the proof is to use the P SL 2-invariance of the resultant to verify that
is a PSL 2-invariant subset of the stable locus of JP2d+ l . Then the existence of the quotient variety Md with affine coordinate ring equal to the ring of invariant functions in Q[Ratd] follows from general theorems of geometric invariant theory [322, Chapter 1]. Further, the fact that M d is connected, integral , and affine follows immediately from the corresponding property of Rat., [322, Section 2, Remark 2]. The dimension of Md is computed as dim M, = dim R atd - dim PSL 2 = (2d
+ 1) -
3 = 2d - 2.
This proves (a), (b), and (c). Finally, (d) follows directly from the description of Md and Q[Md] in (c), since the morphism T induces a map T* : Ov - t qRatd], and the assumption that T satisfies T( tfyf) = T(tfy) implies that the image ofT* lies in the ring qRatd]PGL2 (C) = qM d] ' 0
Remark 4.37. Theorem 4.36 says that the quotient Rat d I PSL 2 is an algebraic variety. Milnor [301,302] shows that Ratd (C)1 PGL 2(C) has a natural structure as a complex orbifold, which roughly means that locally it looks like the quotient of a complex manifold by the action of a finite group . Thus its singularities are of a fairly moderate type, although they can still be quite complicated. However, for rational maps of degree 2, we will see in Section 4.6 that not only is M 2 nonsingular, it has a particularly simple structure.
4.4. The Moduli Space Md of DynamicalSystems
177
Remark 4.38. According to Theorem 4.36(c), the affine coordinate ring of Md is the ring ofPSL 2-invariant functions Q[RatdjPSL2. However, this is the same as the ring of PGL 2-invariant functions, since it suffices to check for invariance by the action ofPGL 2(C) = PSL 2(C) on an element ofQ[Ratd]' Remark 4.39. The moduli space Md is the quotient of Rat., by the conjugation action of the group PSL 2 . In particular, if K is any algebraically closed field, then the set of points Md(K) is exactly the collection of cosets of Ratd(K) by the conjugation action ofPSL 2(K) = PGL 2(K). However, if K is not algebraically closed, then the natural map
is generally neither injective nor surjective. The correct description of Md(K), at least if K is a perfect field, is
M (K) - {(¢) d
E
-
M (K)' for every T E _Gal(K/ K) there is an d · iT E PGL2(K) such that T(¢) = ¢f
} T
•
The field of moduli of a rational map ¢ E Ratd(K) is the smallest field L such that (¢) E Md (L). Fields of moduli and related questions are studied in detail in Section 4.10. In particular, see Example 4.85 for a map ¢ whose field of moduli is Q, yet ¢ is not PGL 2(C)-conjugate to any map in Ratd(Q). The moduli space Md classifies rational maps up to conjugation equivalence, just as we want, but it has the defect that it is an affine variety. It is well known that if possible, it is generally preferable to work with projective varieties. How might we naturally complete Md by filling in extra points "at infinity"? Note that we should not do this in an arbitrary fashion. Instead, we would like these extra points to correspond naturally to degenerate maps of degree d. One possibility is simply to start with all of ]p2d+l and take the quotient by the conjugation action of PSL z. Unfortunately, there is no natural way to give the quotient ]p2d+l / PSL 2 any kind of reasonable structure. For example, it is not a variety. So ]p2d+l is too large. Ideally, we would like to find a subset S c ]p2d+l with the following properties:
1. S contains Ratd. 2. PSL 2 acts on S via conjugation. 3. There is a variety T and a morphism S
----+
T that induces a bijection
4. The quotient variety T is projective. Geometric invariant theory gives us two candidates for S. The smaller candidate is the largest variety satisfying (3), but its quotient T may fail to be projective. The larger candidate has a projective quotient, but the map S(C)/ PSL 2(C) ----+ T(C) in (4) may fail to be injective. The following somewhat lengthy theorem describes the application of geometric invariant theory to our situation, that is, to the conjugation action of PSL 2 on ]p2d+ 1 .
178
4. Families of Dynamical Systems
Theorem 4.40. There are algebraic sets
Ratd C Rat d C Rat dS c jp'2d+l with the following properties: (a) The conjugation action ofPSL 2 on Ra t., extends to an action ofPSL 2 on Rat d and Rat ds . (b) There are varieties M S and M SS and morphisms
( . ) : Rat d ---+ M d
(4.27)
and
that are invariant for the action ofPSL2 on Rat d and Rat ds . The varieties M S and M SS and morphisms (4.27) are defined over iQ. (c) Twopoints [a, b] and [a' , b'] in Ratd(C) have the same image in Md(C) if and only if there is an f E PSL 2 (C) satisfying
[a',b'] = [a, b]f. Thus as a set, M;i(c) is equal to the quotient ofRatd(C) by PSL 2 (C). (d) Two points [a, b] and [a', b'] in RatdS(C) have the same image in M;iS(C) if and only if the Zariski closures oftheir PSL 2 (C)-orbits have a point in common,
{[a, bJI : f
E PSL 2(C)}
n {[a', b/]I : f
Equivalently, they have the same image if and only map f : {t E C : 0 < It I < I} ---+ SL2 (C) such that lim
t-O
E PSL 2(C)}
if there
=I 0.
is a holomorphic
[a, b ]f t = [a', b/].
(e) M;i is a quasiprojective variety and M;is is a projective variety. (f) (Numerical Criterion) A point [a, b] E jp'2d+I (C) is not in RatdS if and only there is an f E PSL 2 (C) such that [a', b'] = [a, bj! satisfies
,
ai
.
= 0 for all z :S
d-l
-2-
and
I
bi
Similarly, [a, b) is not in Rat dif and only [a', b'] = [a, bj! satisfies ai
= 0 for all i <
d-l
-2-
(g) M;i is isomorphic to M;is
and
.
= 0 for all z :S
d+l
-2-'
if
(4.28)
if there is an f E PSL 2(C) such that
b, = 0 for all i
d+l < -2-'
(4.29)
if and only if d is even.
Proof The proofofthis theorem is beyond the scope of this book. However, we note that half of (g), namely M d ~ M':t for even d, follows directly from the numerical criterion in (f), since for even d the criteria (4.28) and (4.29) are the same. See [416] for a proof of a general version of Theorem 4.40 over Z. See also [30 I, 302] for a similar construction over Co D
4.5. Periodic Points,Multipliers, and Multiplier Spectra
179
Remark 4.41. In Theorem 4.40, the set denoted by Ratd is called the set of stable rational maps, and the set denoted by Ratd8 is called the set of semis table rational maps. Note that points in these sets need not represent actual rational maps of degree d on pl. The intuition is that points in Rat d and Ratd8 that are not in Ratd correspond to maps that want to be of degree d but have degenerated in some reasonably nice way into maps of lower degree. Remark 4.42. Theorem 4.40(c) says that the stable quotient Md(C) has the natural quotient property, since its points correspond exactly to the PSL 2(C)-orbits of points in Ratd(C). Quotient varieties with this agreeable property are calledgeometric quotients. The semistable quotient M d8 (C) has a much subtler quotient property. According to Theorem 4.40(d), points in Ratd8(C) with distinct PSL 2(C)-orbits give the same point in M d8 (C) if their orbits approach one another in the limit. Quotients of this kind are called categorical quotients. As compensation for the lessintuitive notion of categorical quotient, Theorem 4.40(e) tells us that M d8 is projective, so M d8 (C) is compact. Finally, Theorem 4.40(g) says that if d is even, then M d and M d8 coincide, so in this case, the moduli space Md has a projective closure with a natural (geometric) quotient structure. Remark 4.43. Applying the full machinery ofgeometric invariant theory to the action of PSL 2 /Z on pid+l, it is possible to prove versions of Theorems 4.36 and 4.40 over Z. In other words, there is a filtration of schemes over Z,
such that the group scheme PSL 2 /Z acts on each of these schemes and such that the quotient schemes Md C M d C M 8 8 exist in a suitable sense. In particular, Theorems 4.36 and 4.40 are true with Q replaced by the finite field JF p- The proof is similar to the proof over Q, but requires Sheshadri's theorem that reductive group schemes are geometricallly reductive. See [416] for details.
4.5
Periodic Points, Multipliers, and Multiplier Spectra
The moduli space Md of rational functions modulo PSL 2-equivalence is an affine variety whose ring of regular functions
consists of all regular functions on Rat., that are invariant under the action of PSL 2 , or equivalently under the action ofPGL 2(C); see Remark 4.38. Abstract invariant theory, as described in the proof sketch of Theorem 4.36, says that there are many such functions. In this section we use periodic points to explicitly construct a large class of regular functions on Md-
4. Families of Dynamical Systems
180
Example 4.44. Let ¢ E Ratd(C) be a rational function of degree d defined over the complex numbers. Associated to each fixed point P E Fix( ¢) is its multiplier Ap (¢) E
The points in Fix( ¢) do not come in any particular order, so the set of multipliers for the fixed points,
is an unordered set of numbers, but as a set, it depends only on {¢), the PGL 2 equivalence class of ¢. Hence if we take any symmetric function of the elements in this set of multipliers, we get a number that depends only on {¢). The elementary symmetric polynomials generate the ring of all symmetric functions, so we define numbers
by the formula
IT PEFix(¢l
d+l
(T + Ap(¢)) = I~>·i(¢)Td+l-i. i=O
In other words, the quantity (Ji (¢) is the i th elementary symmetric polynomial of the multipliers ApI (¢), ... ,APd+1 (¢). From this construction, it is clear that a, (¢f) = (J i ( ¢) for all f E PG L 2 • Further, if we treat the coefficients of ¢ = [Fa, Pb ] as indeterminates, then the fixed points Pi and the multipliers APi ( ¢) of ¢ are algebraic over Q(ao,... , bd) and form a complete set of Galois conjugates, from which it follows that symmetric expressions in the Api (¢), for example the functions a, (¢), are in the field Q( ao, ... ,bd)' With a bit more work, which we describe in greater generality later in this section, one can show that the multipliers Api (¢) are integral over the ring
and hence that (Ji(¢) E Q[Md]' Thus symmetric polynomials in the multipliers of the fixed points of ¢ are regular functions on the moduli space Md.
4.5. Periodic Points,Multipliers, and Multiplier Spectra
181
Example 4.45. We illustrate the construction of Example 4.44 for rational maps of degree 2. As usual, we write
The map rP has three fixed points P1 , P2 , P2 , and after much algebraic manipulation one finds that the first elementary symmetric function of the multipliers,
is given explicitly by the horrendous-looking formula
Notice that the denominator of al (rP) is Res(Fa , F b ) , so al (rP) is in iQ[Ratd]. It is far less obvious that this expression for al (rP) is PGL 2-invariant. One can verify directly that al (rP) = al (rP f ) by checking that al (rP) does not change when ao, ... .bz are replaced by the expressions a~, ... ,b; described in Example 4.34. We leave this task to the interested reader who has, we hope, access to a suitably robust computer algebra system. We have used the set of multipliers of the fixed points of a rational map rP to create PGL 2-invariant functions on Ratd' More generally, we can use the multipliers associated to periodic points of any order to create such functions. Recall from Section 4.1 (page 149) that for any rP E Rat, we write
with homogeneous polynomials F¢,n, G¢,n E K[X, Y] ofdegree d", (See also Exercise 4.9.) Then the set Per., (rP) of n-periodic points of rP are the roots of the n-period polynomial
¢,n(X, Y)
= YF¢,n(X, Y)
- XG¢,n(X, Y),
and the set Per~ (rP) of formal n-periodic points of rP are the roots ofthe nthdynatomic polynomial of rP, ¢,n * (X , Y) = II(YF¢,k, (X Y)-XG ¢,k, (X y))Jl(n/k) =
kin
II ¢,k, (X y)Jl(n/k) , kin
where we proved in Theorem 4.5 that ;;',n is a polynomial. The polynomial ¢,n is homogeneous of degree d" + 1. For the purposes of this section, it is convenient to let Per n ( rP) be a "set with multiplicity" in the sense that a point appears in Per n ( rP) according to its multiplicity as a root of ¢, rr- Similarly, we denote the degree of ;;',n by Vd( n) (see Remark 4.3) and we assume that points appear in Per~ (rP) according to their multiplicity as roots of ;;',n'
182
4. Families of Dynamical Systems
Definition. Let ¢ E Rat d. The n-multiplier spectrum of¢ is the collection of values
The formal n-multiplierspectrum of ¢ is the analogous set of values A~( ¢) = {>. p(¢) : P E Per~ ( ¢)}.
In both sets, the multipliers are taken with the appropriate multiplicity.
Example 4.46. Let ¢(z) = zd with d ~ 2. Then Pern (¢ ) = {O, oo} U JLdn-l consists of the points 0, 00, and the (dn - 1) lh roots of unity. It is easy to check that >'o(¢) = >'oo(¢) = 0, and for ( E JLdn-l we have
dn_l copies An (¢) = {O ,O,dn ,dn , .. . , dn }.
Hence
~
And if n ~ 2, then A~ (¢) consists of
Example 4.47. Let ¢(z) = z2 + bz. Then Perl (¢) = {O,l - b,oo}
and
A1 (¢) = {b, 2 - b, O}.
Next we compute * ¢2(z) - Z 2 <1>4>,2= ¢(z) -z =z + (b+l )z+b +1.
The two points of formal period 2 are the roots of <1>;;,2'
Letting
0:
and (3 denote these two values, we substitute them into
to compute their multipliers, which tum out to be identical ,
Thus
A; (¢) = {4 + 2b -
b2 , 4 + 2b - b2 }
.
The multiplier spectra A n (¢) and A~ ( ¢) depend only on the PGL 2-equivalence class of ¢, so we can use them to define functions on M d.
4.5. PeriodicPoints, Multipliers, and Multiplier Spectra
183
n) Definition. Let ¢ E Ratd and n 2:: 1. Define quantities O"i (¢) for 0 :S i :S d" by the relation dn+l dn+ l-i. n) (T +.\) = O"i (¢)T
+1
II
L
AEAn(q,)
i=O
Similarly, define quantities ;~n) (¢) for 0 :S i :S Vd(n) by Vd(n)
II
(T +.\) =
AEA;;(q,)
L
;~n)(¢)Tdn+1-i.
i=O
Example 4.48. Continuingwith Example 4.46, let ¢(z) = zd. Then
II
(T +.\) = T 2(T + dn)dn-l
and
AEAn(q,)
II
(T +.\) = (T + dn)'PW).
AEA;;(q,)
Example 4.49. Continuing with Example 4.47, let ¢(z) = z2 + bz. We computed Al (¢) = {b,2 - b,O}, so
II
(T +.\)
=
(T + b)(T + 2 - b)T = T 3 + 2T 2 + (2b - b2)T,
AEAt{ q,)
which gives l) O"i
O"~l)
= 2,
= 2b-
b2,
O"~l)
= O.
Similarly, using the set A2(¢) computed in (4.30), we find that
II
(T + .\) = (T + 4 + 2b - b2)2,
AEA;(q,)
so
(;-i2 ) =
2b2 - 4b - 8
and
(;-~2) = (4 + 2b _ b2)2.
Theorem 4.50. For ¢ E Ratd, n 2:: 1, and i in the appropriate range, let
0"; n) (¢)
and (;- ~ n) (¢) be the symmetric polynomials ofthe n-multiplier spectra of ¢. (a) Thefunctions and
(4.31 )
are in Q[Ratd], i.e., they are rational functions in the coefficients ao,... , bd of the map ¢ = [Fa, Fb ] with denominators that are a power ofRes(Fa , F b ) . (b) The functions (4.31) are PGL 2-invariant, and hence are in the ring of regular functions Q[Md] ofthe dynamical moduli space Md. Proof We sketch the proof for O";n)(¢) and leave ;~n\¢) as an exercise for the
reader (Exercise 4.26). (a) We write
184
4. Families of Dynamical Systems
using homogeneous polynomials of degree d", Formal properties of the resultant and an easy induction imply that
(See Exercise 4.9.) Thus the rings
oP)
(¢) are equal. This allows us to replace ¢ by ¢n and consider the quantities associated to the fixed points of ¢. To ease notation, for the remainder of this proof we write a, (¢) instead of O"?) (¢). Let L denote the field Q(ao, ... , bd), where we treat ao, ... ,bd as indeterminates. The fixed points of ¢ = [Fa, F b ] are the roots of a polynomial with coefficients in L, so they are defined over an algebraic extension of L. It follows that the fixed points Fix( ¢) and their set of multipliers Al (¢) are Gal(L j L )-invariant sets, so the symmetric polynomials O"i( ¢) of the fixed points of ¢ are Gal(Lj L )-invariant elements of L. This proves that each a, (¢) is in L. Further, it is clear from the construction that a, (¢) is homogeneous in a and b, in the sense that a, (¢) gives the same value for [Fea, Feb] for any nonzero constant c. Thus the a, (¢) are in L(O), where the 0 denotes rational functions of ao, ... ,bd whose numerator and denominator are homogeneous of the same degree. Our next task is to prove that they are regular functions on Ratd, i.e., that they are well-defined at every point of Ratd. In order to do this, we must show that their only poles occur when Res(Fa , F b ) = O. Let
be the ring of polynomials in the indeterminates ao, ... ,bd , and to simplify notation, let r = Res(Fa , Fb ) E A denote the resultant of Fa and Fb. With this notation, Theorem 4.27 says that the coordinate ring of Rat., is Q[Ratd], which equals A[r- 1 ](0), where again the 0 indicates that we take rational functions of degree O. We also note that the polynomial r is irreducible in A. 5 We are first going to prove that O"i(¢) is in the ring A[bdl, r- 1 j. We dehomogenize by setting [X, Y] = [z, 1] and write ¢(z) = Fa(z)j Fb(z). The fixed points of ¢ are the roots of the polynomial ¢( z) - z, or equivalently, the roots of the polynomial
zFb(z)-Fa(z) = bdZd+l+(bd_l-ad)zd+(bd_2-ad_dzd-l+. ·+(bo-adz+ao. (4.32) 5 Indeed, the resultant polynomial is geometrically irreducible, which means that it is irreducible in K[aa, ... , bd ] for any field K, see [436, §5.9].
4.5. Periodic Points, Multipliers, and Multiplier Spectra
185
The roots a 1, . . . , adH ofthis polynomial in L are integral over the ring A [bd1]. For any such root a, the corresponding multiplier is
¢'(a) = Fb(a )Fa (a ) - F~ (a)Fb (a) . Fb (a )2 Let B be the integral closure of A[bdI ] in the field L (a ). It is clear that the numerator of ¢' (a) is in B. We claim that its denominator is a unit in B [r- I ] . Suppose not. Then we can find a maximal ideallfJ C B[r- I ] with Fb(a ) E 1fJ. But a is a root of (4.32) , so Thus z = a is a simultaneous root of
so a standard property of resultants (Proposition 2.13(a)) implies that
But this is a contradiction, since r is a unit in B[r- 1 ] , so it cannot be an element of a maximal ideal. Hence Fb(a ) is a unit in B[r- I ] , and therefore the multiplier ¢' (a ) is in B [r- I ] . We have now shown that each of the multipliers ¢' (a 1), ... , ¢' (ad+ 1) E L is integral over the ring A [b d1 , r - 1 ]. Hence the symmetric polynomial s in these quantities, i.e., (JI (¢)"" , (Jd+1(¢ ), are in L CO) and are integral over A [b d1, r - I ]. It follows that they are in A [b d!, r - 1j CO) , since the ring A [b d1, r - 1] is integrally closed in its fraction field L. A similar argument dehomogenizing [X, Y] = [1, w] shows that the a, are in the ring A[ad1 , r - I ]CO). Therefore
(b) Our earlier calculation (Propos ition 1.9) showed that for every complex point jF2d+I (C) with Res(Pa , Pb) i- 0, the set of multipliers of ¢ = [Fa, Pb] is P GL 2(C)-invariant. Hence the same is true for the quantities a, (¢) for all i. This invariance says the following : Let f (z) = (az + (3) / (ry z + J) be a linear fractional transformation with indeterminate coefficients a, (3, " J and consider the difference
[a, b] E
(4.33)
n
Clearly (4.33) is a rational function in Q(ao, . .. , bd , a , (3" , J) , and the PGL 2 (C)invariance of Si(¢) says that (4.33) vanishes for all choices of ( ~ E GL 2 (C). It follows that (4.33) is identically zero. Finally, Theorem 4.36(c) says that Q[Md ] is the subring ofPGLrinvariant functions in Q[RatdJ (see also Remark 4.38), so in particular the function s (4.3 1) are
in Q[M d]'
0
4. Families of Dynamical Systems
186
Remark 4.51. As noted earlier in Remark 4.43, the moduli space M d exists as a scheme over Z. Theorem 4.50 is also valid over Z, so in particular the affine coordinate ring Z[Md] is the ring of PSL 2-invariant functions in Z[Ratd]' See [416, Theorems 4.2 and 4.5]. The functions O";n) and ~~n) constructed in Theorem 4.50 are regular functions on Md' so they can be used to map Md to affine space. For example, in the next section we show that * (I ) * (I ) ) ' M .(>2 . 2--+& ( 0"1,0"2
is an isomorphism. In general, Md is not isomorphic to A,2d-2, but we might ask whether using a sufficient number of the O";n) or ~~n) gives an embedding of Md into affine space. In particular, do the values of all O";n) or ~~n)(¢) determine the PGL 2 (C)conjugacy class of ¢? The answer is no. The Lanes maps that we studied in Section 1.6.3 provide nontrivial families of rational maps whose multiplier spectra, and hence whose O";n) and ~~n) values, are all the same.
-I -
Example 4.52. For each t E C* with t
x4 ¢t(x) =
2;,consider the rational map
2tx 2 - Stx + t 2 4x3 + 4tx + 4t -
It is the Lattes map associated to multiplication-by-2 on the elliptic curve
E t : y2 = x 3 + tx + t; see Section 1.6.3. Following standard notation, we write Edm] for the points of E; of order m. Then the n-periodic points of ¢ are given by
and it is not hard to compute the multipliers at these points, if a E x(Ed2n - 1]) and a if a E x(Ed2n if a
+ 1]) and a
-I 00, -I 00,
= 00.
(For proofs of these statements, see Proposition 6.52 in Section 6.5.) For any m, the set E[m] of m-torsion points has order m 2, and for odd m the map x : E[m] --+ ]p'1 is exactly 2-to-l except at x- 1 ( 00 ) , so we can use the listed values of >'0: (¢t) to compute
II
(T + >.)
=
(T
+ 2n)22n-l_2n (T - 2n)22n-l+2n (T + 22n).
).,EAn(t)
In particular, we see that every map ¢t has the same set ofmultipliers, so 0"; n) (¢t) does not depend on t. (A similar statement holds for ~~n) (¢t).) Hence no matter how
4.5. Periodic Points, Multipliers, and Multiplier Spectra
ai
187
n
k ) we use, the resulting map M 4 -+ A always compresses all of many functions the maps ¢t down to a single point. On the other hand, we will later prove that the ¢t are not PGL 2-conjugate to one another (see Theorem 6.46). Hence the image ofthe map C*" {_2J} ~ M 4, t r-----t (¢t),
is a curve in M 4 that is compressed to a single point in A k. An important theorem n says that aside from Lattes examples of this kind, the map on Md defined by the ) is finite-to-one.
ai
Theorem 4.53. (McMullen [294, §2]) Fix d 2: 2, andfor each N 2: 1 let (4.34)
ai
n
be the map defined using all ofthe functions n) with 1 ~ ~ N. If N is sufficiently large, then the map U d,N is finite-to-one on Md(C) except for certain families of Lanes maps that it compresses down to a single point. In particular, it isfinite-to-one if d is not a perfect square. Further, the same statement is true for the map U
*d,N .. M d
Ai
~ fl
defined using all ofthe functions ;;-~n) with 1 ~ n ~ N. (The flexible Lattes mapsfor which U d,N and U d,N' are not finite-to-one are discussed in detail in Section 6.5.)
McMullen's theorem says that aside from the flexible Lattes maps, the maps U d,N and U d,N' are finite-to-one onto their image. One might hope that they are actually injective ifwe avoid the flexible Lattes maps, but it turns out that this is far from true. Theorem 4.54. Define the degree of U d,N to be the number ofpoints in U d,~ (P) for a generic point P in the image U d,N(Md)' One can show that the degree of U d,N stabilizes as N -+ 00. We write deg(u d)for this value. Thenfor every E > 0 there is a constant CE such that for all d. In particular, the multipliers of a rational function ¢ E Rat., determine the conjugacy class of ¢ only up to DE (d~ -E) possibilities. Proof We will prove this in Chapter 6 using Lanes maps associated to elliptic curves with complex multiplication; see Theorem 6.62. 0
Definition. Following Milnor [300], we define the multiplier spectrum of a rational map ¢ to be the function A (¢) that to each positive integer ti assigns the set An(¢). In other words, A( ¢) is the set-valued function
A( ¢) : N
~
Sets with Multiplicities,
188
4. Families of Dynamical Systems
Two rational maps are said to be isospectral if they have the same multiplier spectrum. Then another way to state McMullen's Theorem 4.53 is to say that aside from the flexible Lattes maps, the multiplier spectrum determines the rational map
is an injection, and in fact it maps M 2 isomorphically to the plane z = x - 2, so M 2 S:' A2 . On the other hand, the existence of the Lattes maps described in Example 4.52 shows that a 4,N cannot be finite-to-one on M 4 . The degree of the map does not appear to be known.
4.6
The Moduli Space M Degree 2
2
of Dynamical Systems of
Theorem 4.50 tells us that symmetric combinations of the numbers in the multiplier spectra give well-defined functions on the moduli space Md of degree d dynamical systems on pl. In this section we describe Milnor's explicit identification of M 2 with A 2 using two of these functions. Theorem 4.56. (Milnor [301], see also [416]) Let (Jl, (J2, (J3 E Q[Rat2J be the three functions constructedfrom the fixed points ofa rational map ofdegree 2, i.e.,
II
3
(T
+ Ap(
PEFix(l
=
L (Ji(
(In the notation from page 183, we have a, (a) (Jl = (J3 + 2. (b) The morphism a
= (J?l for 0 ~ i ~
3.)
= (Jl, (J2) : Rat2 -----t A 2
has the following three properties:
t7(
=
Hence a induces a bijection a : M 2 (C) ~ A 2 (C).
4.6. The Moduli Space
M 2 of Dynamical Systems of Degree 2
189
(c) Thefunctions iTl and iT2 are in Ql[M2 ] and the map (T
= (iTl' iT2) :
M 2 ---7 A 2
is an isomorphism of algebraic varieties defined over Ql. Equivalently, the inducedmap
is an isomorphism ofrings.
n
(d) All ofthe functions iTi ) and (;.In) can be expressed as polynomials in iTl and iT2 with rational coefficients. (e) For any field extension K/Ql, the map (T : M2 -> A2 in (c) induces a bijection M 2(K) +--+ A2(K). (Note that Md(K) is not the same as Ratd(K)/ PSL 2(K ); see Remark 4.39 and Section 4.10.)
Remark 4.57. For some applications it is useful to have an explicit description of the map 2 (T = (iTl' iT2) : Rat2 ---7 A described in Theorem 4.56(b). Let
p(a, b)
= a§b6 - ala2bObl + aoa2br + arbOb2 - 2aOa2bob2 - aoa1b1b2 + a6b§
denote the resultant of Fa and Fb . Then, after some algebraic manipulation, one finds that iTl and iT2 are given by the expressions
= a{bo - 4aOa1a2bo - 6a§b6 - aOarbl + 4a6a2bl + 4a 1a2bob1
p(a, b)iTl(
+ a 2b{ - 2arbob2 + 4aOa2bOb2 - 4a2bObl b2 - a1brb2 + 2a6b§ + 4a 1bob§, p(a, b)o-2(
Remark 4.58. According to Theorem 4.56(c), every function (;.In) E Ql[M2 ] is a polynomial in iTl and iT2. In practice, it can be quite challenging to find explicit expressions. Milnor [301] gives the examples
* (2) iT 1
= 2iTl + iT2,
(;.i ) = iTl (2iTl + iT2) + 3iTl + 3, 3
(;.~3)
= ( iTl
+ iT2)2(2iTl + iT2) -
iTl(iTl
+ 2iT2) + 12iTl + 28.
Notice that these expressions are in Z[iTl'iT2], rather than merely in Ql[iTl, iT2]' This reflects the fact that the map (T : M 2 -> A 2 is actually an isomorphism of schemes over Z as described in Remarks 4.43 and 4.51. The functions (;.In) are regular functions on the scheme M 2 /Z, so they are in Z[iTl'iT2]' See [416] for details.
190
4. Families of Dynamical Systems
ProofofTheorem 4.56. (a) Let ¢ E Rat2(iC) and let AI, A2' A3 be the multipliers of the fixed points of ¢. If we assume that none of AI, A2' A3 is equal to 1, then we can apply Theorem 1.14 to deduce that
111 1 - Al 1 - A2 1 - A3
--+--+--=1. After some algebraic manipulation this becomes
which completes the proof that 0"1 (¢) = 0"3 (¢) + 2 for all ¢ E Rat-, (iC) whose multipliers are not equal to 1. It is not hard to see that such ¢ are dense in Rat2 (C) (see Exercise 4.21), from which it follows that the function
is identically zero. (b) The first property O'(¢f) = O'(¢) is a special case of Theorem 4.50, or more directly it is an immediate consequence of our earlier calculation (Proposition 1.9) showing that the multipliers of a rational map are PGL 2 -invariant. In order to prove the second property, it is convenient to show that every rational map of degree 2 is PGL 2 -equivalent to a map of a particular shape. Lemma 4.59. (Normal Forms Lemma) Let ¢ E Rat2(iC) be a rational map of degree 2 and let AI, A2' A3 be the multipliers ofits fixed points. (a) If>'1>"2 -I- 1, then there is an f E PGL 2 (C) such that
¢f (z) = z2 + A1 Z . A2Z + 1 Further, Res(z2 + A1Z, >"2Z + 1) = 1 - A1A2. (b) If A1A2 = 1, then Al = A2 = 1 and there is an
(4.35)
f E PGL 2(iC) such that
¢f (z) = z + ~ + ~.
(4.36)
z
Proof We recall that if a E Fix( ¢) has multiplier An, then the Taylor expansion of ¢ around a looks like
¢(z)
= ¢(a) + ¢'(a)(z - a) + O(z - a)2 = a + An(Z - a) + O(z - a)2.
Hence
= ¢(z) - z = (An - 1)(z - a) + O(z - a)2, so ¢ has multiplicity 1 at the fixed point a if and only if An i- 1. <1>,l(Z)
We also note the formal identity
(X _1)2 - (XY -1)(XZ -1)
= X(X + Y + Z
- 2 - XYZ)
4.6. The Moduli Space M 2 of Dynamical Systems of Degree 2
191
and apply it using the relation
from (a). This yields the useful formulas
(AI - 1)2 = (AIA2 - 1)(AIA3 - 1),
(A2 - 1)2 = (A2AI - 1)(A2A3 - 1),
(4.37)
(A3 - 1)2 = (A3AI - 1)(A3A2 - 1). We now start with a rational map
¢(Z)
=
2
aoz + alz + a2 boz2 + bIZ + bz
and change coordinates in order to put ¢ into the desired form. (a) For this part we assume that AIA2 f- 1, so (4.37) tells us that Al f- 1 and A2 f- 1. Thus the fixed points associated to Al and A2 have multiplicity I, so in particular they are distinct and we can find an element of PGL 2 (q that moves them to 0 and 00, respectively. After this change of variables has been made, the rational map ¢ satisfies ¢(O) = 0 and ¢( 00) = 00, so it has the form with Since ao f- 0, we can dehomogenize by setting ao = 1, and then a simple calculation yields
¢'(O) = al = Al b2 Thus
f
and
has the form
¢(z) = z2 + b2AIZ A2Z + b2
with
Finally, replacing ¢(z) by b:;I¢(b2z) yields the desired form (4.35), and we calculate
1 Al 0 Res(z2 + AIZ, A2Z + 1) = det A2 1 0 o A2 1
= 1-
AIA2,
which completes the proof of (a). (b) The proof of this part is similar. We begin by moving the fixed point associated to Al to 00, so the map ¢ has the form with The assumption that AIA2 = 1 combined with (4.37) tells us that Al we have bl = ao. Dehomogenizing ao = 1 puts ¢ into the form
= A2 = 1, so
192
4. Families of Dynamical Systems
Next we replace ¢(z) by ¢(z - b2 )
+ b2 , so now ¢(z)
¢(z) = Z2 +a1 z+ a2 z
looks like
with
(Of course, the values of a1 and a2 have changed.) Finally, replacing ¢(z) by ¢ (;a2 z)/ ;a2 gives ¢(z) the desired form
,1,( ) _ Z2 + a1Z + 1 _ z - z + a1 z
'f/
1 + -. z
(4.38)
We can compute the value of a1 by observing that ¢ has a double fixed point at 00 and that its other fixed point is at -all. (If a1 = 0, then there is a triple fixed point at 00.) Thus the third multiplier is
so we find that a1 = ~. (This is also correct if a1 of a1 into (4.38) completes the proof of (b).
= 0.) Substituting this value 0
We resume the proof of Theorem 4.56(b,ii). Let ¢1 and ¢2 be rational maps of degree 2 satisfying a (¢1) = a (¢2). Thus and and then the relation (73 = (71 - 2 from (a) implies that also (73(¢d = (73(¢2). It follows that the set of multipliers of the fixed points of ¢1 and ¢2 are the same, say {>'1, >'2, >'3}, since they are the three roots (counted with multiplicity) of the polynomial
T 3 - (71 T 2 + (72 T - (73.
We consider two cases. Suppose first that >'1>'2 -# 1. Then Lemma 4.59(a) says that ¢1 and ¢2 are 2-equivalent to the function (Z2 + >, 1z)/ (>'2 Z + 1), so they are PGL 2-equivalent to each other, and similarly if >'1>'3 -# lor if >'2>'3 -# 1. Weare left to consider the case >'1>'2 = >'1>'3 = >'2>'3 = 1. Then Lemma4.59(b) tells us that >'1 = >'2 = >'3 = 1 and that ¢1 and ¢2 are both PGL 2-equivalent to the rational map z + z-l. This completes the proof of (b,ii), so we turn to (b,iii), the surjectivity of a . Given (81,82) E A 2(C), we set 83 = 81 - 2 and factor the polynomial
Notice that the condition 83
= 81 - 2 gives the familiar relation (4.39)
4.6. The Moduli Space M 2 of Dynamical Systems of Degree 2 which in tum implies the usual formal identities (4.37) for A1'A2'A3. Suppose first that some Ai is not equal to 1, say A1 =I=- 1. Then A1A2 from (4.37), so we can set
193
=I=-
1
¢(Z)=Z2+ A1Z, A2Z + 1 since Res(z2 + A1Z, A2Z + 1) = 1 - A1A2 from Lemma 4.59(a). The fixed points of ¢ are 0:1 = 0, 0:2 = 00, and 0:3 = i=~~. Its multipliers at 0:1 and 0:2 are easily computed, and and the multiplier at
0:3
is
where we use (4.39) for the middle equality. Hence a1 (¢) = 81 and a2 (¢) = 82. We are left with the case A1 = A2 = A3 = 1, which corresponds to the values 81 = 82 = 3. It is easy to check that the rational map ¢(z) = Z + z~l has a triple-order fixed point at 00 and that all three multipliers are equal to 1, so O"(¢) = (3,3). (c) We briefly sketch the proof, which uses basic methods from algebraic geometry. We refer the reader to [416, §5] for further details. The first step is to verify that the map 0" : M 2 ----t A,2 is proper. This follows easily from the valuative criterion [198, 11.4.7] using the fact proven in (b) that every fiber O"~ 1 (t) consists of a single point. (Roughly speaking, a morphism of varieties over C is proper if its fibers are complete.) Next one checks that 0" is finite, which can be proven using the fact that both M 2 and A,2 are affine varieties (cf. [416, Lemma 5.6]). Alternatively, one can show in general that a proper quasifinite map is finite ([299, I, Proposition 1.10]). Then one uses the fact that ft,2 is nonsingular and the bijectivity of 0" on complex points to prove that 0" is an isomorphism (cf. [416, Lemma 5.7]). Finally, it is clear from the explicit formulas for a1 and a2 in Remark 4.57 that 0" is defined over Q. (d) This is immediate from (c), which says that Q[M 2 J = Q[a1,a2], since we already know from Theorem 4.50(b) that a;n) and ;~n) are in Q[M 2 ]. (e) This is a consequence of the fact that the isomorphism 0" : M 2 ----t A 2 in (c) is defined over Q. 0 Remark 4.60. Regarding the proof sketch of Theorem 4.56(c), we observe that a morphism of irreducible varieties V ----t W may be bijective on complex points, yet not be an isomorphism. A simple example of this phenomenon is the cuspidal cubic map Thus in the proof that 0" : M 2 ----t A 2 is an isomorphism, it is crucial that 0" maps M 2 onto the nonsingular variety A 2 . It does not appear to be known in general whether Md is nonsingular.
194
4. Families of Dynamical Systems
The content of Theorem 4.56 is that the moduli space M 2 of dynamical systems of degree 2 is isomorphic to A 2 and an explicit isomorphism is provided by the pair of functions (0"1, 0"2) created from the multipliers of the fixed points of a rational map. From our general theory (Theorem 4.40), the space M 2 sits naturally inside two larger spaces M~ and M~8, but since d = 2 is even, these two spaces coincide and will be denoted by M 2 . We conclude this section with a description of M 2 •
Theorem 4.61. Let M2 = M~ = M~8 be the completion ofM 2 constructed using geometric invariant theory in Theorem 4.40. Then the isomorphism
extends to an isomorphism ir : M mutes:
2 -+ jp'2
such that the following diagram com-
M2~A2
1 M2 ~
1
(x,y)
[x,t,l]
jp'2
The points in M 2 (C) that are not in M 2 (C) correspond to degenerate maps of degree 2 that may be informally described as maps ofthe form (4.40) The point [A, BJ is uniquely determined up to reversing A and B. Proof See [302] and [416, Theorem 6.1 and Lemmas 6.2 and 6.3].
D
Remark 4.62. We expand briefly on what it means to say that the points in the set (M 2 "M 2)(C) correspond to the maps cPA,B given by (4.40). Let U C Al be a neighborhood of and let
°
be a rational map that is a morphism away from 0. We denote the image of t E U by cPt to help remind the reader that cPt is itself a map, i.e., cPt : jp'l -+ jp'l. Consider the composition
which by abuse of notation we again denote by cP. It is a rational map, and since U CA l and M 2 is complete, we see that it is a morphism from U to M 2 . In particular, the point cPo is a well-defined point in M 2(C) . If cPo E M 2(C) , there is nothing to say, so we suppose that cPo ~ M 2(C) . Then the second part of Theorem 4.61 means that possibly after choosing a smaller neighborhood of 0, there exists a morphism
195
4.7. Automorphisms and Twists
1 : U -----+ PGL 2 such that the conjugate map
4/
has the form
with ao, .. . , b2 regular functions on U and satisfying and
a1(0) , b2(0) not both O.
In other words, the degeneration of ¢/ at t = 0 has the form [AXY, XY + By 2 ] . Further, the map determines the point [A, B] = [a1(0) , b2(0)] E JlI'l (C) up to switching the two coordinates. In this way we identify
[A, B]
f---l
[AXY, XY
+ By 2 ],
where the symmetric group on two letters 5 2 acts on JlI'l by interchanging the coordinates . (It is an exercise to show that JlI'l /52 is isomorphic to JlI'l.)
4.7 Automorphisms and Twists As we have repeatedly seen, from a dynamical perspective the geometric properties ofa rational map (z) and its conjugates 1 = 1- 1 1 are the same, since conjugation by 1 E PGL 2 (C) is simply an invertible change of variables. However, matters become more complicated if we restrict the coefficients of 1 to lie in a field that is not algebraically closed. This leads to a notion of conjugation equivalence relative to a particular field, as in the following definitions.
Definition. Two rational maps (z) , 'ljJ(z ) E K (z) are PGL 2-equivalent if there is a linear fractional transformation 1 E PGL 2 (k ) such that 'ljJ = I . More generally, we say that and 1/J are PGL 2 (K) -equivalent if there is a linear fractional transformation 1 E PGL 2 (K ) such that 1/J = I. When PGL 2 is clear from context, we refer more simply to ic-equivalence and K -equivalence. We leave as an exercise (Exercise 4.31) the proof that these are equivalence relations . We denote the set of rational maps that are k -equivalent to by
and similarly the set of rational maps that are K -equivalent by
Remark 4.63. In some sense we already have a notation (. However, we generally view ( . ) as a map Rat., --t M d and (] as a set of rational maps. This is the reason for the notational distinction.
196
4. Families of Dynamical Systems
Not all conjugates
qi
of ¢ need be distinct. For example , the rational map
¢ (z ) = az
b z
+-
- ¢( - z ) = ¢ (z ),
satisfies
so ¢ I = ¢ for the linear fractional transformation f (z ) = - z. The set of transformations f E PGL 2 (K ) that fix ¢ is an interesting group whose properties play an important role both geometrically and arithmetically. Definition. Let ¢ (z ) E K (z ) be a rational map. Theautomorphismgroupof¢ is the group Aut(¢ ) = {f E PGL 2 (K) : ¢ I (z) = ¢ (z )}. (Another common name for Aut (¢ ) is the group ofselfsimilarities of ¢ .) Remark 4.64. It is easy to check that Aut(¢) is a subgroup ofPGL 2(K) and that for any n « PGL 2(K) the map
Aut(¢)
----t
Aut(¢h),
f
f------+
h - I fh ,
is an isomorphism (see Exercise 4.32). Thus as an abstract group, Aut( ¢) depends only on the PGL 2-conjugacy equivalence class of ¢ ; more precisely, the Kequivalence class of ¢ determines Aut (¢ ) in PGL 2 up to conjugation. Proposition 4.65. Let ¢ (z ) E K (z ) be a rational map ofdegree d ~ 2. Then Aut(¢ ) is afinite subgroup ofPGL 2 (K), and its order is bounded by a function ofd. Proof Let
f
E Aut (¢) . Then for any point
P E pI (K) and any n ~ 1 we have
¢ n( p) = (¢ / )n (p ) = (j-I ¢n J) (P ),
and hence f (¢ n (p ))
= ¢n (j( p)).
In particular, if P is a periodic point of (primitive) period n, then f (P) is also a periodic point of (primitive) period n. Thus each f E Aut( ¢) induces a permutation of each of the sets Per n (¢) and Per~* (¢ ). Choose three distinct integers n l, n2, n3 such that ¢ contains primitive n periodic points for each value of n. Corollary 4.7 says that we can find such values, and further that their magnitude can be bounded solely in terms of d. More precisely, they may be chosen from among the first d + 5 primes. Letting fori
=
1,2,3,
the action of ¢ on the sets of primitive periodic points yields a homomorphism (4.41) from Aut (¢) into a product of three symmetric groups. We claim that the homomorphism (4.41) is injective. To see this, we observe that any f in the kernel of (4.41) fixes each Per~* ( ¢); hence f fixes at least three points in pI (K) ; hence f is the identity map. Thus'the homomorphism (4.41) is injective, which clearly implies that Au t (¢) is finite. D
197
4.7. Automorphisms and Twists
Remark 4.66. Propo sition 4.65 tells us that the automorphism group Aut(¢) of a rational map is a finite subgroup of PGL 2 (K ). For K = C, or more generall y for any field of characteristic 0, the classical description offinite subgroup s ofPGL 2 (C) says that every such subgroup is conjug ate to either a cyclic group, a dihedral group , or the symmetry group of a regular three-dimensional solid (i.e., the tetrahedral, octahedral, and icosohedral groups) . Sec, e.g., [414].
tc
Example 4.67. Let ( be an n th root of unity, let 'ljJ( z ) E (z) be any rational map, and let ¢(z ) = z'ljJ(zn). Then Aut(¢) contains a cyclic subgroup of order n generated by the map f (z ) = ( z. Example 4.68. The map ¢(z ) = (Z 2 - 2z )/ (-2z + 1) has an automorphism group Au t (¢ ) that is isomorphic to the symmetric group S3 on three letters. More precisely (see Exercise 4.36), the automorphi sm group of¢ consists of the following six linear fractional transformations:
1 z -1 1 z } Aut(¢ ) = { z , -, - - , - - , - - ,1- z ~ S3' Z z 1- z z-1 Example 4.69. Consider the rational map
¢b(Z) =
Z
b z
+-,
whose automorphism group Aut( ¢b) = {z, - z} has order 2. These maps are all PGL 2-equivalent, since the linear fractional transformation f (z ) = z/bfC gives ¢c = ¢{ . Thus the geometric dynamical properties of ¢b are the same for all b. However, the arithmetic properties of ¢b may change quite substantially depending on the value of b, since the change of variable involves a square root. So although ¢b and ¢ c are always PGL 2 (K)-equi valent, they are not P GL 2 (K )-equivalent unless b]« is a perfec t square in K . The underlying reason for the existence of these "twists" is the fact that Aut (¢ l) is nontrivial. Definition. Let ¢ ( z) E K (z) be a rational map. The set oftwists of ¢ over K is the set Twist (¢/K ) = {K-eqUiValen~e ~Iasse~ of maps 'ljJ SUCh } . that 'ljJ IS K -equivalent to ¢ Remark 4.70. As noted earlier, the geometric dynamical properties of the maps in Twist (¢ / K ) are identical , but their arithmetic properties may be significantly different. For example , if two rational maps are K -isomorphic, then the field extensions of K generated by their periodi c points are the same. This is clear from the fact that Pern(¢ f ) = U -1 (P ) : P E P er n (¢ )} ,
so if f E PGL 2 (K), the fields generated by Per n (¢ ) and Pern(¢ f ) are identical. However, if ¢ and 'ljJ are only K -isomorphic, these fields may well be different. This often provides a convenient method for proving that two maps are not K -isomorphic, as in the following example.
4. Families of Dynamical Systems
198
Example 4.71. Continuing with Example 4.69, for each b E K * we let
¢b(Z)
b
= Z + -. Z
We saw that these maps are all tc -isomorphic , which gives a map of sets
K*
---+
T wist (¢dK ),
(4.42)
Note that if b/ c is a square in K , say b/ c = a 2, then ¢b and ¢c are K -isomorphic , since ¢ c = for the map f (z) = az E PGL2 (K ). Thus (4.42) induces a welldefined map K* / K *2 ---+ Twist(¢d K) , (4.43)
¢t
We can use Remark 4.70 to prove that the map (4.43) is injective. (We assume that K does not have characteristic 2.) A quick computation shows that iI> ~b ,2(X, Y) = 2X 2 + by 2, so the primitive 2-periodic points of ¢b are -b/2.
J
±J J
Hence if ¢b and ¢c are K-isomorphic, then the fields K ( -b/2 ) and K( -c/2) are the same, so b/c must be a square" in K. Notice that we could not use Perl (¢b) to prove this result, since the only fixed point of ¢b is 00 . These quadratic twists of ¢ l (z) are analogous to quadratic twists of elliptic curves (cf. [410, §X.5]). Thus fix A, B E K and, for each D E K *, let ED be the elliptic curve ED : Dy2 = X 3 + AX + B. Then all of the ED are isomorphic over but E D 1 and E D 2 are isomorphic over K if and only if the ratio D, / D2 is a square in K. This gives a natural map K * / (K * )2 --+ Twi st(E d K ).
x,
Remark 4.72. How does the theory of twists fit in with the moduli spaces Md constructed in Section 4.4? The answer is that if ¢ and 'l/J are twists of one another, then their corresponding points (¢) and ('l/J) in the moduli space M d are equal. This is true because points in M d(f< ) classify rational maps of degree d modulo PGL 2 (f<) conjugation, so points in M d(f<) do not detect whether the conjugat ion is defined over K . In other words, there is a natural map
(4.44) but this map is not one-to-one. It fails to be injective precisely for those maps in Ratd(K) that have nontrivial twists; cf. Remark 4.39. On the other hand, the next proposition and Exercise 4.38 imply that the map (4.44) is injective on a Zariski open subset of Ratd. We now prove that a rational map with no automorphisms has no nontrivial twists. Later, in Theorem 4.79, we prove a much stronger result. Proposition 4.73. Let ¢( z) E K (z) be a rational map ofdegree d 2: 2 and assume that its automorphism group Aut(¢) is trivial. Then ¢ has no nontrivial twists, i.e., Twist (¢/ K ) has only one element. the K -equivalence class of ¢ itself 6It is an easy exercise to prove that K ( VA) and K (..;B) are isomorphic if and only if AlB is a square in K, assuming that K does not have characterist ic 2.
4.8. General Theory of Twists
199
Proof Suppose that 't/J E K (z ) is a twist of , so there is an j E PGL2(k ) such that 't/J = /. We let an element a E Gal (k/ K ) act on j (z) and (z) in the natural way by applying a to each of the coefficients. Notice that a( and a('t/J ) = 't/J, since the coefficients of and 't/J are in K. Hence /
= 't/J = a('t/J ) = a( / ) = a(Jj-l ) = a(J )a(a(J)-I = <1(1).
Hence a(J )j-l E Aut(/ is K -isomorphic to , so it represents the trivial twist. 0
Remark 4.74. As we have seen in Section 4.3, the set Rat., of all rational maps of degree d is a Zariski open subset of]P'2dH. It is the complement of the hypersurface described by the vanishing of the resultant Res (Fa , F b ) = O. One can show that the set of rational maps E Ratd having nontrivial automorphism group forms a proper Zariski closed subset of Ratd; see Exercise 4.38. Thus most rational maps have no nontrivial automorphisms, and those that do, fall into finitely many irreducible algebraic families . Further, since for all j E PGL 2 the automorphism groups of and / are isomorphic as abstract groups, there is a proper Zariski closed subset of Md determined by the conjugacy classes of rational maps with nontrivial automorphisms. It is an interesting geometric problem to describe the irreducible subvarieties making up this set and an interesting arithmetic question to describe their rational and integral points . See Exercises 4.28 and 4.41.
4.8 General Theory of Twists In this section we develop the basic theory of Galois twists in an abstract setting. Although we apply this material only to rational maps, in Section 4.9, and to ]P'I , in Section 4.10, we develop the theory in some generality in order to clarify the underlying principles. Let X and Y be objects defined over the field K. For example , X and Y might be curves or algebraic varieties or rational maps.f We consider X and Y to be equivalent if they are isomorphic over K . However, it may happen that they are isomorphic over tc, but not over K. This leads us to consider the following set. Definition. Let X be an object defined over the field K .The set of twists of X / K is the set 7The proof that PGL2(K ) is the subgroup ofPGL2 (K ) fixed by Gal(K / K ) is not hard, although it does use Hilbert 's Theorem 90. 8Formally, X and Y should be objects in a category on which the Galois group Gal(K / K ) acts in an appropriate way. We do not concern ourselves with such formalism and leave it to the reader either to formulate the correct abstract concepts or to restrict attention to those situations, namely algebraic varieties and rational maps, in which the Galois action is clear.
4. Families of Dynamical Systems
200
Twist(Xf K)
=
K -isomorphism classes of objects Y} such that Y is defined over K ~nd . { Y is isomorphic to X over K
In other words, an element of Twist (X f K) is an object Y that is defined over K such that there is an isomorphism i : X ----+ Y, but the isomorphism i might be defined only over an extension of K. Two elements Y and Y' in Twist(Xf K) are considered to be equivalent if there is an isomorphism j : Y ----+ Y' that is defined over K. The following examples should help to clarify this definition.
Example 4.75. Our first example deals with twists ofrational maps. Let
Example 4.76. Our second example deals with twists of the variety lP'1. For any nonzero a E K*, let C« be the plane curve C a : x 2 + y2 All of these curves are isomorphic over
=
a.
K via the explicit isomorphism
i(x, y) =
(x~, y~) .
(4.46)
Further, if bfa is a square in K, say bfa = e2 , then C a and Cb are isomorphic over K via the isomorphism i(x, y) = (ex, ey). Thus exactly as in Example 4.75, we obtain a natural map
K* f K*2
----+
Twist( C 1 f K),
where [CalK denotes the K-isomorphism class of Ca. Notice that iftwo curves C and C' are K-isomorphic, then the K-isomorphism i : C ----+ C' identifies their K-rational points i : C(K) ----+ C'(K). This suffices to prove that the curves C 1 and C- 1 are not isomorphic over Q, since and
C- 1 (Q) = 0.
Hence C -1 represents a nontrivial element of Twist (C 1 f Q), and indeed a nontrivial element of Twist ( CdlR.). More generally, a famous theorem of Fermat says that ifp is an odd prime number, then
201
4.8. General Theory of Twists
if and only if
p
== 1 (mod 4).
Hence ifp == 3 (mod 4), then Cp represents a nontrivial element ofTwist(CdQ). It is not hard to show that different primes p == 3 (mod 4) yield distinct elements of Twist (Cd Q); see Exercise 4.44. Returning to the general situation, let X be an object defined over K and let Y represent an element of Twist( XI K ). This means that there is a K -isomorphism
i : Y --+ X. We wish to determine whether X and Yare K -isomorphic. If i is itself a K -isomorphism, then we are done; but even if i is not a K -isomorphism, it may be possible to modify i and tum it into a K -isornorphism. In order to measure the extent to which i fails to be a K -isomorphism, it is natural to make use of Galois theory, since i is defined over K if and only O'(i) = i for every 0' E Gal(KIK) . Thus for each element 0' E Gal(KI K), we consider the composition of maps 9a : X
a(i-I) Y
i
------+
------+
X
.
The map 9a = iO' (i- l ) is a K -automorphism of X, i.e., it is an isomorphism from X to itself defined over K. If i is already defined over K, then 9a (X) = x is the identity map, but in general 9a will be a nontrivial automorphism. Proposition 4.77. Let X be an object defined over K , let Y be a twist of X I I{, choose a K -isomorphism i : Y --7 X , and define a map 9: Gal(K I K )
--+
A ut (X ),
(a) The map
Gal(K / K ) ~ A ut (X ), satisfies for all 0' , T E Gal(K I K ).
9ar = 9 aO' (9r )
Maps satis fying 9ar = 9aO' (9r ) are called l -cocycles. (b) Y is the trivial twist of X if and only if there is an f E Aut (X ) satisfying fo r all a E Gal(K I K ). Maps ofthe fo rm f O'(j -l ) are called l -coboundaries.
Proof (a) We have gar = i 0 (O'T)(i- l) = i commutative diagram of maps:
X
a(r (i)- I) )
Y
a(i)
O' (T(i )- I ). Consider the following
i
------+
1
X
0
X
il
~Y
202
4. Families of Dynamical Systems
The top row is gun but if we travel around the diagram the long way, we get i
0
a(C 1) 0 a(i ) 0 a(T(i )-l ) = goo 0 a(gT)'
(b) Suppose first that 9 is a coboundary, say goo = la(f- l ) for some IE Aut (X ). We verify that the isomorphism I-I i : Y --; X is defined over K. For any a E Gal (K/ K ) we have
a(f-l i ) = a(f -l )a(i ) = 1-1 gua(i ) = 1-1 ia(i-1 )a(i) = I -I i. This proves that 1-1 i is defined over K, so Y is K -isomorphic to X. Next suppose that Y is K -isomorphic to X, say j : Y --; X is a K -isomorphism. Then u(j ) = j for every a E Gal (K/ K), so we have
900
= i 0 u(i- 1) = i 0 rI o aU) 0 u(C 1) = (i 0 r 1) 0 u((i 0 r1)-1).
Thus if we let I = i r 1 E Aut(X) , then goo = lu(f -l), which proves that 9 is a D coboundary and completes the proof of the proposition. As the terminology suggests, there is a cohomology theory underlying Proposition 4.77. Definition. Let is a map 9 :r
r be a group that acts on another group A . A l -cocycle (from r to A) ---. A
satisfying
gUT = gUU (gT)
for all a , T E I',
and a l-coboundary (from r to A ) is a map of the form
I' ---. A ,
o
f--->
lu(f-l ) for some I E A.
The cohomology set H1(r, A) is defined to be the collection of l-cocycles r --; A modulo the equivalence relation that two l-cocycles gl and 92 are cohomologous if the map gl"lg2 is a l-coboundary. Remark 4.78. If the group A is abelian, then H1(r, A) is itself an abelian group, and in this situation it is possible to define cohomology groups Hn(r , A) for all n ~ O. For the general theory of group (and Galois) cohomology, with many important applications to class field theory, Diophantine equations, and arithmetic geometry, see for example [97, 396, 410]. A slight elaboration of the proof of Proposition 4.77 shows that if Au t (X) is abelian, then there is a well-defined one-to-one map
i y u(i y1)) , (4.47) where iy : Y --; X is a K-isomorphism; see Exercise 4.42. In some cases, for example when X is an algebraic variety, the map (4.47) is an isomorphism.
Twist (X /K ) ---. HI (Gal(K/K ), Aut (X)) ,
[Y] K
f--->
(u
f->
4.9. Twists of RationalMaps
203
4.9 Twists of Rational Maps According to Proposition 4.77, every twist corresponds to a l-cocycle, and a twist is trivial if and only if its l-cocycle is a l-eoboundary. The natural question that arises is whether every l-cocycle comes from a twist. It turns out that the answer depends on the category from which the objects are being chosen. For example, in the category of algebraic varieties, every l-cocycle does come from a twist. We will not prove this general result, but in the next section we treat the case of twists of pI . In this section we describe what happens for rational maps.
Definition. We define an action of the Galois group Gal(K I K) on the group of linear fractional transformations PGL2(K) in the natural way, thus
a(f) = a (az + b) = a(a)z + a(b) cz + d a(c)z + a(d) az + b for f = - - E PGL 2(K) and a cz+d
-
E Gal(K
I K).
The automorphism group Aut(
g: Gal(KIK)
----7
Aut(
with values in Aut(
Gal(KIK) ~ Aut(
'--*
PGL2(K)
(4.48)
with values in PGL2(K). This way of extending g provides the key to describing which l-cocycles correspond to actual twists of
Theorem 4.79. Let
-+
Aut(
be a Y-cocyde with values in Aut(
{~E
HI (Gal(KIK), Aut(
~ becomes trivial in HI (Gal(K I K), PGL2(K)) }. Proof Suppose first that there is a twist
204
4. Families of Dynamical Systems
Hence 9 is the PGL 2 (K) l-coboundary associated to the element f E PGL 2 (R ). This proves that (a) implies (b). Next suppose that 9 is an Aut (¢» l-cocycle that is a PGL 2 (K) l-coboundary. This means that there is an f E PGL 2 (K) with the property that 917 = fO"(f-l ) for all 0" E Gal ( K j K ). We claim that ¢>f is a twist of ¢> j K with l-cocycle 9. What we need to check is that ¢>f is defined over K , i.e., that ¢>f E K (z ), since once we know that , it is clear from the definitions that 9 is its associated I-cocycle. Let 0" E Gal(KjK ). Then
= ¢> ¢>9a = 9a¢>
since we are given that 917 E Aut(¢»,
¢>9"
by definition of ¢>9" ,
¢> f O"(f - l ) = fO"(f -l) ¢>
since 917
rl¢>f = O"(f-l) ¢>O"(I) ¢>f = ¢>a(f) ¢>f
= fO"(f-l ) by assumption,
multiplying by since O"(f-l)
= O"(¢>)"(f ) = O" (¢>f )
i:' and by 0" (f) ,
= 0"(1)-1,
since ¢>(z) E K (z), so O"(¢» = cPo
We have proven that ¢>f = 0"(¢> f) for all 0" E Gal( K j K ), which shows that ¢> f (z) E K (z) and thus completes the proof that (b) implies (a). In order to prove the cohomological description of Twist (¢>j K ), we first use Proposition 4.77 , which says that there is a natural embedding of Twist (¢>j K ) into the cohomology set Hi (Gal (K j K ), Aut (¢») (cf. also Remark 4.78). Then the (Gal (K j K ), Aut (¢») comes equivalence of (a) and (b) tells us that an element of from an element of T wis t (¢> j J() if and only if it becomes trivial when mapped to (G al (Kj K ), PGL 2 (K)). 0
u:
u:
Remark 4.80. A formal argument with commutative diagrams shows that
where Br(K ) is the Brauer group of K. The Brauer group plays an important role in class field theory and many other areas of number theory and arithmetic geometry. For example, Br(iQp) = iQjZ, so Br(iQp)[2] has only two elements, and the same is true for finite extensions of iQp. The Brauer group of a number field is more complicated.
Example 4.81. Let K be a field of characteristic not dividing n, and let ¢>(z ) E K(z) be a rational map whose automorphism group is Aut (¢» where we recall that JLn C isomorphism
K *jK*n
= {( z : (
E JLn},
K denotes the set of nth roots of unity. Then there is an
~Twist ( ¢>jK ) ,
(4.49)
4.9. Twists of Rational Maps
205
Note that by assumption we have ¢(z) = (-l¢((Z) for all ( E J.t n . In particular, the function Z-l¢(Z) is invariant under the substitution z ----+ (z, so it has the form ¢(z) = z'lj;(zn) for some 'lj;(z) E K(z) (Exercise 4.37). Thus the twist of ¢(z) given in (4.49) is equal to z'lj;(bz n), so it is indeed in K(z). Since we are assuming that the automorphism group of ¢ is isomorphic to J.t n , not merely as an abstract group, but also in terms of the way in which Gal( R / K) acts on Aut(¢) and on J.t n , a standard result in Galois cohomology? says that there is an isomorphism
b I-------t
(CTf---+ 0-( V'b))
V'b
.
(4.50)
This allows us to identify Twist (¢ / K) with a subset of K*/ K* n. In order to show that they are isomorphic, Theorem 4.79 says that we must show that every cocycle in (4.50) becomes a coboundary when we consider it as a cocycle with values in PGL 2(R). But this is clear, since with our identification of J.t n with Aut(¢), the cocycle in (4.50) is equal to
z
V'b
with j(z) =
-
E PGL 2 (K ).
Note that this example generalizes Example 4.75, which dealt with the case n = 2.
Example 4.82. Let ¢(z) E K(z) be a rational map with automorphism group Aut( ¢) = {z, Z-l}. (See Exercise 4.35.) The Galois group Gal(R/ K) acts trivially on Aut( ¢), so we have
The isomorphism is given explicitly by associating to any b E K* / K* 2 the cocycle
0-
I-------t
{
z
if 0-( Vb) =
z-l
if 0-( Vb) =
Vb, -Vb,
(4.51)
To ease notation, we let (3 = Vb and let 9 be the cocycle described by (4.51). Then Theorem 4.79 says that 9 comes from a twist of ¢ if and only if there is some j E PGL 2(R) satisfying ga = jo-(1-1). Using thefact that ga(z) E {z, Z-l }, we are looking for an j E PGL 2(R) satisfying 9This statement is equivalent to H 1 (Gal( K / K), K*) = 0, which is a version of Hilbert's Theorem 90. Using this and taking Galois invariants of the Kummer sequence
1 ------+ J-t n ------+ K yields the cohomology long exact sequence
*
Q:t---+Q:n
-------+
K
*
------+
1
206
4. Families of Dynamical Systems
if a({3) = (3, f (Z) a(f)(z ) = { l/f(z ) ifa({3) = - (3. It is not hard to construct such an
f , for example
f( z ) = {3z + 1 - {3z + 1 Note that det( ! {3 ~) = 2{3
:I
0, so
f
is in PGL 2 (k ). Hence every 9 gives a twist
of ¢>, and indeed we can write the twist associated to b = using f ,
{32
E K */ K * 2 explicitly
1 {3¢> ( ~) + {3 ¢> (z) = _J... ( {3z+1 ) + 1 . If'
-{3z+1
For example, let Md(z) = zd be the £ilh-power map. Then a judicious use of the b) binomial theorem yields a formula for the b-twist Md of Md'
Mdb )(z) =
I: (2~)bkz2k /I: k
k
(2k
~ 1)bkz2k+
1 .
In particular, the first few b-twists of M d are
1 + bz
2
2 M (b)(Z) _ 1 + 3bz 3 3 - 3z + bz '
M 2(b) ( Z )
_ -
4.10
Fields of Definition and the Field of Moduli
2z
'
If ¢>(z) is a rational map whose coefficients lie in a field K , then it is interesting to study the orbits of points whose coord inates lie in K or in finite extensions of K . The smallest field K containing the coefficients of ¢> can generally be determined by inspection. For example, the map ¢>(z ) = Z3 + 1 is clearly in Q(z) and the map ¢>(z) = Z3 + i is just as clearly in Q(i)(z ). However, if we make a change of variables f (z) = iz in the latter map ¢>(z ) = z3 + i, we find that
= r 1 (¢>(f (z ))) = - i¢>(iz) = -i( -iz 3 + i) = _ z3 + 1 E Q(z). map ¢>(z) = z3 + i is really a Q(z) map that has been altered by
¢>1 (z)
Thus the injudicious change of variables.
an
Definition. Let K be a field of characteristic 0, let ¢>( z) E k (z) be a rational map with coefficients in an algebraic closure of K, and let K' / K be an extension field. We say that K' is afield 0/definition/or ¢> ifthere is a linear fractional transformation f (z) E PGL2 (k ) such that
In other words, K' is a field of definition for ¢> (or an FOD for short) if, after a change of variables, ¢> has coefficients in K'.
4.10. Fields of Definition and the Field of Moduli
207
As in Section 4.9, we can use Galois theory to investigate the possible fields of definition of a given rational function ¢( z) E K (z). We let Gal( K j K) act on K (z) in the natural way by applying (J E Gal (K j K) to the coefficients of ¢ E K (z),
+ adzd) (J(ao) + (J(a1)z + + bdzd = (J(b o) + (J(bdz +
() (a o + a1Z + (J ¢ = (J bo + b1z +
+ (J(ad)zd + (J(bd)zd .
Then Galois theory (and Hilbert's Theorem 90) tell us that
¢
E
K' = (fixed field of{ (J
K' (z) for the field
E Gal(K j K)
: (J( ¢) = ¢} ).
Suppose instead that ¢ is merely equivalent to a map whose coefficients are in some field K'. That is, suppose that there is a linear fractional transformation f E PGL2(K) such that e/ E K'(z). Then (J(¢f) = ¢f for all (J E Gal(KjK'),so
f-1¢f = ¢f = (J(¢f) = (J(r1¢J) = (J(J-1)(J(¢)(J(J) = (J(J)-1(J(¢)(J(J). Solving for (J( ¢) yields
(J(¢) = (J(J)f-1¢f(J(J-1) = (J(J(J-1)f1¢(J(J(J-1)) = ¢fa(J-l). Thus (J(¢) is equal to ¢ conjugated by the map f(J(J-1) E PGL2(K), so in particular (J( ¢) and ¢ are equivalent. We have proven that
K ' is a field Of) ( definition for ¢
===}
(for every (J E ~al(K j K') there exists ) a ga E PGL 2 (K) such that (J (¢) = ¢gu .
(4.52)
Turning this around, we start with an arbitrary rational map ¢(z) E K(z) and study the automorphisms (J E Gal( K j K) for which d ¢) is equivalent to ¢.
Definition. Let rjJ(z) E K(z) be a rational map. We associate to rjJ a subgroup G¢ of Gal(K j K) and a field K¢ defined by G¢ = {(J E Gal(KjK): d¢) = ¢gu for some ga E PGL 2(Kn,
K¢ = fixed field ofC¢ = {o
E
K: (J(O:) = 0: for all (J
E
G¢}.
The field K ¢ is called the field ofmoduli (FOM) of ¢.
Remark 4.83. The group PGL2(K) acts on the space of rational maps via the usual conjugation action, ¢f = ¢ f. If we consider as usual the equivalence class
r:'
consisting of all maps that are conjugate to a particular map ¢, then C ¢ is the subgroup of Gal(K j K) consisting of elements that map the set [¢J to itself. Equivalently, if ¢ has degree d, let (¢) be the image of ¢ in the moduli space Md = Rat., j PSL 2 that we defined and studied in Section 4.4. The space Md is defined over Q (Theorem 4.36) and the field of moduli of ¢ is exactly equal to the field generated by the coordinates of the point (¢) E Md.
208
4. Families of Dynamical Systems
Webeginby provingsome elementarypropertiesofFOM and FOD, in particular, the important fact that the field of moduli is contained in every field of definition. Proposition 4.84. Let ¢(z) E K (z ). (a) The set Gel> is a subgroup ofGal(K / K ). (b) Let K' be afield of definition for ¢. Then K eI> FOM ~ FOD.
~
K'. Informally, we say that
Proof The proof of this proposition is simply a matter of unsorting definitions. Thus let (J,T E Gel> . Then
Thus ((JT) (¢) is equivalentto ¢, so (JT E Gel>. Similarly,
which shows that (J-l(¢) is equivalent to ¢, and hence that (J-l E Gel>. This proves that Gel> is a subgroup of Gal(K/ K ). Next let K' be a field of definition for ¢. Under this assumption, we proved earlier (4.52) that for every (J E Gal(K/ K' ) there is a g(7 E PGL 2 (K ) such that (J (¢) = ¢9" . In other words, (J(¢) is equivalent to ¢, so (J E Gel> = Gal(K / K eI» by definition. This provesthat Gal(K/ K' ) c Gal(K/ KeI», and henceby Galoistheory, that KeI> c K'. (Weare also using the fact that K eI> is a finite extension of K .) 0 It follows from Proposition 4.84 that the smallestpossiblefield of definition for ¢ is the field of moduli of ¢, but it is not clear whether the field of moduli is always a field of definition. The following example shows that we can have FOM i=- FOD.
Example 4.85. Let
¢(Z) =i (: ~ ~ r Clearly Q(i) is a field of definition for ¢. Let (J be complex conjugation, so Gal(Q(i)/Q) = {I, (J}, and let g(z) = -1/ z. Then we obtain 1 ¢(-I/z)
=i
1
i(-I / Z-I)3 -I / z + 1
(- 1/z _ 1)3= - i (z + 1)3= (J (¢)(z). - I/z + I
z - I
This shows that (J E Gel> . so Gel> = {I , (J} and K eI> = Q. In other words,Q is the field of moduli of ¢. Now suppose that Q is actually a field of definition for ¢. This means that we can find some f E PGL 2 (Q) such that ¢! E Q(z). In particular, letting (J denote complex conjugation, we have
4.10. Fields of Definition and the Field of Moduli
209
It is not hard to verify (see Exercise 4.39) that Aut(¢) = 1, so we deduce that f = ga(f). Using the fact that g(z) = -1/ z, this equation says that
az+b cz +d where we use an overscore to denote complex conjugation. The fact that these two linear fractional transformations are equal means that there is a A E Q* such that
a=-AC,
b=-Ad,
c=Aa,
d=Ab.
Multiplying the first and third equations gives
This is a contradiction, since A -I- 0 and we cannot have a = C = O. Hence Q is not a field of definition for ¢, so we have an example of a map with FOM -I- FOD. In order to investigate more closely the question of when the field of moduli of a rational map ¢ is a field of definition, we make the simplifying assumption that Aut(¢) = 1.
(4.53)
Replacing K by K p, we may assume that K is the field of moduli of ¢. This means that for every a E Gal(K/ K) there exists a ga E PGL 2(K) satisfying
Note that the assumption (4.53) that Aut(¢) = 1 implies that ga is uniquely determined by a. The next proposition describes some of the properties of the map a ~ gao Proposition 4.86. Let ¢ E K (z) be a rational map of degree d 2: 2 with field of moduli K and satisfying Aut(¢) = 1, andfor each a E Gal( K / K) write a( ¢) = ¢9rr as above. (a) The map
is a l-cocycle, i.e., it satisfies for all a, T E Gal(K/ K). (b) K is a field ofdefinition for ¢ if and only if there is an f E PGL 2(K) such that
if g is a l-coboundary,
for all a E Gal(K/ K).
i.e., if and only
210
4. Families of Dynamical Systems
Proof (a) Let a, r E Gal (K j K ). Then ¢9u r
= a (r (¢ )) = a (¢9r ) = a (¢t
(9r ) = ¢9uU(9r) .
Since Au t (¢ ) = 1, we conclude that gUT = gua (gT)' (b) Suppose first that K is a field of definition for ¢ . This means that there is an f E PGL2(K ) such that ¢f E K (z ). Hence for any a E Gal(K j K ) we have ¢f
= a (¢ f ) = a ( ¢ t(f ) = ¢9u u (f) .
Again using the fact that Au t (¢ ) = 1, we find that f = gua(J), and hence gu =
fa (J-1 ). Conversely, suppose that there is an 1 E PGL2(K ) such that gu Then a( ¢f) = a( ¢t(f) = ¢9uu (f ) = ¢f.
= la(J-1 ).
This is true for every a E Gal( K j K), so ¢f E K (z). Hence K is a field of definition for ¢ . 0 The criterion for FOM = FOD given in Proposition 4.86 may seem complicated, but it represents a tremendous simplification. lfwe try to use the definition ofFOD directly, we need to search for an 1 E PGL2 (K ) that makes the coefficients of ¢f lie in K. The substitution ¢f (z) =
(az+b) - b cz+d , - c¢ (a z+b) + a cz+d d¢
even for a rational function ¢ (z) of small degree , has coefficients that are very complicated expressions in the quantities a, b, c, d. It is thus difficult to determine whether there is some choice of a, b, c, d that makes the coefficients lie in K. On the other hand, the cocycle--eoboundary criterion FOM
= FOD
{:::=:}
gu has the form la (r 1)
in Proposition 4.86 involves only linear functions, i.e., elements of PGL2(K), so it is often considerably easier to apply. The 1-cocycles that arise in the FOD = FOM question have the form g : Gal (K j K ) -+
PGL2(K ).
The group PGL2 (K ) is the automorphism group of the projective line p1 , so these cocycles should be associated to twists of p l. We saw in Section 4.8 that every twist gives rise to a 1-cocycle , but in general it is a delicate question to determine whether every 1-cocycle comes from a twist. It turns out that this is true for algebraic varieties , but since we do not need the most general result , we are content to construct the twists of p 1 that are needed to answer our question about FOD = FOM . (See also Example 4.87.)
211
4.10. Fields of Definition and the Field of Moduli Proposition 4.87. Let
g: Gal(K/K)
--+
PGL 2 (K )
be a 1-cocycle, and assume that g has the property that there is a finite extension L / K such that grr = 1 for all IJ E Gal( K / L). 10 Then there is an algebraic curve C defined over K and an isomorphism i : C -+ ]p>1 defined over K such that the 1cocycle Gal(K/K) --+ PGL 2 (K ), IJ f--------' iIJ(i- I ) , is equal to the 1-cocycle g. Hence C is a twist of]p>1 / K, and
C is the trivial twist of]p>1 / K
if and only if
g is a I-co boundary.
Proof We construct the curve C by describing its field of rational functions. Note that the field of rational functions for the curve ]p>1 is the field K (z) and that the Galois group Gal( K / K) acts naturally on K (z) by acting on K and leaving z fixed. Now consider another field of rational functions in one variable K(w), but this time with a "twisted action" of Gal(K/ K). We define this twisted action by letting Gal(K/ K) act on K as usual, but setting
IJ(w) = g;;I(W). In other words, if gl7(z) = (az + b)/(cz + d), then
IJ(w) = g;;I(W) =
dw -b , -cw+a
and for any rational function
we have
IJ(F)(w) = IJ(ao) + IJ(at}g;;l(w) IJ(bo) + IJ(bt}g;;I(W)
+ IJ(a2)g;;l(w)2 + + IJ(b2)g;;I(W)2 +
+ IJ(ad)g;;l(w)d. + IJ(bd)g;;l(w)d
It is clear that IJ(F + G) = IJ(F) + IJ( G) and that IJ(FG) = IJ(F)IJ(G). However, to be an action, we must also have (IJr)(F) = IJ(r(F)). We use the cocycle relation to verify this condition,
IJ(r(w)) = IJ(g;lw) = IJ(g;I)a(w) = IJ(g;l)g;;I(W) = g;;;(w) = (IJr)(w). We now look at the subfield of K (w) that is fixed by the twisted action,
K = {F E K(w) : IJ(F) = F for all IJ E Gal(K/ K)}. We prove that the field K has the following properties: lOOne says that 9 is a continuous l-cocyc1e for the profinite topology on Gal( K / K) and the discrete topology on PGLz(K).
212
4. Families of Dynamical Systems
(i) K n K
= K.
(ii) K has transcendence degree lover K.
(iii) KK = K (z). To verify (i), let a E K n ic. Then a E K, so the action of Gal(K j K ) on a is the usual Galois action. On the other hand, the fact that a E K means that the Galois action is trivial. Hence a E K. This shows that K n K c K , and the opposite inclusion is obvious. Next let L j K be a finite Galois extensionwith the property that gT = 1 for every T E Gal(K j L). This implies that the action of an element <J E Gal(K j K ) on W depends only on the image of <J in the quotient group Gal(L jK ) = Gal(KjK)jGal(K jL) ,
since if r E Gal(KjL), then
(<JT)(W) = <J (gT(W)) = <J (w). It follows that the polynomial P (T ) =
II
II
(T - ..\(w )) =
>'EGa l(L/ K )
(T - g>: l (W)) E K (w )[T]
(4.54)
>'EG al (L/ K )
is well-defined. Further,we claim that the coefficients of P (T ) are in K. To see this, for any <J E Gal( K j K ) we note that
<J(P )(T ) =
II
(T - <J..\(w)) = P (T ),
>'E Gal(L/K )
since the effect of replacing ..\ with <J..\ is simply to permute the order of the factors in the product. Wehavenow constructeda polynomial P(T) E K[TJ, and we observethat W is a root of P(T). Hence by definition, the extension K(w)j K is an algebraic extension. Since the element W is transcendental over K, this proves that K has transcendence degree at least lover K. On the other hand, K is contained in the field K (w), and it is clear that K (w)j K has transcendence degree lover K . Therefore K has transcendence degree exactly lover K , which proves (ii). Finally, consider the splitting field I:- over K of the polynomial P (T). As already noted, we have wE I:-. The definition of P (T ) showsthat there is a naturalsurjection Gal(L j K ) ~ Gal(l:- j q , which implies in particularthat I:- = LK. Hence w E L K, which proves that K (z) c KK . This gives (iii), since the other inclusion is true from the definitionof K. We now have the tools needed to complete the proof of Proposition 4.87, but we pause briefly for an example illustratingthe general construction.
213
4.10. Fields of Definition and the Field of Moduli
Example 4.88. Define a map g by the rule
z
ga(z)
=
{
-
II Z
ifcr(i)=i, ·f (.) . 1 cr Z --to
It is easy to check that g is a l-cocyc1e, and indeed it is the l-cocycle described in Example 4.85. Fix an embedding ofQl into C and let p E Gal(Ql/Q) denote complex conjugation. The twisted action on Ql(w) is given by
p(a) =
a
for a E Ql
p(w) = -l/w.
and
The field K. is the subfield of Ql(w) consisting of elements that are fixed by the twisted action. The coefficients of the polynomial P(T) defined by equation (4.54) give us elements in K;
This yields one interesting element in K; namely u = w - w- I , and we observe that the quantity v = i( w + w- I ) gives another element in K. It is not hard to show that u and v generate K,
x=
K(u, v) = K ( w -
~,i ( w + ~) ) .
(See Exercise 4.45.) Of course, u and v are not independent, since
u2
+ v2
= (
W _
~
) 2
+
(i ( + ~ ) ) w
2
= -4.
The field K. is the function field of the curve C : u 2 + v 2 = -4. Notice that C is defined over Q, and C is Ql-isomorphic to ]pI, but C is not Q-isomorphic to ]pI, since C(Q) = 0. From our general theory, the fact that C is a nontrivial twist of ]pIIQ is equivalent to the fact (proven by a direct calculation in Example 4.85) that the l-cocyc1e g is not a I-coboundary. Resuming the proof of Proposition 4.87, we have constructed a field K that is the function field of a curve C I K, and we have an isomorphism
KlC = K(w) ~ K(z),
W
f------+
Z,
that induces a K-isomorphism i : C ---+ ]pl. In other words, the functions w on C and z on]pI are related by the formula w = z 0 i. The curve C is a twist of ]pII K, and its associated cocyc1e is given by a ---+ icr(i- I). We compute
214
4. Familiesof Dynamical Systems
z 0 a (i ) = a(z 0 i) = a(w)
since a (z ) = z, since w
= g;;I (W)
= z 0 i,
by definition of the twisted action on
dw - b - cw +a dz 0 i - b - cz 0 i + a - 1 . = Z 0 ga 02 .
az
since w =
+b
= --d
where ga(z )
cz+
R (w),
-
E PGL 2(K ),
z 0 i,
s:'
I Thus a (i) = 0 i, which proves that ga = ia(i- ) is the l-cocycle associated to the R -isomorphism i : C --> pl . This completes the proof that the algebraic curve C is a twist of pI / K whose associated l-cocycle is g. Finally, Proposition 4.77 tells us that C is the trivial twist if and only if its associated l-cocycle is a l-coboundary. 0
Returning to the question ofFOD = FOM, let ¢(z) E R( z) be a rational function with field of moduli K and trivial automorphism group. We have constructed a
I-cocycle g : Gal(R/ K)
----t
PGL 2 (R)
that is associated to ¢ (Proposition 4.86) and a twist C'" of p I/ K that is associated to the l-cocycle 9 (Proposition 4.87). We have also proven the following chain of equivalences: K is a field of definition for ¢ {::=}
9 is a l-coboundary
(Proposition 4.86),
{::=}
C'" is K -isomorphic to p I
(Proposition 4.87).
It remains to find a way of determining whether a twist of pi is K -isomorphic to pl . We will use the Riemann-Roch theorem to provide two sufficient conditions for resolving this problem. For the convenience of the reader, we recall the general statement of the Riemann-Roch theorem, although we will need it only for curves of genus O.
Theorem 4.89. (Riemann-Roch Theorem) Let C / K be a smooth projective curve ofgenus 9 defined over K. (a) There is a divisor on C ofdegree 2g - 2 that is defined over K. (b) Let D be a divisor on C that is defined over K and assume that the degree ofD satisfies deg ( D ) ~ 29 - 1. Then there is a function
div(J) + D
~
f
E K (C) satisfying
o.
Corollary 4.90. (Riemann-Roch in Genus 0) Let C / K be a smooth projective curve ofgenus 0 defined over K. (a) There is a K -rational divisor ofdegree 2 on C .
215
4.10. Fields of Definition and the Field of Moduli
(b) Let D E Div(C) be a divisor defined over K and satisfying deg(D) 2 1. Then there is afunction f E K(C) with div(f) + D 2 o. Proof The proof of the Riemann-Roch theorem over an algebraically closed field is given in most introductory texts on algebraic geometry, such as [198, IV.1.3], or see [255, Chapter I] for an elementary proof due to Weil and [410, II, §5] for an overview. The divisor of degree 2g - 2 in (a) is a canonical divisor, i.e., the divisor of any K -rational differential form such as df for any nonzero f E K (C). In particular, for curves of genus 0 we get a K -rational divisor D of degree - 2, so - D is a Krational divisor of degree 2. 0
Proposition 4.91. Let C be a twist of pI / K. The following are equivalent: (a) C is the trivial twist of pI / K, i.e., Cis K -isomorphic to pl. (b) C(K) is nonempty, i.e., C has a point with coordinates in K. (c) There is a divisor D = 2:= n;(Pi) on pI (K) ofodd degree such that D is defined over K, i.e.i for all a E Gal(K/K) we have 2:=ni(a(Pi)) = 2:=ni(Pi) as a formal sum ofpoints. Proof If C is the trivial twist of pI / K, then there is a K -isomorphism j : C --t pl. In particular, j : C (K) --t pI (K) is a bijection, so C (K) is certainly nonempty. This proves that (a) implies (b). Next, it is clear that (b) implies (c), since if P E C(K), then the divisor D = (P) has odd degree (one is an odd number!) and is clearly defined over K. Finally, suppose that the degree n = deg(D) = 2:= ni of D = 2:= ni(Pi) is odd and that D is defined over K. We also use the fact that C and pI are K -isomorphic, so in particular C is a curve of genus zero. It follows from the Riemann-Roch theorem (Corollary 4.90(a)) that there is a K-rational divisor on C having degree -2, say
Consider the divisor
E=D
n-1
n-1
+ -2- D' = (Pd + (P2) + ... + (Pn) - -2-((Qd + (Q2))'
The divisor E is defined over K and has degree 1, so the Riemann-Roch theorem (Corollary 4.90(b)) tells us that there is a rational function 'l/J on C such that 'l/J is defined over K and 'l/J has degree 1. In other words, 'l/J is a map 'l/J : C --t pI of degree 1 defined over K, and hence C is K-isomorphic to pl. This shows that (c) implies (a) and completes the proof of the theorem. 0 We have now assembled all of the tools that we need to prove the main theorem of this section. We state the theorem in full generality, but give the proof only for rational maps with trivial automorphism groups. The general case is proven similarly, but there are many additional technical complications. Theorem 4.92. Let ¢( z) E K (z) be a rational map ofdegree d 2 2. Then the field ofmoduli of ¢ is a field ofdefinition for ¢ in the following two situations:
4. Families of Dynamical Systems
216 (a) ¢(z) has even degree. (b) ¢( z) is a polynomial.
Proof We prove the theorem under the assumption that Aut(¢) = 1. See [414] for a proof in the general case. Without loss of generality, we may assume that K is the field of moduli of ¢. Let g : Gal( K / K) --+ PGL 2 (K) be the l-cocycle associated to ¢ (Proposition 4.86) and let C be the twist of pI / K associated to the l-cocycle g (Proposition 4.87). This means that there is a K -isomorphism such that Using many of the results proven in this chapter, we have the following chain of equivalences:
K is a field of definition for ¢ {=::} g is a l-coboundary
(Proposition 4.86),
{=::}
C is K -isomorphic to pI
(Proposition 4.87),
{=::}
C(K) is not empty
(Proposition 4.91),
(There is a divisor on C(K) Of) odd degree and defined over K
(Proposition 4.91).
{=::}
We are going to produce divisors and points on the curve C using the map --+ C defined by the composition
'ljJ : C
We begin by checking that the map 'ljJ = i- 1¢i is defined over K. To verify this, we let a E Gal(K/ K) and compute
a('ljJ)
= a(i- 1¢i) = a(C 1)a(¢)a(i) = a(i- 1)¢g"a(i) 1)a(i) = a(i- 1)g;;I¢gaa(i) = a(i- 1)(ia(C 1))-I¢ia(i= C 1¢i = 'ljJ.
Hence e : C --+ C is defined over K. We also note that ¢ and 'ljJ have the same degree, since i is an isomorphism. (a) Let D'lj; be the divisor of fixed points of'ljJ, that is, the collection of fixed points of'ljJ counted with appropriate multiplicities. In the language of algebraic geometry, D'lj; is the pullback by the diagonal map
Pf------t(P,P), of the graph {(x,'ljJ(x)): x E C} of'ljJ. A map of degree d has exactly d + 1 fixed points (counted with multiplicities), and if the map is defined over K, then the divisor of fixed points is defined over K, i.e., it is fixed by the Galois group. Thus D'lj; is a divisor of degree d + 1 on C
4.10. Fields of Definition and the Field of Moduli
217
and D'ljJ is defined over K. By assumption, d is even, so D'ljJ has odd degree. Hence by the chain of equivalences derived earlier, the field K is a field of definition for ¢. (b) The map ¢ : pI ----t pI is a polynomial by assumption, so it has a totally ramified fixed point P. In other words, ¢( P) = P and the ramification index of ¢ at P satisfies e p ( ¢) = d. The map i : C<jJ ----t pI is an isomorphism, so the point
is a totally ramified fixed point of 7/J. Suppose first that P is the only fixed point of ¢ in pI (K) with ramification index d. Then Q is the only fixed point of 7/J in C<jJ (K) with ramification index d. However, for any 0" E Gal(K/ K) and any point R E C<jJ(K) we have
since 7/J is defined over K. In particular, taking R = Q to be the given fixed point of ramification index d, we see that 7/J(O"(Q)) = O"(Q) and d = eQ(7/J) = ef7(Q)(7/J), so 0" ( Q) is also a fixed point of 7/J of ramification index d. Hence 0" ( Q) = Q, and since this holds for every 0" E Gal(K/ K), we conclude that Q E C<jJ(K). Hence C<jJ(K) is not empty, so we again conclude that K is a field of definition for ¢. We are left to consider the case that ¢ has a second fixed point of ramification index d, say ¢( PI) = P' and e pi (¢) = d. (Note that the Riemann-Hurwitz formula, Theorem 1.1, precludes more than two such points.) Choose some linear fractional transformation h E PGL 2(K) satisfying h(oo) = P and h(O) = Pl. Then ¢h satisfies
SO
¢h must have the form ¢h (z)
= czd
for some c E K*.
But any rational map of this form has a nontrivial automorphism group. Indeed, its automorphism group is a dihedral group of order 2(d - 1) generated by the maps Z
f----+
(z
and
Z
f----+
a/z,
where (d-I = 1 and a d-I = 1/c. In any case, we have ruled out this case by our assumption that Aut( ¢) = 1, which completes the proof of Theorem 4.92. D
Remark 4.93. The distinction between the field of moduli and fields of definition is important in the study of abelian varieties; see for example the work of Shimura [398]. More recently, the FOM = FOD question has been investigated for the collection of of covering maps ¢ : X ----t B up to automorphism of the base B. In particular, this is much studied for B = pI (cf. Grothendieck's "dessins d'enfant"). If also X = B = pI, then one studies the set of rational maps ¢(z) E K (z) under the left composition equivalence relation ¢ rv f¢ for f E PGL 2(K). This bears
4. Families of Dynamical Systems
218
considerable resemblance to the material in this section, where we use instead the equivalence relation ¢ rv j -l ¢j, but there are significant differences. For example, Couveignes [111] shows that using the relation equivalence ¢ rv j ¢, there are polynomials in Q[z] with FOM =I FaD, in direct contrast to Theorem 4.92. For further results, see for example work of Debes and Douai [117, 118, 119] and Debes and Harbater [120].
4.11
Minimal Resultants and Minimal Models
Let P (X ,Y ) = l: aij X iy j E K[X, Y] be a polynomial and let p be a prime of K . We define the order of P at p to be ord, (P)
= min ord, (aij ). t,)
Notice that ord, (P) = 0 if and only if all of the coefficients of ¢ are p-integral and at least one coefficient is a p-unit. In Section 2.4 we defined the resultant of a rational map ¢ (with respect to p) by writing ¢ = [F, G] using homogeneous polynomials F, G E K [X , Y] satisfying min {ord, (F), ord, (G)} = aand setting Res p (¢) = Res(F, G).
The resultant Res, (¢) is well-defined up to multiplication by the 2lf h power of a p-adic unit, so in particular ord, (R esp (¢)) depends only on ¢ and p. We also recall that ¢ has good reduction at p if and only if its resultant is a p-adic unit.
Example 4.94. Even if ¢ has bad reduction at p, it may be possible to change coordinates and achieve good reduction. In other words, there may be some j E PG L2 (K) such that Res(¢/ ) is a p-unit, For example, the map ¢(z ) = z + p2z - 1 has bad reduction at p, since
However, ifwelet j(z) = pz, then ¢I (z) = z + z-l, which has good reduction at p. Our aim in this section is to study this phenomenon. In particular, we study the extent to which we can eliminate, or at least ameliorate, bad reduction in ¢ by conjugating ¢ with a linear fractional transformation in PGL2 (K ). Definition. Let ¢ = [F, G] be a rational map given by homogeneous polynomials F, G E K[X, Y], let j E PGL2 (K ), and choose a matrix A=
(~ ~)
E
GL2 (K )
representing f. We define polynomials FA,GA E K[X ,Y ] by the formulas
4.11. Minimal Resultants and Minimal Models
219
FA(X , Y ) = e5F(aX + (3Y" X + e5Y ) - (3G(a X + (3Y, , X + e5Y) , G A(X , Y) = - ,F(aX + (3Y, , X + e5Y ) + aG(aX + (3Y, , X + e5Y).
(4.55) (4.56)
It is often convenient to write this in matrix notation as
[~~] where
Aadj = (
1- 10 ¢ 0 I
=
Aadj
[~ :~] ,
!-y -!) is the adjoint matrix to A. Note that the conjugate ¢t
=
of ¢ by I is equal to
Proposition 4.95. Let ¢ = [F, G] be a rational map ofdegree d described by homogeneous polynomials F, G E K[X,Y] and let p be a prime ideal. (a) The valuation ofthe minimal resultant of ¢ is given by the formula ordp(Res p (¢)) = ord p(Res( F , G)) - 2dmin{ord p (F), ord p(G)}.
(4.57)
Note that there is no requirement that the coefficients ofF and G be p-integral or that some coefficient be a p-adic unit. (b) Let A E GL 2 (K ). Then with FA and G A defined by (4.55) and (4.56),
ord, (Res(FA, G A)) = ord, (Res(F, G)) + (d2
+ d) ord, (de t A ),
min{ordp (FA ),ordp (G A )} ~ min{ ordp (F), ordp (G )}
+ (d+ l )ord p (A ),
where ord, (A) denot es the minimum ofthe order ofthe coordinates ofthe matrix A. (c) In particular, ifU E GL2 (Rp ), then
ord p( Res(Fu, G u ))
= ordp(Res(F, G )) ,
min{ ordp (Fu ), ord p( Gu )} = min{ ord, (F) , ord, (G) }. Proof (a) Choose a constant c E K * satisfying
ord, (c) = min{ ord; (F), ord p( G)}. Then
where we have made use of the homogeneity property of the resultant (Proposition 2.13(d)). This gives the desired result (4.57) . (b) An elementary calculation (see Exercise 2.7(c)) shows that 2+d
Res(FA,G A) = (det A)d
Res(F,G) ,
so taking ord; gives the first part of (b). For the second part , we observe that every coefficient of FA and G A is a sum of terms of the form
4. Families of Dynamical Systems
220
coefficient ) x ( homogeneous polynomial of ) ( of For G degree d + 1 in Z[a, (3, 'I, 8] . Hence min{ ord, (FA), ord, (G A)} 2: min{ ord, (F), ord, (G)}
(c) The assumption that U
+ (d + 1) ord, (A).
E GL 2 (R p ) is equivalent to the two conditions
ordp(U) 2: 0
and
ord, (det U)
= 0,
i.e., the coefficients of U are p-adic integers and the determinant is a p-adic unit. Applying (b) with A = U gives
ord, (Res(Fu, Gu») = ord, (Res(F, G»), min{ ordp(Fu), ordp(Gu)} 2: min{ ordp(F), ordp(G)}.
(4.58)
This almost completes (c). We next apply (b) to the polynomials Fu and Gu and to the matrix A = U- 1 E GL 2(R p ) . Using the fact that (Fu )u-1 = F and similarly for G, we find from (b) that min{ ord, (F), ord; (G)} = min{ ord, ((Fu )u-1), ord, (( Gu )u-1)}
2: min{ ord, (Fu ), ord, (Gu )}. This gives the opposite inequality to (4.58), which completes the proof of (c).
0
Definition. Let K be a number field and let ¢(z) E K(z) be a rational map. For each prime p of K, define a nonnegative integer by Ep(¢) =
min !EPGL 2(K)
ord, Res p(¢!).
In other words, Ep (¢) is the exponent of the power of p dividing the resultant of the conjugate ¢! that is closest to having good reduction at p. Then the (global) minimal resultant of ¢ is the integral ideal (4.59) We say that (F, G) is a minimal model for ¢ at p if ¢ = [F, G], the coefficients of F and G are p-integral and
ord, Res(F, G) = ord, 9t.p. (See Section 6.3.5 for the analogous definition of mimimal models of an elliptic curve.) Remark 4.96. The product (4.59) defining 9t.p makes sense, since Ep(¢) = 0 for all but finitely primes p. To see this, write ¢ = [F, G] for any F and G having coefficients in the ring of integers of K. Then there are only finitely many primes p with ord; Res( F, G) > 0, and it is clear that Ep(¢) = 0 for all other primes.
4.11. Minimal Resultants and Minimal Models
221
The minimal resultant of a rational map is clearly invariant under PGL 2(K)conjugation. It measures the extent to which the conjugates of cj; have bad reduction, so provides a convenient measure of the arithmetic complexity ofthe conjugacy class of cj;. A coarser way to measure arithmetic complexity is simply to take the product of the primes with bad reduction, which we denote by (4.60)
It is clear that !)1¢ divides 9l¢, and it is tempting to conjecture an inequality in the opposite direction that would be a dynamical analogue of Szpiro's conjecture [205, FJ.2] for elliptic curves.
Conjecture 4.97. Let KjQ be a number field and d ~ 2. There is a constant c = c(K, d) such that for all rational maps cj;(z) E K (z) ofdegree d,
N K / Q 9l¢ < (N K / Q !)1¢t The minimal resultant gives one way to measure the arithmetic complexity of a rational map, but note that there are infinitely many PGL 2(K)-inequivalent rational maps of a given degree whose minimal resultants are the same. For example, the minimal resultants of the polynomials
are the same as u ranges over all units in the ring of integers of K. The moduli space Md provides an alternative way to measure the arithmetic complexity of the conjugacy class of a rational map. If we fix a projective embedding M d '---7 lPN, then cj; E K(z) determines a point (cj;) E Md(K) and we can take the height of the corresponding point in JP'N (K). However, this way of measuring arithmetic complexity is also not entirely satisfactory, since twists of a rational map give the same point in Md, yet are arithmetically quite different. Note that the same situation arises in the theory of elliptic curves, where curves with the same jinvariant need not be arithmetically identical. This suggests combining the primes of bad reduction with the height coming from moduli space. We do this and formulate a dynamical version of a conjecture of Lang (cf. [202], [254, page 92], or [410, VIII.9.9]). Recall that the canonical height ofa point P satisfies h¢(P) = 0 if and only if Pis preperiodic for cj; (Theorem 3.22). The following conjecture says that the height of nonpreperiodic points grows as cj; becomes more arithmetically complicated.
Conjecture4.98. Fix an embedding ofthe moduli space Md in projective space and let h M d denote the associated heightfunction. Let K be a numberfield and d ~ 2 an integer. Then there is a positive constant c = c(K, d) such that for all rational maps
cj; E K(z) ofdegree d and all wandering (i.e.. non-preperiodic) points P E JP'I (K),
222
4. Families of Dynamical Systems
In the theory of elliptic curves , the notion of global miminal Weierstrass equation is extremely useful; see the discussion in Section 6.3.5 and [410, VIII, §8] for further details. We briefly discuss a dynamical analogue. Definition. Let K be a number field and let ¢(z) E K (z ) be a rational map . Then ¢ has a global minimal model if there is a linear fractional transformation f E PGL 2 (K ) and homogeneous polynomials F and G satisfying ¢f = [F , G] with the property that the coefficients of F and G are in the ring of integers of K and
ord; (Res( F,G)) = ord; (rytc/»
for every prime p.
In other words, the pair (F, G ) is simultaneously a minimal model for
rytc/>=
Res (F , G )Q¥ G '
{ Res(F, G)a'j;.,G
if d is odd, (4.61)
if d is even.
If d is odd,
then the ideal class of QF ,G depends only on ¢, independent of the choice ofF and G. (c) If d is even, then QF ,G depends only on ¢ up to multiplication by the square ofa principal ideal.
(b)
Proof For any f E PGL 2 (K) with corresponding A E GL 2(K) , we use Proposi tion 4.95(a,b) to compute
ord, (Resp(¢f)) = ord, (Res( F A, G A)) - 2dmin{ ord, (FA) , ord; (G A)} = ord, (Res(F, G)) + (d2
+ d) ordp(det A)
- 2dmin{ ordp(FA), ordp(GA)} .
(4.62)
For each prime ideal p we choose a linear fractional transformation fp E PGL 2(K) and corresponding matrix A p so as to minimize the resultant of ¢f p. In other words, (4.63) Combining (4.62) and (4.63) yields
ord, (rytc/» - ord, (Res(F , G))
= d [(d + 1) ordp (d et A p )
-
2min{ordp (FAp) ,ordp (G Ap)}] '
(4.64)
4.11. Minimal Resultants and Minimal Models
Hence if we define an ideal
UF ,G
223
by the rule dodd,
d even, then (4.64) says that UF ,G satisfies the desired formula (4.61). It remains to determine the extent to which the ideal UF ,G depends on the choice of F and G. Let ¢ = [F, G] = [F ' , G'l be two lifts of ¢. Then there is a constant c E K * satisfying F' = c F and G ' = c G. Suppose first that d is odd. We use (4.61) to compute c
2d
Res(F, G)u~i,cG
= Res(cF, cG) u~i,cG = 9'-tcl> = Res(F, G) u~G '
This is an equality of ideals, so by unique factorization of ideals we conclude that COcF ,cG = OF,G. Hence the ideals 0 cF ,cG and OF,G differ by a principal ideal. Next suppose that d is even. Then by a similar calculation we find that c
Hence ideal.
2d
Res(F, G)U~F,cG
2) (C OcF ,cG
=
OF,G ,
= Res(cF, cG) O~F,cG = 9'-tcl> = Res(F, G)oi,G' so
OcF ,cG
and
OF,G
differ by the square of a principal 0
Definition. Let K be a number field, let ¢(z) E K (z) be a rational map of degree d, and write ¢ = [F,G] with polynomials F and G as usual. If d is odd, we write o.cI> / K for the ideal class of OF ,G in the ideal class group of K. If d is even, we write aN K for the image of OF,G in the group of fractional ideals modulo squares of principal ideals . In both cases, by analogy with the theory of minimal equations of elliptic curves (cf. [410, VIII §8]), we call acl> / K the Weierstrass class of ¢ over thefield K. By Proposition 4.99 the Weierstrass class acl> / K depends only on ¢, independent of the chosen lift [F,G].
The triviality of the Weierstrass class gives a necessary condition for the existence of a global minimal model. Proposition 4.100. Let K be a number field and ¢( z) E K (z ) a rational map of degree d ~ 2. If ¢ has a global minimal model over K. then its Weierstrass class acl>/ K is trivial.
Proof Replacing ¢ by ¢f for an appropriate choice of f E PGL 2(K), we may assume that ¢ = [F, G] with polynomials F and G having coefficients in the ring of integers of K and satisfying for every prime p.
It follows from the defining equation (4.61) of OF,G that for every prime p, Hence OF,G = (1), so its image in the ideal class group (if d is odd) or in the group of ideals modulo squares of principal ideals (if d is even) is also trivial. 0
Exercises
224
Exercises Section 4.1. Dynatomic Polynomials 4.1. Let¢(z) = z + liz, or inhomogeneous form, ¢([X, YJ) = [X 2 + y 2 , X Y J. (a) Compute the first few dynatomic polynomials iI>;,n (X, Y) for ¢, say for n = 1,2,3,4. (b) Prove that for all n 2:: 2, the dynatomic polynomial iI>;,n satisfies iI>;,n(±X, ±Y) = 2 2 iI>;,n(X, Y). Deduce that iI>;,n(X, Y) E Z[X , y ]. (c) Prove that the field 1Ql2,¢ generated by the points of exact period 2 is the field IQl ( A). (d) Prove that the field 1Ql3,¢ generated by the points of exact period 3 is an 53 extension oflQl. (Hint. Show that the roots of iI>¢,3 (1, VW,) E Z[w] generate a cyclic cubic extension oflQl.) (e) Prove that iI>;,4(X, Y) factors into a polynomial of degree 4 and a polynomial of degree 8. (See Exercise 4.40 for a more general result.) Describe the fields generated by the roots of each factor.
4.2. Prove the following elementary properties of the Mobius function. (See (4.2) on page 148 for the definition of the Mobius function.) (a) Let n 2:: 1 be an integer. Then ifn
= 1,
ifn 2:: 2. (b) Let g(n) be a function whose domain is the positive integers, and define a new function f(n) by f(n) = Ldln g(d). Prove that
g(n)
=
L
f(d)J1
G) .
din
This is called the Mobius inversion formula. (c) Prove that the nth cyclotomic polynomial ~(Zk
_ l)~(n/k)
kin is indeed a polynomial. (Hint. Use (b) and the fact Zk - 1 has distinct complex roots.)
4.3. Let l/d(n) = deg(iI>;,n) be the number offormal n-periodic points ofa rational map of degree d, counted with multiplicity (see Remark 4.3). (a) Prove that l/d(l) = d + 1 and
z:= J1 (iD dk
for n 2:: 2. kin (b) Make a table of values ofl/d(n) for some small values ofn and d. (c) Prove that l/d(n) =
formally as power series. For what range of x
> 0 do the series converge?
225
Exercises I Type of Point
I Char K I a;'(n) values
Multiplier
a;'(n) = 0 for all n
Wandering
a;'(n) > o for exactly one n
Periodic
not root of unity
Periodic
root of unity
0
Periodic
root of unity
p>O
a;'(n) > ofor exactly two n's a;' (n) > 0 for n = ip", exactly two t's and all k ::0: 0
Table4.4: Values of ap(n).
4.4. Let ¢(z) E K(z) be a rational function of degree d ::0: 2 and suppose that z = 0 is a fixed point of ¢ with multiplier A = 1. Write ¢(z) = z + ze'ljJ(z) with e ::0: 2 and'ljJ(O) =1= 0, where e = ao(¢, 1) in the notation of Theorem 4.5. (a) Prove that
¢n(z)
=
Z + nze'ljJ(z)
(b) Assume further that K has characteristic p
+ O(z2e-l).
> O. Prove that for all k ::0: 1.
4.5. Verify that the description of the values of a;'(n) given in Table 4.4 is correct. 4.6. Let ¢( z) be a rational function, let n ::0: 1, let p be a prime with p t n, and let P E pl. (a) Prove that
a;'(n, ¢P) = a;'(n, ¢) + a;' (np, ¢). (b) Deduce that the nth dynatomic polynomial for ¢P factors as
(c) More generally, ifgcd(n,k) = 1, prove that
*nk,¢ -- Il(*n,¢J. )p(kfj) . jlk
4.7.
* Let ¢(z) = Zd
+ ad_lzd-1 + ... + a2z 2 + alZ + ao
be a monic polynomial of degree d and let ~,¢(z) be its nth dynatomic polynomial. Each root a of ~,¢(z) has an associated multiplier A¢(a) = (¢n)' (a). (a) Prove that there is a (unique) monic polynomial
whose nth power satisfies
IT
(x-A¢(a))=Resz(~,¢(z),x-(¢n),(z)),
<1>;"". (a) =O
where Res, means to take the resultant with respect to the z variable. (Hint. All n of the points in the orbit 0 ¢ (C,lC) have the same multiplier.)
226
Exercises
(b) Let Ck(X) E Z[x] denote the kth cyclotomic polynomial, and for integers min with m < n , define
b.n,m = Res(Cn/ m,8 m,¢) E Z [ao ,a l, ... ,ad- d· Prove that the discriminantof ~ , ¢ ( x ) is given by the formula Disc ~,¢
= ±8n ,¢(1t
II b.;;"r;:, . min m -e -a
(c) Let min with m
< n. Provethat the resultant of ~ ,q, and ;'",q, is given by the formula
= ±b. ~: m '
Res (~ ,¢ , ;'",¢)
Conclude that 1> has a point of formal period n whose exact period m is strictly smaller than n if and only if there is a point of formal period m whose multiplier is a primitive (n/m)lh root of unity. 4.8. This exercise describes an analogue of Theorem 4.5 for automorphisms of projective space. Let 1> : lP'N -> lP'N be an automorphism defined over a field K, i.e., 1> E PGL N+I (K). We say that 1> is nondegenerate if the equation 1>(P ) = P has only finitely many solutions in lP'N(K ). (a) Let A E GL N+I (K) be an invertible matrix with coefficients in K representing the map 1>. Prove that 1> is nondegenerate if and only if every eigenspace of A has dimension 1. (b) Assume that 1>n is nondegenerate, let r q,n C lP'N X lP'N be the graph of 1>n , and denote the diagonalmap by b. : lP'N -> lP'N X lP'N . Following Remark4.4, we definethe O-cycle of n-periodic points of 1> to be the pullback
¢,n = b.' (r q, ) =
L
ap (1), n)(P )
P EpN
and the O-cycle of formal n-periodic points to be ; ,n =
Lit (~) ¢,n = L k in
ap(1) , n)( P).
PE p N
Let P E lP'N be a point of primitive period m for 1>. Prove that
ap(n) = {ap(m)
o
if mln, ifmfn,
* (n ) -_ {ap(m) ap
and
o
ifm=n,
ifm =I- n.
In particular, ; ,n is an effective (i.e., positive) cycle. 4.9. Let F, G E K[X, Y] be homogeneous polynomials of degree d :::: 2 with no common factors and let 1> = IF,G] : lP'1 -> lP'1 be the associated rational map. Define two sequences of polynomials inductively starting with Fo(X, Y ) = X and Go(X , Y ) = Y and then for
n:::: 0,
Fn+1 = Fn(F(X,Y),G(X ,Y))
and
Gn + 1
(a) Prove that F n and G« have no common factors.
= Gn (F (X , Y ),G(X , Y»).
227
Exercises (b) More precisely, prove that the resultant of F n and G n is given by
Res(Fn , G n )
= Res(F, G) C2n-l)dn~1 .
(Hint. Use Exercise 2.12.) (c) Prove that ¢>n = [Fn, G n]. (d) For all n, m ~ 0, prove that
Fn+m(X, Y) Gn+m(X, Y)
= Fn(Fm(X, Y),Gm(X, Y)), = Gn(Fm(X, Y),Gm(X, Y)).
4.10. With notation as in Exercise 4.9, we define the (generalized) (m, n )-periodpolynomial of¢> to be the polynomial
= n(Fm(X, Y),Gm(X, y)),
m,n(X, Y) where n(X, Y) (a) Prove that
= YFn -
m,n(X, Y)
XG n is the usual n-periodpolynomial of¢>.
= Gm(X, Y)Fn+m(X, Y)
- Fm(X, Y)Gn+m(X, Y).
(b) LetPElP'l(K).Provethat
m,n(P)
=0
if and only if ¢>m+n(p)
= ¢>m(p).
Thus P is a root of m,n if and only if P is a preperiodic point with "tail" oflength at most m and with period dividing n. (c) Prove that for all m, n ~ 1, the quotient
m,n m-l,n is a polynomial.
4.11. We continue with the notation from Exercises 4.9 and4.10. Let ~(X, Y) be the nth dynatomic polynomial of ¢>. Then for m, n ~ 1 we define the (generalized) (m, n)-dynatomic "polynomial" of ¢ to be * ~(Fm(X, Y),Gm(X, Y)) m n(X, Y) = ( ). , ;; Fm-1(X,Y),Gm-1(X,Y) (a) Prove that ::",n(P) = 0 if and only if ¢>m(p) has formal period n. Points satisfying ::",n (P) = 0 are called preperiodic points with formal preperiod (m, n). (b) ** Prove (or disprove) that ::",n(X, Y) is a polynomial.
Section 4.2. Quadratic Polynomials and Dynatomic Modular Curves
4.12. We continue with the notation from Exercise 4.7. Thus for any monic polynomial¢>(z), we define a monic polynomialon,¢(x) by
On,¢(xt=
II
(x-A¢(a))=
4>n,
and for min with m
II
(x-(¢>n)'(a)),
(4.65)
4>n,
< n we set ~n,m
= Res(Cn/m(X),Om,¢(X)),
where Ck(X) is the k th cyclotomic polynomial. We now specialize to ¢>c(z) write on(e,x) = on,¢c (x) and ~n,m (c) to indicate the dependence on e.
= z2 + e and
228
Exercises
(a) Prove that bn(c,x) E Z[c,x] and that ~n,m(c) E Z[c]. Then use Exercise 4.7(c) to explain the powers that appear in Table 4.3. (b) Prove that 8n (c, x) = T deg , ot>;'bn(C, 2nx) is in Z[c, x]. (This eliminates many powers of 2 from the coefficients of bn (c,x).) Compute
8n (c,x)
forn = 1,2,3,4. (c)
** Prove that there is a monic polynomial Wn,m(X)
E Z[x] such that
~n,m(c) = Wn ,m(4c),
and deduce that the set {c E
Q: ~n,m(C) = 0 for some m < n with min}
is a set of bounded height. Conclude that the set of bifurcation points in the Mandelbrot set M is a set of bounded height. (Hint. The formula ~n,m(C) = wn ,m(4c) bounds the denominator of c, then mimic the proof of Proposition 4.22.) (d) Prove that ~n,l (c) and ~2n,2 (c) are irreducible in lQi[c] for all n 2 1. (e) ** Prove the following conjecture of Morton-Vivaldi [314]: The polynomial ~n,m( c) is irreducible in lQi[c] for all min with m < n.
4.13. Let cPe(Z) = Z2 + c and write ~ (c, z) for the nth dynatomic polynomial of cPe. Continuing with the notation from Exercise 4.12, we let bn(c,x) E Z[c,x] be the polynomial defined by (4.65) and 8n (c, x) = T degot>;'bn(C, 2n x) the normalized polynomial in Exercise 4.l2(b). (a) Recall that the dynatomic curve Y1 (n) C A2 attached to the polynomial
Z(n):
8n (y, x ) = o.
Prove that there is a morphism
(b) Define an action of i E Z j nZ on Y1 (n) by
(y,Z) = (y,cP~(z)). (See Section 4.2.3.) Prove that the map F in (b) is invariant for this action, i.e., prove that F(y,cP~(z)) = F(y,z) for all i E ZjnZ. Deduce that there is a unique morphism Yo(n)
--->
Z(n) such that the composition
is the map F. (c) Prove that the map Yo(n) ---> Z (n) in (b) has degree 1, so the equation 8n (y, x) = 0 gives a (possibly singular) model for Yo(n). (d) Prove that Z(l) and Z(2) are nonsingular and that Z(3) and Z(4) are singular. Resolve the singularities of Z (3) and Z (4) and check directly that Yo(3) and Yo (4) are curves of genus O.
229
Exercises
4.14. Let ¢c(z) = Z2 + c and let ;'",n be the generalized dynatomic polynomial defined in Exercise 4.11. (a) Use a computer algebra system to compute the first few generalized dynatomic polynomials ;'",n for ¢c, say for (m , n ) in the list
{ (I, 1), (2, 1), (3, 1), (4, 1), (1,2 ), (2, 2), (3,2) , (4, 2), (1,3), (2, 3), (1, 4) } . (b) From your list, it should be apparent that many of the leading terms of ip;;' , 1 and <1>;'",2 coincide . Prove that for all m
~ 1.
(Note that this is a special property of the generalized dynatomic polynomi als for ¢c .) 4.15. Let c E C and let ¢c(z ) holomorphic function
= z2 + c. Prove
that there is a number R c
'l/Jc : {z E C : Izl > R c }
----t
>
0 and a
C
satisfying log I'l/Jc (z) I rv log
and
[z]
as
(Hint. Show that there is a consistent way to choose square roots so that
Izi ...... 00 . 'l/Jc can be defined as
limn ~ oo 2\!¢6' (Z) .) 4.16. Consider the doubl ing map
D : Q/ 'l. ----t Q/ 'l.,
D(t ) = 2t mod Z.
Fix t E Q/Z and let m ~ 0 and n ~ 1 be the smallest integers such that the denom inator of t divides 2m (2n - 1). (a) Prove that t is periodi c for D if m = 0 and that t is strictly preperiodic for D if m ~ 1. (b) Prove that m ~ 0 and n ~ 1 are the smallest positive integers such that t can be written as a (not necessarily reduced ) fraction of the form
(c) If t is preperiodic for D , prove that the preperiod of t is equal to m and that D'" (t ) is periodic with exact period n. (d) For t as above, we say that t is of type (m , n) for the doubling map D . For a given pair of positive integers (m, n) , how many t E Q/ 'l. are of type (m, n)? The map D is used in the analytic characterization of Misiurewicz points; see Theorem 4.25. However, note that (d) does not count the exaet number of Misiurewicz points of type (m, n) , because distinct t may give the same Misiur ewicz point. 4.17. ** Let ¢c(z ) (of c) by
(c) =
F
m,n
IT ( kin
= z2 + c as usual, and for integers ¢;;,+k(O) - ¢;;'( O)
¢~ - I+ k ( O ) _ ¢~ - I (O )
) J1o(n/ k) =
m , n ~ 1, define a rational function
IT (-I,m- l+k(O) + -I,m- l (O))J1o( n/ k). kin 'Pc
'Pc
Then set
Gmn(C) ,
= {Fm,n(C)
ifm ,!-1 (mod n ) orm = 1, Fm,n(C)/F1,n(c) ifm=l(mod n)andm=l=-l.
230
Exercises
(a) Prove that Gm,n(e) is a polynomial in e. More precisely, prove that Gm,n(e) is in :E[e]. (b) If n ~ 2, prove that the roots of O ,«,« are the Misiurewiczpoints of type (m , n). (c) Provethat G,« ,« (e) is irreduciblein lQ>[e). 4.18. Let V be a variety and suppose that the points of V algebraically parameterize a family of quadratic polynomials 'Ij; together with a marked point Aof formal period n. Theorem4.11 says that there is a unique morphism TJ : V -+ YI (n) satisfying TJ(?)
= ('Ij;p (z), A(P ))
E Formal(n).
Note that by construction, each point I = (e, Q;) E YI ( n) is identified with a quadratic polynomial
---->
PGL 2
such that the followingidentitiesare true for all P E VCR): (a)
U;;l 0 'lj;p 0
jp)( z) =
ri o
(b) (TJ 0 A)(P) = (jJ. 0 TJ)(P). In other words, prove that the followingtwo diagrams commute: ]p~
.pI ----->
]p~
1"'1
V
1"'1 4>
]ph (n) ----->
]pI
YI (n )
/ -1 0 )..
----->
]p~
1"'1 YI (n)
1"'1 I-'
----->
]pI
YI (n)
(Hint . During the proof of Theorem 4.11 we defineda linear fractional transformation f p( z). Provethat fp is uniquely determinedby P . Deducethat the map P f-+ fp is a morphism.)
4.19. In Example 4.14 we gave an explicit description of the quotient map YI(2) Performa similar analysis and describe the map YI (3) -+ Yo(3).
-+
Yo(2).
4.20. Weprovedthat the dynatomicmodularcurves X I (1), X I (2), and X I (3) are isomorphic to ]pl . This exerciseasks you to investigate X I (n) for other small values of n. (a) Prove that X I ( 4) is a curve of genus 2 and X 0 ( 4) is a curve of genus l. (b) Prove that Xo (4) (IQ» is finite, and Yo (4) (IQ» is the empty set. (c) Prove that X I(5) has genus 14 and X o(5) has genus 2. (d) * Prove that X o(5)(IQ» is finite, and Yo (5) (IQ» is the empty set. (e) Computethe genera of Xl (6) and X o(6 ). (f) ** Find all rational points in Yo(6)(IQ» . Section 4.3. The Space Ra t d of Rational Functions
4.21. Let
Exercises
231
Deduce that there is a nonzero homogeneous polynomial D( a, b) E Q[ao, ... ,bd ] such that
(
c/> has a fixed point Whose) ' 1rer ' equals 1 mu1tip
{=?
D(a, b) =0.
Hence
{ c/> E Ratd : c/> has a fixed point whose multiplier equals 1} is a proper Zariski closed subset of Ratd.
Section4.4. The Moduli Space Md of Dynamical Systems 4.22. Let GL n be the group of n x n matrices with nonzero determinant, let SL n be its subgroup of matrices with determinant 1, let PGL n be the quotient of Cl.; by its subgroup of diagonal matrices, and similarly let PSL n be the quotient of SL n by its subgroup of diagonal matrices. (a) Let K be an algebraically closed field. Prove that the natural map from PSLn(K) to PGLn(K) is an isomorphism. (b) More generally, prove that for any field K there is an exact sequence
1 ~ PSLn(K) ~ PGLn(K) ~ K*IK*n ~ 1. 4.23. We say that a separable rational map c/>(z) E K(z) is very highly ramified if there is a point P E pI such that the ramification index of c/> satisfies ep (c/» ;::: 3. Let
V = {c/> E Ratd : c/> is very highly ramified}. (a) Prove that V is a proper Zariski closed subset of Ratd. Hence "most" rational maps are not very highly ramified. (One says that a generic map of degree d is not very highly ramified.) (b) Prove that V is invariant under the conjugation action of PGL 2 • (c) Prove that the quotient VI PSL 2 is a Zariski closed subset of Md = Rat., I PSL 2 . 4.24. Let c/>(z) E K(z) be a nonconstant rational function. The Schwarzian derivative Sc/> of c/> is the function
(Sc/»(z)
= c/>"'(Z) c/>'(z)
_
~
2
(c/>II(Z)) 2 c/>'(z)
It measures the difference between c/> and the best approximation to c/> by linear fractional transformations. (a) If j(z) = (az+b)/(cz+d) is a linear fractional transformation, prove that (Sf)(z) = O. (b) Let j be a linear fractional transformation. Prove that c/> and j 0 c/> have the same Schwarzian derivative. (c) Let c/>(z), 'lj;(z) E K(z) be nonconstant rational functions. Prove that
(S(c/>o'lj;))(z) = (Sc/»('lj;(z)) . ('lj;'(Z))2
+ (S'lj;)(z).
(d) In particular, if j is a linear fractional transformation, prove that
and deduce that the quadratic differential form
232
Exercises w¢(z)
= (S¢>)(z) (dZ)2
t :'
is invariant under the substitution (¢>, z ) f---> (¢>!, z ). Thus the map ¢> a natural map M d ----> (quadratic differential forms on jp'1 ) . (e) Suppose that ¢>( z) has a multiple zero or pole at z
¢>(z)
= a(z -
f--->
w¢ induces
= a, say
a) m + .. . with a =1= 0 and Iml
~ 2.
Prove that S ¢> has a double pole at a . More precisely, prove that the Laurent series expansion of S ¢> around a looks like
1 - m2
(S ¢» (z ) = - 2 - (z - a)
- 2
+ .. . .
(f) Prove that the map
(¢>, P)
e----+
(S ¢» (P),
is a morphism. 4.25. ** An algebraic variety V is caIIed unirational if there is a rational map jp'N ----> V whose image is Zariski dense in V, and the variety V is called rational if there is a rational map jp'N ----> V that is an isomorphi sm from an open subset of jp'N to an open subset of V . It is clear that every moduli space M d is unirational, since Rat ., is an open subset of jp'2 d+1 and the map Ratd ----> A1d is surjective. We also know that M 2 is rational, since Theorem 4.56 says that M 2 ~ It. 2 . For which values of d is M d a rational variety? In particular, is M 3 rational ? Section 4.5. Periodic Points, Multipliers, and the Multiplier Spectrum
4.26. This exercise asks you to prove the part of Theorem 4.50 that was left undone in the text. Prove that the map
defines a function in Q[Ratd] . Prove that it is PGL 2 -invariant and deduce that it defines a function in Q[M d] . 4.27.
** What is the degree of the map
for sufficiently large N? Similarly, what is the degree of (J' 4 ,N on M locus on which it is constant?
4
away from the Lattes
Section 4 .6 . The Moduli Space M 2 of Dynamical Systems of Degree 2
4.28. Prove that ¢> E Ra t 2 is conju gate to a polynomial if and only polynomial maps in M 2 ~ It. 2 trace out the line x = 2.
0"1
(¢»
= 2. Thus the
4.29. Let ¢> E Rat2(C) be a rational map of degree 2 and suppose that one of its fixed points has a nonzero multiplier A.
233
Exercises
f
(a) Prove that there are an
E PGL 2 (iC) and a c E C such that
q/ (z) = ~
(z + c+ ~) .
Th is generalizes the normal form given in Lemma 4.59. (b) Verify that the multiplier of 1/ at the fixed point z = 00 is Aoo (1/
(4.66)
) = A.
(c) Let AI, A2, A3 be the mult ipliers of the fixed poin ts of 1> with Al numb er c in (4.66) satisfies c 2 = 4 - Al (2 + A2 + A3). 4.30. Fix d
2
= A. Prove that the
2 and consi der the subset of Rat., defined by
{1> E Ra t d : 1> has exactly two critical points}.
Bi Crit d =
For d = 2 we have Bi C rit 2 of Ratz .
=
R at», but for larger d the set Bi Crit d is a proper subset
(a) Prove that BiCrit., is an algebraic variety of dimension 5. (b) Prove that conjugation induces a natural action of PGL 2 on Bi Crit d, i.e., there is a morphism of varieties P G L 2 x Bi Crit d (c) Suppose that
---->
B iCritd,
1> E BrC rit ., has critical points at 0 and 00. Prove that 1>(z) has the form (4.67)
(d) Let 1> E BrC ri t ., and apply a conjugation to move the critica l points of ¢ to 0 and 00, so ¢ has the form (4.67) describ ed in (c) . Prove that the following quantities depend only on the conjugacy class of ¢:
w(c/»
=
Q
J
(i+1(3d -1
(3/ -
(3'Y '
a( ¢)
=
(oJ - Ih)d'
(e) Let ¢ , 'Ij; E Bi C ri t d and suppo se that
w(¢)
= w('Ij;) ,
a (¢ ) = a('Ij; ),
Prove that ¢ and 'Ij; are PGL2 conjugate (working over an algebra ically closed field). (I) Prove that the three quantities (4.68) described in (d) satisfy the relation
o-r
= W d - 1 (w + 1) d+ 1
and no other relati ons. Conclud e that M~iCrit is isomorphic to A 2. (This generalizes Theore m 4.56, since M ~iCrit = M 2.) (g)
Describ e the stable and semista ble completions of jvt ~iCril coming from geo metric invariant theory.
**
Section 4 .7 . Automorphisms and Twists 4.31. In Section 4.7 we defined two rational maps ¢ and 'Ij; to be equivalent if there is an f E PG L2(K ) such that 'Ij; = 1>f , and similarly to be K -equivalent if there is an f E P GL 2 (K) such that 'Ij; = 1>f .
234
Exercises
(a) Prove that these definitions do indeed define equivalence relations on the set of rational maps. (b) More generally, suppose that a group G acts on a set X . Define a relation on X by setting y '" x if there exists a a E G such that y = u (x ). Prove that this is an equivalence relation. 4.32. Let <jJ(z) E K (z) be a rational map. (a) Prove that Aut (<jJ) is a subgroup ofPGL 2(K ). (b) Let hE PGL2(K). Prove that Aut(<jJh) = h- I Aut (<jJ)h, so Aut(<jJ) and Aut(<jJh) are conjugate subgroups ofPGL 2(K ). 4.33. Let <jJ(z) E K (z), let f E Au t (<jJ), and let 0: be a critical point of <jJ (i.e., <jJ' (o:) = 0). Prove that f (o:) is also a critical point of <jJ . More generally, prove that 0: and f (o: ) have the same ramification index. 4.34. Describe all polynomials <jJ(z) E K[z] whose automorphism group Aut(<jJ) is nontrivial. (Hint. If f E Aut( <jJ), what does f do to the totally ramified fixed point(s) of <jJ?) 4.35. Let <jJ(z) E K (z) be a rational map of degree d and write <jJ(z) as a quotient of polynomials
+ adz d <jJ(z) + bdZd Prove that f (z) = Z-1 is in Aut(<jJ) if and only if b, = ad-i for all i. 4.36. Let <jJ(z) = (Z2 - 2z)/( -2z + 1). + al Z + = bo + bI Z + co
(a) Prove that Aut (<jJ) contains the maps
I z- I I z } { z ' -z ' - z - ' -l - z' -z--l, l - z , and that they form a group isomorphic to 5 3. Prove that Aut( <jJ) ~ 53. (Hint. Find the fixed points of <jJ .) (b) Compute the values of UI (<jJ), U2(<jJ), and U3(<jJ).
#
Aut(<jJ)
6, so in fact
4.37. Let ( E K be a primitive nth root of unity and let <jJ(z) E K (z) be a nonconstant rational map. (a) Prove that Aut (<jJ) contains the map f (z ) = (z if and only if there is a rational map 1/J(z) E K (z) such that <jJ (z) = z1/J(zn). (b) Prove that Au t (<jJ) contains both of the maps f (z) = ( z and g(z) = liz if and only if there is a polynomial F( z) E K[z] such that the rational map 1/J(z) E K(z) in (a) has the form 1/J(z) = F(z) / (zdF(z-I) ), where d = deg(F). Verify that the group generated by f and 9 is the dihedral group of order 2n. (c) Let <jJ(z) = zd. Prove that Au t (<jJ) is a dihedral group of order 2d - 2. 4.38. We identify the set of rational functions Rat., of degree d with an open subset of lP'2d+1 as described in Section 4.3. (a) Let f E PGL2. Prove that the set { <jJ E Rat., : <jJI = <jJ } is a (possibly empty) Zariski closed subset of Ratz. If f( z) ¥ z, prove that it is a proper subset of Ratd . (b) Let A C PGL 2 (K ) be a nontrivial finite subgroup . Prove that the set
{ <jJ E Rat d : <jJI is a proper Zariski closed subset of Rat d.
= <jJ for all f
E A}
235
Exercises
(c) Prove that up to conjugation, PGL2(K) has only finitely many distin ct finite subgroups of any given order. (d) Prove that the set {¢ E Rat., : Aut(¢) i= I} is aproperZariski closed subset of'Ratz. (Hint. Note that the order of Allt (¢ ) is bounded by Proposition 4.65. ) (e) Prove that the set {¢ E Rat., : Aut ]¢) i= I} is PGL 2-invariant and defines a proper Zariski closed subset of M d. (Hint. The groups Allt (¢ ) and All t (¢ i ) are conjugate subgroups ofPGL 2; see Remark 4.64 and Exercise 4.32.) 4.39. Let a E C * and d ~ 1 and consider the rational function
¢ (z ) = a (Z + l ) d z- l Prove that Aut(¢)
= {f E PGL 2(C) : ¢i = ¢}
is trivial.
4.40. Let ¢( z) E K (z) be a rational map of degree d ~ 2 and assume that Aut (¢) contains the element h(z) = -zoforder2. (a) Suppose that d is even. Prove that at least one of the fixed points of ¢ ( z) is defined over K. (b) Suppose further that d = 2. Prove that there is an f E PGL 2 (K ) such that hi = hand such that ¢i has the form ¢I (z)
= az +~ . z
bz- I . Prove that for all c E K *, the maps ¢ a,bc2 and ¢ a,b
(c) Write ¢ a,b(Z) = az + are PGL2(K)-equivalent.
2
(d) In homogeneous form we have ¢a,b ([X, YJ) = [aX 2 + by , X Y]. Let ~ (a , b; X , Y) be the associated dynatomic polynomial. Prove that ~ ( a , b; X , Y) E Z[a, b, X , Y ]. (e)
* If n
~ 4 is even, prove that ~ (a, b; X , Y) is reducible. More preci sely, prove that there are nonconstant polynomials 'li n, An E Z[a, b, X , Y] such that
~ (a ,
bj X , Y) = 'li n(a, b; X , Y )An(a, b; X , Y) .
(Hint. Divide the set of points P of formal period n into two subsets depending on whether the involution h permutes the orbit O
4.41. Let ¢
= az + bz- 1 E Rah
(a) Prove that (al(¢),a2(¢)) depends only on a. (b) We saw in Exercise 4.40 that Aut(¢) contains {±z}, so it has order at least 2. Find all values of a for which Aut(¢ ) is strictly larger than {±z} . (c) As a varies, prove that the set of points
describes the curve
C : 2x 3
-
8x y + x 2y - 4y 2 - x 2 + 12x + 12y - 36
in A.? ~ M2 . (d) Prove that the curve C in (c) is singular at the point (- 6, 12) .
=0
236
Exercises
(e) Move the singular point to the origin and perform a further change of variables to prove that the curve C is isomorphic to the curve y2
= 4x 3 + x4 .
Thus the singularity of C is a cubic cusp. (f) The point (- 6, 12) is the unique singular point of the curve C, so the rational map ¢ satisfying (()1(¢) , ()2 (¢») = (- 6, 12) should be special in some way. How is it special? (Hint. Compare with the answer to (a). See also Exercise 4.36.) Section 4.8 . General Theory of Twists 4.42. Let X be an object defined over K , and for each twist Y of X, fix a K-isomorphism iy : Y ----> X. Assume that Aut (X ) is abelian. Prove that Twist(X/K)
----t
H 1 (Gal(K /K ), Aut(X)),
is a well-defined one-to-one map of sets. (See Remark 4.78.) Section 4.10. Fields of Definition and the Field of Moduli 4.43. Let ¢ E K (z ) be a rational function. We know from Proposition 4.84 that the field of moduli 1(.1> is contained in every field of definition of ¢ . Prove that K <j> is equal to the intersection over all fields of definition of ¢. 4.44. Let C« : x 2 + y2 = a be the family of curves studied in Exampl e 4.76. (a) Let p be a prime numb er satisfying p == 1 (mod 4). Prove that Cp is lQl-isomorphic to C 1 . (b) Let p be a prime number satisfying p == 3 (mod 4). Prove that Cp(lQl) = 0. Deduce that Cp is not lQi-isomorphic to C l , and hence that [Cp]K represents a nontrivial element of T wist (C 1/ lQl). (c) Let p and q be distinct prime numbers that are congruent to 3 modulo 4. Prove that C p and Cq are not lQl-isomorphic. Deduce that T wist(C 1/ lQl) is an infinite set. 4.45. Let 9 be the cocycle
z
ga (Z ) = { -l / z
ifl7 (i ) ifl7 (i )
= i, = -i,
described in Example 4.88, and let K. be the associated fixed field in Q(w ). (a) Prove that K. = Q(u, v) , where u = w - l/w and v = i (w + l /w). Deduce that C : u 2 + v 2 = -4 is the twist of lP'1 / lQl associated to the cocycle g. (b) Let ¢ (z ) = i((z - l )/ (z + 1))3 be the rational map from Example 4.85 satisfying l7(J) = ¢ga. Our general theory says that K is a field of definition for ¢ if and only if C( K ) =1= 0, where C is the curve in (a). For example, C(lQl(H )) =1= 0, so lQl(H ) must be a field of definition for ¢(z ). Find an explicit linear fractional transformation f E PGL 2 (Q ) such that ¢! (z) E lQl( H)( z). Section 4.11. Minimal Resultants and Minimal Models 4.46. ** This exercise raises some natural questions concerning global minimal models . (a) Is it true that every rational map ¢( z ) E lQi (z) of (odd) degree d ~ 2 has a global minimal model over 1Qi?
237
Exercises
(b) Let K be a number field, let R be the ring of integers of K, let 5 be a finite set of primes, and let ¢(z) E K(z) be a rational map of degree d 2: 2. We say that ¢ has a global Ssminimal model if there is a linear fractional transformation f E PGL2 (K) and homogeneous polynomials F and G satisfying ¢f = [F, G] with the property that the coefficients of F and G are in the ring of 5-integers Rs and for every prime p r:J. 5. Suppose that Rs is a principal ideal domain. Is it true that every ¢ of (odd) degree has a global 5-minimal model? (c) Let K be a number field and ¢(z) E K (z) a rational map of (odd) degree d 2: 2. Suppose that the Weierstrass class ii1>/ K is trivial. Is it then necessarily true that ¢ has a global minimal model? (This is true in an analogous situation for elliptic curves; see [410, VIII.8.2].) 4.47. ** Let K be a number field and let ¢ E K(z) be a rational map of degree d be a finite set of primes of K. Prove that
2: 2. Let 5
{ 1jJ E Twist( ¢/ K) : 1jJ has good reduction at all p r:J. 5} is a finite set. 4.48. ** Fix an embedding of the moduli space Md in projective space and let hM d denote the associated height function. Let K /Q be a number field, let d 2: 2 be an integer, and let B 2: 1 be a number. Prove that the set
contains only finitely many PGL 2(K)-conjugacy classes of rational maps. 4.49. * Let K be a number field, let R be the ring of integers of K, let 5 be a finite set of primes of R, and let d 2: 2 be an integer. Prove that there is a finite set of rational maps l3K ,S,d C Ratd(K) such that if 4> E Ratd(K) is a rational map satisfying (I) ¢ has three or more critical points, (2) the critical points of ¢ remain distinct modulo all primes not in 5, (3) the critical values of ¢ remain distinct modulo all primes not in 5, where we recall that a critical value of¢ is the image of a critical point of ¢, then there are automorphisms t, 9 E P GL 2 (Rs) such that
fo¢og E BK,s,d. (Note that when we say that the critical points or values are distinct modulo a prime p, we really mean that they are distinct modulo \J3 for all primes \J3 lying above p in a suitable finite extension of K.)
Chapter 5
Dynamics over Local Fields: Bad Reduction In this chapter we return to the study of dynamical systems over complete local fields such as iQ>p. We saw in Chapter 2 that if a rational map ¢(z) E iQ>p( z) has good reduction , then its Julia set is empty, in which case considerable information about the dynamics of ¢ on jp'l (iQ>p) may be deduced by studying the dynamics of the reduct ion ¢ on jp'l (JFp). But if ¢( z) has bad reduction, then the situation is far more complicated. Indeed , since "interesting" unpredictable dynamic s occurs only in the Julia set, we might say that the good reduction scenario studied in Chapter 2 is the uninteresting situation. This chapter is devoted to the interesting case! The field iQ>p and its finite extensions have the agreeable property that they are complete, but they are not algebraically closed , so they are more analogous to JR than they are to C. This suggests that we work instead with an algebra ic closure Qp oflQ!p, but unfortunately we then lose the completeness property! Going one step further, we take the completion of Qp, and it turns out that this field, denoted by
Cp
= the completion of the algebraic
closure ofiQ>p,
is both complete and algebraically closed. For proofs of the basic properties of Qp and c; see for example [81, 175, 183,249, 382]. The fields C and C p share many common properties, but they also differ in crucial ways. In particular, the field of p-adic complex number s C p is not locally compact! However, it is often essential to work in Cp, rather than in a finite extension ofiQ>p, for example if we want to guarantee the existence of large numbers of periodi c points. Table 5.1 compares some of the properties of the complete fields iQ>p, Qp, Cp, JR, and C. The subject of p-adic and more general nonarchimedean dynamics is relatively new. After some early articles [50,201 ,278,433,438] in the 1980s, there was an explosion of interest that put the subject on a firm footing with a body of significant theorems and, just as importantl y, an array of fascinating conjectures. In this brief 239
240
5. Dynamics over Local Fields: Bad Reduction
Nonarchimedean metric
.;
Algebraically closed Complete Locally compact Totally disconnected
.; .; .;
.; .;
.;
.; .; .;
.; .;
.; .; .;
.;
Table 5.1: Comparison of complete fields.
chapter we can provide only a glimpse into this active area of current research, with many important topics omitted entirely in order to keep the chapter at a manageable length. For example, we do not touch on the important concept of local conjugacy, nor do we describe Rivera-Letelier's classification of Fatou domains [373, 375]. For the reader desiring further information, we mention the following articles on p-adic dynamics that are listed in the references: [4,5, 13, 14,22,23,26,30,29,50,63,53, 54,56,57,58,59,60,62,70,71,72,73,82,84,104,116,170, 168, 169, 185, 188, 189,201,206,208,220,222,236,237,238,239,240,241,242,243,244,245,262, 266,267,268,271,272,273,274,278,280,282,283,320,318,319,334,337,338, 339,340,344,348,352,355,372,373,374,375,376,378,389,427,433,438,449].
5.1 Absolute Values and Completions We recall from Section 2.1 that a valued field is a pair (K, I . I) consisting of a field K and an absolute value I . I on K. In this section we briefly remind the reader of the construction and basic properties of the completion of a valued field.
(K, I . I) be a valued field. A sequence aI, a2, a3,'" Cauchy if for every E > 0 there exists an N = N (E) such that
Definition. Let
for all m,n
~
E
K
is
N.
In other words, the sequence {an} is Cauchy if Ian - ami ~ 0 as m, n ~ 00. It is clear that every convergent sequence is Cauchy. A valued field K is said to be complete if every Cauchy sequence in K converges.
Example 5.1. The real numbers JR and the complex numbers C are complete with respect to their usual absolute values.
Theorem 5.2. Let (K, I . IK) be a valued field. Then there exists a valued field (k, I . Itc), unique up to isomorphism of valued fields, with the following properties: (a) There is an inclusion K (b) k is complete.
ck
as valuedfields.
5.1. AbsoluteValues and Completions
241
(c) K is a dense subset of k. Further, k is the smallest valuedfield satisfy ing (a) and (b) in the follow ing sense: Let (L , I . IL) be any comp lete valued field containing K. Then there is a unique inclusion k <---+ L ofvalued fi elds that respects the inclusions of K into k and L. Thefield k is called the completion of K with respect to the absolute value I . IK . Proof (Sketch) The field k may be constructed as follows. Let C be the set of all Cauchy sequences in K and make C into a ring by setting and We define an absolute value on C by setting
and we define an equivalence relation
r-
on C by
Then one checks that these definitions are consistent and that the quotient
k = Cj '" is a complete field with the desired properties. Note that K is identified with the subfield of k consisting of constant sequences. For further details on completions, see for example [190,4 §7] or [217, IV §l ]. 0 Theorem 5.3. Let (K , I . I K ) be a comp letefie ld and let L I K be a finite extension. Then there is a unique absolute value 1 . IL on L extending the absolute value on K. Thefield L is compl ete with respect to 1 . I L . Proof (Sketch) One checks that
lalL
I . IL =
is given by
INL/K(a)I~[L:K].
o
See [259, Proposition XII.2.6] or [382, II §4]. Example 5.4. The following table gives some examples of complet ions. Field
Absolute Value
Completion
Q
lal oe = max{a , - a} ja + bil = Ja2 + b2 lal p = p - or dp (a )
lR
Q(i) Q
C Qp
242
5. Dynamics over Local Fields: Bad Reduction
Remark 5.5. Theorem 5.3 says that for a finite extension LI K, there is a unique absolute value on L extending the absolute value on K, and further that L is complete with respect to this extended absolute value. If we go to the algebraic closure k of K, then we still get a unique absolute value, since k is the compositum of the finite extensions of K, but unfortunately k may not be complete. (See, e.g., [249, 111.4, Theorem 12] for a proof that Qp is not complete.) So we use Theorem 5.2 to form the completion of the valued field K as in the next theorem.
Theorem 5.6. Let C p = Qp
= the completion ofthe algebraic closure ofQp-
Then C p is both complete and algebraically closed. It is the smallest complete algebraically closedfield containing Qp. Proof More generally, the completion of an algebraically closed nonarchimedean field is algebraically closed. We briefly sketch the proof. Let a be algebraic over Cp, say a root of f(X) = L: aiXi E Cp[X]. Choose b, E Qp that are very close to ai, which can be done since Qp is dense in Cp. Then the roots of g(X) = L: biX i can be matched with the roots of f(X), so we can find a root f3 of g(X) that is very close to a. Note that f3 E Qp, since Qp is algebraically closed. Thus we can find elements in Qp (so a fortiori in Cp) that are arbitrarily close to a. This gives a sequence of elements of C p that converges to a, so the fact that C p is complete implies that a E Cp. For details of the proof, see [249, 111.4, Theorem 13] or [382, D Lemma 11.4.2].
5.2 A Primer on Nonarchimedean Analysis Throughout this section we take K to be a field that is complete with respect to an absolute value I . I satisfying the nonarchimedean (ultrametric) triangle inequality
la + f31 ::; max{ lal, 1f31} for all a, f3 E K.
(5.1)
We recall from Lemma 2.3 that if lal t- 1f31, then (5.1) is an equality. We set the following notation for the ring of integers of K, its unit group, and its maximal ideal:
R=RK ={aEK:lal::;l}, R*=R'K ={aEK:lal=l},
9J1 = 9J1K = {a E K: lal < I}. We define "open disks," "closed disks," and "circles" in K by the usual formulas: D(a, r) D(a, r)
= {u = {u
S(a,r) = {u
lu E K : lu E K: lu E K :
al
< r} = open disk of radius r at a,
al ::; r}
= closed disk of radius r at a,
al = r} = circle of radius rata.
243
5.2. A Primer on Nonarchimedean Analysis
However, it is important to note that in the nonarchimedean setting, the disks D( a, r) and D(a, r) and the circle S(a, r) are simultaneously open and closed sets! To see this, let x E S( a, r). Then for any s < r we have D(x, s) C S( a, r), since any y E D(x,s) satisfies
< s < r,
Iy - z]
so
Iy - al = max{ly - z], Ix - al}
= r.
This shows that S (a, r) is open, and it is clearly closed by definition. It follows that D(a, r) = D(a, r) U S(a, r) is open and that D(a, r) = D(a, r) "S(a, r) is closed. Notice that D(O, r) and D(O, r) are groups under addition, i.e., they are closed under addition and negation, with a nontopological meaning of the word "closed." More precisely, the closed unit disk D(O, 1) is the ring of integers R of K and the open unit disk D(O, 1) is the maximal ideal9J1 of R. Recall that the chordal distance between two points P1 = [Xl, Yd and P2 = [X 2 , Y2 ] in jp'l(K) is the quantity (cf. Section 2.1)
(P P.) _ IX1Y2 - X 2Y11 p 1, 2 - max{IX11,IY11} max{IX21, 1Y21}'
°::;
The chordal metric p satisfies p(P1 , P2 ) ::; 1, so it is bounded. In particular, if K is a finite extension of Qlp, so K is locally compact, then jp'1 (K) is compact, since it is a locally compact bounded metric space. On the other hand, the field C p is not locally compact, so although jp'1 (C p) is a bounded space, it is not compact, nor even locally compact. We can define open and closed disks in jp'1 (K) using the chordal metric. We will use the same notation as above:
D( Q, r)
= {P E jp'l(K)
D(Q,r) = {P
E
: p(P, Q) < r}
= open disk of radius r in jp'1 (K),
jp'l(K): p(P,Q)::::: r} = closed disk of radius r injp'l(K).
It should be clear from context whether we are working in K or in jp'1 (K). We observe that jp'1 (K) is equal to the union of the two unit disks
jp'l(K) = {[x,I]: Ixl::;
1} U {[I,y]:
Iyl::; I},
each of which is metrically isomorphic to RK.
Remark 5.7. It is sometimes convenient to make the assumption that the radius r of a disk is equal to the absolute value of some element of K* . In this case we say that the disk has rational radius. We note that if K/Qlp is a finite extension of ramification degree e, then the value set IK* I has the form {pn/e : n E Z}. Similarly, the value set IC;I is equal to {pT : r E Ql}. Thus rational radius really means that the radius is a rational power of p. The nonarchimedean nature ofthe absolute value implies that a sequence {aih>o in K is Cauchy if and only if
lim lai+l - ail = O. "->(X)
244
5. Dynamics over Local Fields: Bad Reduction
This follows immediately from the inequality
This observation and the completeness of K tell us that a power series 00
¢(z) =
L ai(z - a)i
E
K[z]
i=O
converges if and only if
lim lai(z - a)il = t--->oo
o.
In other words, the "nth term test" from elementary calculus becomes both necessary and sufficient in the nonarchimedean setting.
Definition. A function ¢ : D(a, r)
----t
K is holomorphic (or analytic) ifit is repre-
sented by a power series 00
¢(z) =
L ai(z - a)i
E
K[z - a]
(5.2)
i=O
that converges for all z E D(a, r). The order of ¢ at a, denoted by orda(¢), is the smallest index i such that ai =I- O. A meromorphic function on D(a, r) is a quotient ¢ = ¢I!¢2 of functions ¢I and ¢2 =I- 0 that are holomorphic functions on D(r, a).l A meromorphic function ¢ = ¢I!¢2 induces a well-defined map
¢ : D( a, r)
-----7
jp'I (K),
The order of ¢ at a is the difference
We say that ¢ has a zero (respectively a pole) at z iford",(¢) < 0).
= a if ord., (¢) > 0 (respectively
The next proposition describes some elementary properties of nonarchimedean holomorphic and meromorphic functions.
Proposition 5.8. (a) Let ¢(z) be a holomorphic function on the closed disk D( a, r) and let bE D(a, r). Then ¢(z) is a holomorphicfunction on D(b, r), i.e., ¢(z) is given by a convergent power series in K[z - b]. (b) Let ¢(z) be a nonzero holomorphicfunction on D(a, r). Then the zeros of ¢(z) in D (a, r) are isolated. This means that if ¢( b) = 0, then there is a disk D(b, E) such that ¢(z) =I- 0 for all z E D(b, E) <, {b}. 1More precisely, a meromorphic function is an equivalence class of pairs (P1,
i= 0, with
245
5.2. A Primer on Nonarchimedean Analysis
(c) Let ¢(z ) be a meromorphicfunction on 15(0., r) and suppose that the only pole of ¢(z ) in 15(0., r}, if any. is z = o.. Then ¢ (z ) is represented by a convergent Laurent series, 00
for all z E D(o.,r) <, {a }.
(5.3)
i= - m
(d) Let ¢ (z ) be a meromorphic f unction on 15(0. , r ). Then for eve,y b E 15(0., r )
there is an s such that ¢(z ) is represented by a convergent Laurent series on 15(b, s). Proof (a) Let ¢(z) formula
= I: o.i(Z -
c )' , and for k ~ 0, define coefficients bk by the
(These values are not mysterious. If K has characteristic 0 then they are the usual Taylor coefficients bk = (l/k !)(dk¢ / dzk)( b).) The series defining bk converges since the convergence of ¢( z) on b (o. , r) implies that
lo.i (b - o.)il :::;
lo.ilri
----t
0 as i
----t
00.
Further, we have the estimate so Hence Ibklrk ----t 0 as k ----t 00, so the power series I: ~ o bk(z - b)k converges on 15(b, r) . Finall y, we check that the series represents ¢ by computing
~ bk(z -
b)k = =
~ ~ G)o.i(b -
f= i =O
o.;
t G)
a/- k(z - b)k
(b - o.)i- k(z - b)k
k =O
00
=
L o.i( z - a)i. i= O
(b) Let b E 15 (o. ,r) be a point with ¢ (b) = O. We will find a deleted neighborhood of b on which ¢ (z ) is nonvanishing. We use (a) to expand ¢ as a power series ¢ (z ) = I: bi(z centered at b and converging on 15(b, r). The assumption that ¢ i- 0 means that some coefficient is nonzero; we let j be the smallest index such that bj i- O. The fact that ¢ converges on 15(b, r ) implies that Ibilr i ----t 0 as i ----t 00 , so there is a constant C such that Ibilri < C for all i. Let E = r j + 1 Ibj l/2C. We claim that the only zero of ¢(z ) on 15(b, E) is z = b. To see this, we take any z E D(b, E) and estimate
W
246 .max ' ~J + l
5. Dynamics over LocalFields: Bad Reduction
Ibil .lz_bli ~ m~ =
Hence for all
G
'~J + l
Z
E
( Iz- bl) i ~ G ( Izr
Glz - bl rj+ l
D(b, E) with
Z
bl) j+1
r
.
GE
.
~ r j+1
·I(z- WI
1
.
·I(z - WI = "2!bj (z - WI·
-I b, the first term in the series 00
¢(z ) = bj (z - b)j +
L
bi(z - b)i
i=j+1 has absolute value strictly larger than any of the other terms , so the nonarchimedean triangle inequality (Lemma 2.3) implies that for all
Z
E D(b, E).
In particular, ¢(z) -10 for all z E D(b, E) with z -I b. (c) Write ¢( z) = (Pt (z) /¢2 (z) as a quotient of functions that are holomorphic on D(a,r ). We give the proof of (c) in the case that ¢2(Z) is a polynomial, which is the only case that we will need. For the general case, see [81, 175] or Exercise 5.7. Let se D(a, r ) with b -I a and ¢2(b) = O. Taking the Taylor series of ¢ l and ¢ 2 around b, we can write and where 'l/Jl and 'l/J2 are nonvanishing holomorphic functions on D(b, r). Then the assumption that ordb(¢) ~ 0 implies that we can write ¢(z) as a quotient
of holomorphic functions on D (a,r ) such that the denominator does not vanish at b. Repeating this process for each of the zeros of ¢2 in D(a, r) other than z = a, we find that ¢(z) is a quotient of holomorphic functions on D(a, r ) such that the denominator does not vanish except possibly at a. Further, this cancellation process must stop, since we have assumed that ¢2 (z) is a polynomial, so it has only finitely many zeros. By abuse of notation, we again write ¢(z ) = (Pt (Z)/¢2(Z), where we may now assume that the polynomial ¢2(Z) has no zeros in D(a, r ) other than z = a. We assume for the moment that K is algebraically closed and factor ¢2(Z ) as n
¢2(Z ) = c(z - a)e
II (z -
Qi Y i .
i= l
By assumption, the roots satisfy Q i rJ. D(a, r), or equivalently 1 ~ i ~ n. The reciprocal 1/¢2 (z) has a partial fraction expansion
IQ i -
a[ > r, for all
5.2. A Primer on Nonarchimedean Analysis
1
e
_ =~
A.
n
J
~ (z - a)j
(l>2( z)
247
B ..
ei
+~~
tJ
~ ~ (z - O:i)j
for certain coefficients A j , B ij E K. (See [259, IV §5] or [436, §5.1O].) The terms with negative powers of z - a form part of the desired Laurent serie s. We claim that all of the other terms are holomorphic on D(a,r ). To verify this, we let 0: E K with 1 0: - al > r , and for each k 2: 0 we consider the function 1/ (z - 0:)k+ 1 • If k = 0 we get a geometric series 1
1
z - 0:
=
(z - a) - (0: - a)
= 1-
-(o: -a) - 1 (0: - a)-I (z - a)
-1
= 0: -
~ ( z-a) n a ~ 0: - a
(5.4) This series converges for all z satisfying Iz - al < 10: - c], so in particular for all z E D(a, r), since 10: - al > r . Hence 1/(z - 0:) is holomorphic on D(a, r). More generally, the same argument works using the identity/
1
1
(z - o:)k+ l
((z - a) - (0: - a))k+l
=
(_ 1)k+ ~ (0: - a)k+ l ~ 1
(n+ k) (~)n 0: - a k
This completes the proofthat 1/lh (z) is holomorphic on D(r,a) in the case that K is algebraically closed. If K is not algebraically closed, we let L be the completion of the algebraic closure of K and use the above argument to write
Remark 5.9. The assumption that
248
5. Dynamics over Local Fields: Bad Reduction
Further, if D (a,r) has rational radius, then r E ing 11 1>11 = Ibl ·
IK*I, so there is a bE
K * satisfy-
The nonarchimedean nature of the absolute value gives immediately the inequality
11>(z)l :::; sup lai!· lz - ali :::; 11 1>11 for all z E D(a,r ). i2: 0
We now show that 11 1>11 is more or less a Lipschitz constant for a holomorphic map 1>. Proposition 5.10. Let 1>(z ) E K[ z] be a power series converging on D(a,r). Then
[1>(z) - 1>(w) l :::; lltll [z-w[ !ora!l z,wED (a,r ). r
Proof We take z, wE D(a,r) and compute 11>( z ) - 1>( w) I = I
=
~ a;((z -
a)i
-
(w -
ani
00 i- I j [z - w i ~ a;.t;(z - a)i- I- (w - a)j
I
I
:::; [z - w isup su p lail ·l z - al i-I - jl w - alj i 2:0
os. «
:::; [z - wi sup lail r i - I ,
since [z - c], Iw - a l :::; r,
i2:0
=
Iz -wl~ . r
D
5.3 Newton Polygons and the Maximum Modulus Principle A powerful tool in complex analysis is the maximum modulus principle.' which asserts that a holomorphic function 1>(z ) on an open set U C C has no maximum on U. Equivalently, if D c U is any closed disk in U, then 11>(z)1 attains its maximum value on the boundary oD of D. In this section we prove a nonarchimedean analogue of the maximum modulus principle that is of similar fundamental importance in the theory of nonarchimedean analysis. However, in the nonarchimedean setting we cannot prove the maximum modulus princ iple using path integrals and Cauch y's residue theorem. In their place we substitute the powerful method of the Newton polygon. 3Indeed, Ahlfors [I ] says that "because of its simple and explicit formulation it is one of the most useful general theorems in the theory of functions. As a rule, all proofs based on the maximum principle are very straightforward, and preference is quite justly given to proofs of this kind."
5.3. Newton Polygons and the Maximum ModulusPrinciple
249
The Newton polygon of a nonarchimedean power series ¢(z) is very easy to describe, and it can be used to give a precise description of the distribution of zeros of the power series. To ease notation, we let
v(z) = -logp Izi for z E Cpo Notice that the valuation v is a surjective homomorphism v : C; have normalized I . I so that Ipl = p-l.
-*
(Ql, since we
Definition. Let ¢(z) = L anzn E Cp[z] be a power series. The Newton polygon of ¢ is the convex hull of the set of points
{(n, v(an ) )
: ti
= 0, 1,2, .. ,},
where by convention we set v(O) = 00. Informally, the Newton polygon is created as follows: take a vertical ray starting at the point (0, v(ao)) and aiming down the yaxis. Then rotate the ray counterclockwise, keeping the point (0, v(ao)) fixed, until it bends around all of the points (n, v(a n ) ) . The Newton polygon consists of a set of line segments that connect the dots required to create the convex hull. A typical Newton polygon is illustrated in Figure 5.1(a). It has a segment from (0,5) to (2, 1), a segment from (2,1) to (4, -1), a segment from (4, -1) to (7, -1), etc. A Newton polygon may have infinitely many line segments, or it may terminate with an infinite ray. A fundamental theorem says that the Newton polygon of an analytic function contains a tremendous amount of information about the roots of the function. It provides a very powerful tool for studying nonarchimedean power series.
Theorem 5.11. Let ¢(z) E Cp[z] be a power series. Suppose that the Newton polygon of ¢ includes a line segment ofslope m whose horizontal length is N, i.e., the Newton polygon has a line segment running from
whose slope is
v(an+N) - v(a n)
m=--'---'-'--'---'--':-c:----'-~
N
.
Suppose further that ¢ converges on the closed disk of radius pm. Then ¢( z) has exactly N roots 0:, counted with multiplicity, satisfying 10:1 = pm. Proof See [249, IVA, Corollary to Theorem 14]. We observe that the proof of this result for polynomials or rational functions is quite easy. For power series, one first proves a p-adic version of the Weierstrass preparation theorem saying, roughly, that ¢(z) factors into the product of a polynomial g(z) and a nonvanishing power series 'IjJ(z) such that the initial parts ofthe Newton polygons of ¢(z) and g(z) coincide. Then the theorem for power series follows immediately by applying the elementary result for polynomials to 9 (z). D
250
5. Dynamics overLocal Fields: Bad Reduction
• •
•
•
• ¢ has no roots in D(O, 1)
(a) Typical Newton polygon
(b) Positive-slope Newton polygon
Figure 5.1: Examples of Newton polygons.
Example 5.12. The Newton polygon ofthe power series
¢(z) = p5+p4 Z+ pZ2 + pz3 +p-l Z4 +p-l Z5 +p3 Z6 +p-l Z7 +p2 ZS +p3 z9 + ... is illustrated in Figure 5.1(a). The leftmost line segment has slope -2 and width 2, so ¢(z) has exactly 2 roots 0: satisfying 10:1 = p-2 (assuming that ¢(z) converges on the appropriate disk). Similarly, ¢(z) has exactly 2 roots satisfying 10:1 = p-l, exactly 3 roots satisfying 10:1 = 1, and exactly 2 roots satisfying 10:1 = p2. Our first application of the Newton polygon is a nonarchimedean version of the classical maximum modulus principle from complex analysis.
Theorem 5.13. (Maximum Modulus Principle) Let ¢( z) E Cp [z] be a power series that converges on a disk D( a, r) ofrational radius. (a) There is a point (3 E D (a, r) satisfying 1¢((3)1 =
syP
1¢(z)1 = II¢II·
zED(a,r)
(b)
If ¢ does not vanish on D( a, r), then 1¢(z)1 = II¢II Jar all z E D(a,r). In other words, either ¢ has a zero in D( a, r), or else it has constant magnitude on D(a, r).
Remark 5.14. For many applications, it suffices to know that the maximum modulus principle is truefor rational functions ¢(z) E Cp(z). In this case, both (a) and (b) are quite easy to prove. We do (b) and leave (a) for the reader. Replacing z by (z - a)/e for some ewith lei = r, we may assume that ¢(z) is well-defined and nonvanishing on the disk D(O, 1). We factor ¢(z) as
¢(z) = zk (1 - O:lz)(1 - 0:2 Z ) ' " (1 - O:r Z ) (1 - (31z)(1 - (32Z)'" (1 - (3sz)
5.3. Newton Polygons and the Maximum Modulus Principle
251
The assumption that ¢ has no zero or pole at z = 0 implies that k = O. Further, we have lail < 1 and 1,6i l < 1 for all i, since otherwise ¢ would have a zero at ai l E D (O , 1) or a pole at fJi l E D (O, 1). It follows that
11 - aiz l = 11- fJiz l = 1 for all z
E
D(O, 1),
and hence 1¢(z)1 = 1 for all z E D (O, 1).
Proofof Theorem 5.13. Write ¢( z ) = I: ai(z - a)i and choose constants b, e E K * with lei = rand Ibl = II¢II (cf. Remark 5.9). Consider the series
'l/J(z ) = b-l ¢(ez + a)
= b- l
00
00
i=O
i= O
L ai(ez) i = L
i
a~e zi.
The convergence of ¢ on D(a, r) clearly implies the convergence of 'l/J on D(O, 1), and we have aiei Jailri I ¢II 11 'l/J11 = ~~~ -b- = ~~~ -Ib-I = = 1.
I
I
1bI
Replacing ¢ by sb, we are reduced to proving the theorem under the assumptions that ¢ converges on the unit disk D (O, 1) and that II¢II = 1. (a) The condition II¢II = 1 says that every coefficient of ¢ lies in R, and the fact that ¢ converges on D(O, 1) implies that all but finitely many coefficients lie in the maximal ideal 9Jl of Cpo So when we reduce ¢( z) modulo 9Jl, we get a nonzero polynomial Let a l , ... , aT E R be representatives for the distinct roots of ¢( z) mod 9Jl in the residue field R/9Jl. The residue field is infinite, since K is algebraically closed, so we can find a fJ E R satisfying
fJ
t
ai
(mod 9Jl)
for all 1 Si S r ,
Then ¢(fJ ) t o (mod 9Jl), so 1¢(fJ)1 = 1 = I ¢II· (b) Write ¢(z) = I: aizi as usual and consider the Newton polygon of ¢(z) . Theorem 5.11 asserts that if some line segment of the Newton polygon of ¢(z ) were to have slope m S 0, then ¢(z ) would have at least one root a satisfying lal = pm S 1, and hence ¢(z ) would have a root in D (O, 1). Thus our assumptionthat ¢(z ) has no roots in 15(0, 1) implies that every line segment of the Newton polygon has strictly positive slope. (An illustrative Newton polygon is given in Figure 5.I (b).) In particular, directly from the definition of the Newton polygon, this implies that
v(ai ) > v(ao) for all i Equivalently, we have lail <
laol for all i
~
1.
~ 1. Wefirst observe that this gives
laol = suplail = II¢II ·
252
5. Dynamics over Local Fields: Bad Reduction
Second, we note that it implies that the constant term ao in the series with z E D(O, 1) has absolute value strictly larger than any of the other terms, so the ultrametric inequality (Lemma 2.3) says that
1¢(z)1 = lao\ = II¢II
for all z E D(O, 1).
o
Remark 5.15. The maximum modulus principle (Theorem 5.13(a)), which we stated over Cp , is true more generally as long as K has infinite residue field, but otherwise it need not be true. For example, let K = Qp and ¢(z) = z - zP on D(O, 1).
Clearly II¢II = 1, but for every (3 E Zp that (3 == (3P (mod p). Hence
= D(O, 1), Fermat's little theorem tells us
1¢((3) I 5: p-l < II¢II
for all (3 E D(O, 1).
We conclude this section with some useful consequences of the maximum modulus principle, including the fundamental fact that p-adic holomorphic and rational functions send closed disks to closed disks. Proposition 5.16. Let ¢(z) E Cp[z] be a nonconstant power series that converges
on a disk D (a, r) ofrational radius. (a) ¢(D(a,r)) is a closed disk. (b) Write¢(D(a,r)) = D(¢(a),s). Then 1¢'(a)l5: sir. (c) If ¢' (a) -I 0, then thereexists a radius t > such that
°
I¢(z) - ¢(w)1 = 1¢'(a)I·lz - wi
for all z, wE D(a, t).
Proof Replacing ¢(z) with ¢(cz + a) - ¢(a) for some c E C p satisfying Icl = r, we may assume that ¢(z) E Cp[z] converges on D(O,I) and that ¢(O) = 0. Write ¢(z) = I:n21 anz n as usual (note that ao = since ¢(O) = 0), and let
°
s=
1¢(z)l.
s~up zED(O,l)
(a) The maximum modulus principle (Theorem 5.13(a)) says that s
= II¢II = sup lanl· n20
Let j 2: 1 be the smallest index such that s = Iaj I. The definition of s clearly implies that ¢(D(O, 1)) ~ D(O, s), so we are reduced to proving the opposite inclusion. Let (3 E D(O, s). Consider the Newton polygon of the power series
5.3. Newton Polygons and the Maximum Modulus Principle
¢(z ) - (3 The fact that 1(31
253
= - (3 + al Z + a2z2 + a3z3 + ... .
:S s = lajl and lanl :S lajl for all n ::::: 1 implie s that and
so the Newton polygon of ¢(z) - (3 includes one or more line segments connecting (0, v((3) ) to (j, v(aj )). Further, since the point (0, v((3 )) is no lower than the point (j, v(aj )) , at least one of those line segments has slope rn :S 0. The fundamental theorem on Newton polygons (Theorem 5.11) then tells us that the power series ¢ (z ) - (3 has a least one root a satisfying [o] = pm :S 1. This proves that there is a point a E D(O , 1) satisfying ¢( a ) = (3, and since (3 E D(O , s) was arbitrary, this completes the proof that ¢ (D(O, 1)) = D(O, s). (b) As in (a), the maximum modulus principle (Theorem 5.13(a)) gives
s=
s~p
1¢(z)1 = II¢II = sup lanl : : : lall·
z ED (O,I )
n2:0
This is the desired result, since r = 1 and a I = ¢' (0). (c) Continuing with the notation from (b), we compute
I¢( z) -
¢(w)1=
I~ an(zn -
= [z -
=
For all n ::::: 2, all
°:S
i
wn) 1
wl 'l~ an ~ ziwn- I- i l ~ anziwn- l- i!.
[z- wl ' !¢'(O) + ;
< n, all t :S 1, and all z, w
E
(5.5)
D(O, t ) we have
Hence if we choose a value oft satisfying 0< t < I¢'(O)I/II¢II, then the double sum in the righthand side of(5.5) has absolute value strictly smaller than I¢' (0) I, and (5.5) reduces to the desired inequality
I¢ (z ) -
¢(w)1:S [z- w\·I¢'(O)I ·
This completes the proof of (c) and provides an explicit value for
t.
o
Corollary 5.17. In each ofthefollowing situations. the indicated map ¢ is both open and continuous: (a) ¢ : D(a, r ) ----+ C p is a nonconstant analytic map. (b) ¢ : b t«, r ) ----+ p I (C p ) is a nonconstant meromorphic map. (c) ¢ : p I (C p ) ----+ pI (C p ) is a nonconstant rational fun ction.
254
5. Dynamics over Local Fields: Bad Reduction
Proof. (a) The continuity is a consequence of Proposition 5.10, which gives the stronger assertion that ¢ is Lipsch itz. The openness of ¢ follows easily from Proposition 5.16, since the collect ion of "closed" disks (which is also a collection of open sets) {D (b , t) : t > 0 and « Cp }
s
forms a base for the topology of Cpo (b) Let bE D(a,r) . Proposition 5.8(d) says that we can write ¢ as a Laurent series in some neighborhood D(b, s) of b. If ¢ does not have a pole at b, then (a) completes the proof. If ¢ does have a pole at b, we consider instead the meromorphic function I j¢(z ). It too can be written as a Laurent series in some neighborhood D(b , s), and it has no pole at b, so again we are done using (a). (c) A rational function is clearly everywhere meromorphic, since it is the ratio of two power series (i.e., polynomials) that converge on all ofC p • (For the point at 00, change coordinates and use I j¢(z ).) Hence we are done from (b). 0
5.4 The Nonarchimedean Julia and Fatou Sets In this section we recall some basic notions of convergence for collections of functions, define the Fatou and Julia sets of a rational map ¢(z) E K (z) over a field K with an absolute value, and use a formula from Chapter 1 to show that in the nonarchimedean setting , the Fatou set is always nonempty. We begin with three definitions. Definition. Let U be an (open) subset of p I (K) and let be a collection of functions ¢ : U - t r l( K) . (a) is equicontinuous on U if for every P E U and every I: > 0 there exists a > 0 such that
o
¢( D(P,o» )
C
D (¢(P), I:)
(b) is uniformly continuous on U if for every I:
¢(D(P,o) n U) C D(¢(P), 1:)
for every ¢ E .
> 0 there exists a 0 > 0 such that
for every ¢ E and every P E U.
(c) is uniformly Lipschitz on U ifthere is a constant C such that
p(¢(P),¢(Q») '5c C· p(P, Q)
for every ¢ E and every P, Q E U.
In the case that = {¢n} is the collection of iterates of a single function, we say simply that ¢ is equicontinuous, uniformly continuous, or uniformly Lipschitz. It is important to understand that equicontinuity is weaker than uniform continuity, because equicontinuity is relative to a particular point, while uniform continuity is uniform with respect to all point s in U . In particular, uniform continuity is an open condition, whereas equicontinuity is not. Similarly, the uniform Lipschitz property
5.4. The Nonarchimedean Julia and Fatou Sets
255
is an open condition, and indeed it is even stronger than uniform continuity. The following implications are easy consequences of the definitions: uniformly Lipschitz
=::}
uniformly continuous
=::}
equicontinuous at every point
As we will discover throughout this chapter, in a nonarchimedean setting it is often just as easy to prove that a family of maps is uniformly Lipschitz as it is to prove that it is equicontinuous. Definition. Assume first that K is algebraically closed. Then the Fatou set F (¢) is the union of all open subsets ofJ!D 1 (K) on which ¢ is equicontinuous, i.e., F( ¢) is the largest open set on which ¢ is equicontinuous. The Julia set .:J(¢) is the complement of the Fatou set. In general, the Fatou set of ¢ over K, which we denote by F( ¢, K), is the intersection of F( ¢, K) with J!Dl (K). Similarly, the Julia set .:J (¢, K) is the complement of F( ¢, K) in J!Dl (K). Proposition 5.18. For every integer n 2: 1,
and Proof We proved this over
'L" PEFix(¢)
1
1 _ .Ap(A-.) -1 -
provided that .Ap (¢) =1= 1 for all P E Fix( ¢).
'f'
The following corollary of this formula has the useful consequence that nonarchimedean Fatou sets are never empty, a fact that is false in the archimedean setting (cf. Theorem 1.30, Example 1.31, and Theorem 1.43). Corollary 5.19. Let K be an algebraically closed field of characteristic 0 that is complete with respect to a nonarchimedean absolute value. Let ¢( z) E K (z) be a rational function ofdegree d 2: 2. Then ¢ has a nonrepellingfixed point.
Proof If some fixed point P has multiplier .Ap( ¢) = 1, then P is nonrepelling and we are done. Otherwise, we can use Theorem 1.14 to estimate
Hence there is at least one fixed point Q satisfying
256
5. Dynamics over Local Fields: Bad Reduction
It follows that IAQ(¢)I :::; max{IAQ(¢) -11, Ill} :::; 1,
o
so Q is nonrepelling.
Proposition 5.20. Let K be an algebraically closedfield of characteristic 0 that is complete with respect to a nonarchimedean absolute value. Let ¢(z) E K (z) be a rational function ofdegree d ::::: 2. (a) Let P E JP'I(K) be a nonrepelling periodic point for ¢. Then P is in the Fatou set F( ¢). (b) Let P E JP'I (K) be a repelling periodic point for ¢. Then P is in the Julia set :1(¢). (c) The Fatou set F( ¢) of ¢ is nonempty. Proof Making a change of variables, we can move P to 0, and then Proposition 5.18 lets us replace ¢ by ¢n, so we may assume that 0 is a fixed point. This puts ¢ into the form
¢(z) =
AZ +
Z2 F(z)
G(z)
with G(O) i= 0 and A = AO(¢) E K. Thus IG(z)1 is bounded away from 0 if we stay away from the roots of G, and clearly F (z) is bounded on any disk around 0, so there are a disk D(O, r) and a constant C such that IF(z)IG(z)1 :::; C for all z E D(O, r). Let 5 = min{r, 1/C}. (a) The assumption that 0 is a nonrepelling fixed point means that IAI :::; 1. Hence for z E D(O, 5) we have 1¢(z)1
=
IAZ + Z~;~) I< Izl max {IAI, Iz~~) I} < 14
This proves that ¢ is nonexpanding on the ball D(O, 5), so its iterates are uniformly Lipschitz on that disk (with Lipschitz constant 1). Hence 0 is in the Fatou set. (b) For this part the assumption that 0 is a repelling fixed point means that 1>\1 > 1. Hence for z E D(0,5) with z i= 0 we have
IZ~;~) I :::;
Izl
<
IAZI·
The strict inequality allows us to conclude that I¢(z) I = I,\z
+ Z~;~) I =
I'\zl
for all z E D(O, 5).
Suppose now that 0 E F( ¢). This implies that there is some E if a E D(O, E), then every iterate satisfies ¢n(a) E D(O, 5). But then l¢n(a)1 = 1,\lnlal
----400
as n
--4
>
0 such that
00,
contradicting ¢n(a) E D(O, 5). Hence 0 E :1(¢). (c) Corollary 5.19 says that ¢ has a nonrepelling fixed point and (a) tells us that 0 nonrepelling periodic points are in the Fatou set. Hence F( ¢) i= 0.
5.5. The Dynamics of (Z2 - Z) I p
257
Remark 5.21. Corollary 5.19 is also true in characteristic p, since it follows directly from the fixed point multiplier formula (Theorem 1.14), which can be proven in characteristic p either by reduction modulo p from the characteristic-O case or by using the abstract theory of residues (see Exercise 5.10). It follows that Proposition 5.20 is also true in characteristic p.
5.5
The Dynamics of (z2 - z) / p
In this section we illustrate the general theory by studying the p-adic dynamics of the map 2
¢(z)=Z -z
for a prime p 2: 3.
p
Note that ¢ has bad reduction at p. An important observation is that the Julia set sits inside two disjoint disks.
Proposition 5.22. Let ¢( z) = (Z2 - z) I p. 1 1 (a) Izl > - and [z - 11 > p P (b) J(¢) c D(O, lip) U D(I, lip).
===}
lim l¢n(z)1
n--->oo
= 00.
Proof (a) We consider two cases. First, if Izl > 1, then Izi = Iz - 11, so we find that j¢(z)j=
I
Z(Z- I ) I p =p·lzl·lz-II=p·!zI2.
Inparticular, I¢( z) I > 1, so we can apply this inequality again. Repeating the process shows that
Next suppose Izi ::; 1. Then max{ Izl, [z - II} = 1, so using the assumption that Izi
> lip and [z - 11 > lip, we compute 1¢(z)1
= Z(Z-p I ) \ = p. max{lzl,lz -
I
II} . min{lzl,lz -II}
> 1.
Now we can apply the earlier result to ¢( z) to conclude that I¢n (z) I ---4 00. (b) From (a) we see that every point outside ofthe two disks D(O, lip) U D(I,Ilp) is attracted to the superattracting fixed point at 00, so such points are in the Fatou
0
~
The proposition tells us that the Julia set is contained in D(O, lip) U D(I, lip), but not every point in these disks is in the Julia set. For example,
¢(poo) = poo 2
-
a == -a (modp),
258
5. Dynamics over Local Fields: Bad Reduction
so if - 0: is not in D(O, lip) U D(l , l ip ), then ¢1I(po:) is attracted to 00, hence it is in the Fatou set. It is not easy to predict which points in the two disks are attracted to 00. However, since only points with bounded orbit can be in J (¢), we let
A
= A(¢) = { z E Cp : [¢1I (z)1is bounded for n 2: o}.
It is clear that A is a completely invariant set, and Proposition 5.22 tells us that A is contained in two disjoint disks, which for notational convenience we henceforth denote by 10 and h:
A c 10 Uh ,
where
10 = D(O , l ip ) and h = D(l , l ip)·
Definition. The orbit of a point z E A is contained within A, so each iterate ¢11 (z) is contained in one of the two disjoint disks 10 , h. The itinerary of z is the sequence [{30{3I!32 ...] of numbers {3n E {O, 1} determined by the condition for n = 0,1,2, ... . In other words, the itinerary [{311 ] specifies how the points ¢n (z) jump back and forth between the disks 10 and h . Our goal is to show that the dynamics of the itineraries of the points in A accurately reflects the actual dynamics of ¢ on the set A. In particular, we will prove that A = J (¢). In order to do this, we make a brief digression to discuss symbolic dynamics.
5.5.1
Symbolic Dynamics
Symbolic dynamics is a tool for modeling seemingly more complicated dynamical systems. It has a long history and is used in many areas of mathematics. See [276] for a thorough introduction to the subject. In this section we briefly develop enough of the theory to complete our analysi s of the dynamic s of (z2 - z)lp. Let S = {0'1 ' 0'2 , . .. , 0' s} be a finite set of symbols and let
SN = { sequence~ Of} elements 10 S be the set of sequences [{30{31{32 . . .], where each {3n E S. (Here N = {O, 1,2, ... } denotes the set of natural numbers.) We put a metric on Sri by fixing a number p > 1 and setting p(0: , {3) = p - (smallest 11 with On i- f3 n ) . Thus the more that the initial terms of 0: and {3 agree, the closer they are to one another. It is easy to check that p is a metric , and indeed a nonarchimedean metric:
• 1 2: p(o: , {3) 2: 0.
• p(0:, {3) = 0 if and only if 0:
= {3.
5.5. The Dynamics of (z2 - z)/p
259
• p(a,')') ~ max{p(a,13), p(;3,')')}. Symbolic dynamics is the study of the dynamics of continuous maps S N ---. SN. An important map on S N is the left shift map L : S N ---. S N, defined by
In other words, L simply discards the first term in the sequence and shifts each of the remaining terms to the left. More formally, the sequence L(;3) is defined by L(l3)n = ;3n+l . The next proposition describes some elementary properties of the map L.
Proposition 5.23. Let SN be the space of S-sequences with associated metric as above and let L : SN ---. SN be the left shift map. (a) If p(a,;3) < 1, then p(L(a) , L(I3)) = p. p(a, ;3). (b) L is continuous (indeed Lipschitz), and it is uniformly expanding on each ofthe "disks"
{a E S N : a o = ad,
i
= 1,2 , . . . , s .
(c) The set SN contains exactly s" points satisfying Ln(a ) = a . In other words. Per n (SN, L) has s" elements. (d) The periodic points ofL are dense in SN. (e) There exists a point rs E SN whose orbit O L(, ) = {L n (, ) : n ~ O} is dense in SN. (Maps with this property are called topologically transitive.) Proof (a) The condition p(a , ;3) < 1 is equivalent to ao = ;30. It is then clear from the definition that p(L(a ), L(;3)) = p. p(a, ;3), since L(a)n i L(l3) n if and only if
O:'n+! i ;3n+1· (b) The metric p satisfies 0 S p SI, so (a) implies that p(£(a ), £ (,6 ») S p-pic» , (3) for all a , (3. Hence L is Lipschitz. Further, (a) says that L is expanding by a factor of p on each of the disks. (c) A sequence a E SN satisfies Ln(a) = a if and only if its first n terms repeat, i.e., it has the form
a
= [aoal . . . an- l a Oa l ... an-l aOal .. . an- I ' . .J "--v----""--v----""--v----" initial n terms
same n terms
same n terms
There are s choices for each of ao, a l , " " an-I, SO s" possible elements. (d,e) We leave the proof of these elementary results as exercises for the reader (Exercise5.12). D
5.5.2 The Dynamics of (z2 - z)/ p Recall that we are studying the p-adic dynamics of the map ¢( z) = (Z2 - z)/ p for a prime p :::: 3, and that we have defined
260
5. Dynamics over Local Fields: Bad Reduction
10 = D(O, lip) and h = D(l, lip), A = {z E C p : ¢n(z) is bounded for all ti 2: O} C 10 U h, (3(z) = [(30(31(32 ... J = (Itinerary of z E A), where (3n is defined by ¢n(z) E I(3n' The sequence (3(z) is an element of the space of binary sequences {O, 1 obtain a map (of sets)
y:l, so we
(3(z) = itinerary of z. Proposition 5.24. With notation as above, the itinerary map (3 : A ----+ {O, 1y:l has the following properties: (a) (3 is injective. (b) (3(A n Qp) = {O, 1}N, i.e., (3 restricted to A n Qp is surjective. (c) (3 respects the metrics on A and {O, 1}N, i.e.,
Iz -
wi = p ((3 (z), (3 (w))
for all z,w E A.
(d) Let L : {O, l}N ----+ {O, l}N be the left shift map on {O, 1}N. Then (3 0 ¢ = L 0 (3. In other words, the following diagram is commutative:
A
~
A
(31
(31
{O, l}N
.L:
{O,l}N
Proof (a) We begin with the following observation. Let u be
Ifz,w
E I u , then
I¢(z) - ¢(w)1
=
°
or 1.
p ·Iz - wi.
(5.6)
To verify (5.6), we use the assumption that z, w E I u to write z = u + px and w = u + py with Ixl :::; 1 and Iyl :::; 1. Then
¢(z) - ¢(w) = (x - y)((2u - 1) + p(x + y)). The quantity 2u - 1 is ±1, so the final factor is a unit and we have
z-wl
1¢(z)-¢(w)I=lx-yl= -p- =p·lz-wl·
I
Now suppose that z,w E A have the same itinerary (3. Then ¢n(z) and ¢n(w) are in the same I(3n for every n 2: 0, so applying (5.6) to these two points, we find that l¢n+l(Z) - ¢n+l(w)1 = p .I¢n(z) _ ¢n(w)1 for all ti 2: 0. Hence by induction, for all
ti
2: 0.
(5.7)
5.5. The Dynamicsof (Z2 - Z) / p
261
The lefthand side of(5.7) is bounded as n ----t 00, since z and ware in A. Hence we must have Iz - wi = O. (b) Let wE 10 U h. We claim that ¢-l(W) consists of two points {zo, zd, one of which is in 10 and one of which is in h, and further, if w E Qp, then Zo and Zl are in Qp. To see why this is true, we fix wand solve
¢(z) = w,
Z2 - Z - pw
or equivalently, solve
=
o.
The quadraticformula,the binomialtheorem, and some elementaryalgebra givesthe two solutions explicitlyas
1± (1 + 4pW)1/2 = ~±~ ~ (1/2)(4 2
2
2 c:
k=O
k
)k = pw
~±~ ~ (_1)k-1 (2k) ( w)k. 2
2 s: 2k - 1 k=O
k
P
Note that the series converge for all lt»] ::; 1. The plus sign givesa solution satisfying Zl == 1 (modp) and the minus sign gives a solution satisfying Z2 == 0 (modp), so one solution is in 10 and the other is in h. Further, if w E Qp, then Zl and Z2 are also in Qp. We are going to apply this observation to (nonempty) sets U c 10 U h. The inverse image ¢ -1 (U) of such a set thus consists of two nonempty disjoint pieces, one in 10 and one in h. And since the inverse images ofQp points in 10 U hare again in Qp, we have
In particular, if U contains a point in Qp, then each of the pieces of ¢-1 (U) contains a point in Qp. Let J o = 10 n Qp and J 1 = t, n Qp, or equivalently, Jo and J 1 are the open unit disks in Zp centered at 0 and 1, respectively. For any binary sequence of 0:00:1 0:2 ... O:n with ti 2: 1, define a set
Jaoa1 ...an = {z E Qp : z E J ao and ¢(z) E J a1 and ¢2(Z) E J a2 ... and ¢n(z) E Jan}
= Qp n Js; n ¢-l(JaJ n ¢-2(JaJ n··· n ¢-n(JaJ. Notice that if z E Jaoa1 ...an n A, then the initial ti terms in the itinerary of z are 0:00:1 .. ·O:n' The sets J ao... an are closed in Qp, since ¢ is continuousand Jo and J 1 are closed, and they are nested in the sense that
We claim that they are also nonempty. To prove this by induction, suppose that we know that J(30(31 ... (3n-1 is nonempty for all sequences (30(31 ... (3n-1 oflength n. We use the equality
262
5. Dynamics over Local Fields: Bad Reduction
and apply the inductive hypothesis to see that Jala2 ... an is nonempty. Then from our earlier remarks we know that ¢-1 (Jala2 ... an) consists of two pieces, one in J o and one in J 1 , so its intersection with J a o is nonempty. Let a = [aOa1a2 ... J be any binary sequence. Then
J« ~
n
Jaoal ... an
n20
is the intersection of a nested sequence of nonempty closed bounded subsets of Qp. By compactness, the set Ja is nonempty, and by construction, any point in Ja has bounded orbit and itinerary a. This proves that the itinerary map
is surjective. We also note that since every point in Ja has itinerary a and since we know from (a) that (3 is injective, it follows that J a consists of a single point. (c) If z = w, there is nothing to prove. Assume that z -I- wand let k be the first index at which their itineraries diverge, so by definition, p((3(z), (3( w)) = p-k. This means that for each 0 :S n < k, the points ¢n (z) and ¢n (w) are in the same disk (either J o or J 1 ) , but ¢k(z) and ¢k (w) are in different disks. Repeated application of (5.6) tells us that for all 0
:S n :S k.
(5.8)
On the other hand, the assumption that ¢k (z) and ¢k (w) are in different disks (one in Jo and one in Jt} implies that (5.9) Combining (5.8) and (5.9) yields
Iz-wl=
l¢k(z)_/k(w)1
=
P
lk =p((3(z),(3(w)). P
(d) It is clear that the itinerary of ¢( z) is the left shift of the itinerary of z.
0
Proposition 5.24 allows us to identify the dynamics of the polynomial map
¢( z) = (Z2 - z)/ p with the dynamics of the shift map on the space of binary sequences. It then becomes an easy matter to read off a great deal of information about the dynamics of ¢( z) from elementary properties of the shift map.
Corollary 5.25. Let p 2 3 be a prime and let
¢(z) = (a) .:J(¢)
2 Z
-
p
z
= A c Qp.
and
A = {z E C p : ¢n (z) is boundedfor all n 2 O}.
5.6. A Nonarchimedean Montel Theorem
263
+ 1 for all n ;::: 1, and asidefrom thefixed point at 00, every periodic point of ¢ is repelling. (c) The repellingperiodic points are dense in .:J (¢). (d) There exists a point w E .:J (¢) such that the orbit 04> (w) is dense in .:J (¢). (Thus ¢ is topologically transitive on .:J (¢).) (b)
# Pern ( ¢)
= 2n
Proof Let {3 : A --+ {O, l}N be the itinerary map. Proposition 5.24 tells us that (3 is injective, and further that it is surjective even when restricted to A n Qp. It follows that A c Qp, which proves one part of (a). For the other part, we note that Proposition 5.24 says that {3 is an isomorphism of metric spaces {3: A --+ {O, l}N. (The proposition says that (3 is bijective and respects the metrics, so its inverse also respects the metrics.) The proposition also tells us that this isomorphism transforms the map ¢ into the left shift map L, i.e., (30¢ = Lo{3. Hence the dynamical properties of ¢ acting on A are identical to the dynamical properties of L acting on the space of binary sequences. We proved earlier that .:J( ¢) c A, so
(3: .:J(¢) ~ .:J(L). However, the shift map L is uniformly expanding on each of the disks (3(10) and (3(h) (Proposition 5.23(b)). A uniformly expanding map is nowhere equicontinuous, so .:J(L) = L, from which we deduce that .:J(¢) = A. (b) The points not in A are attracted to 00, so the only periodic point in pI (Cp ) <, A is the fixed point at 00. The periodic points in A are determined by the isomorphism
and we proved (Proposition 5.23(c)) that # Per n (L) = 2n . Further, we know that the shift map L is uniformly expanding (by a constant factor p) on each of (3(10) and (3(h), so via the metric isomorphism (3 and the identification of ¢ with L, we find that I(¢n)' (z) I = pn for every z E Per., (¢) except z = 00. In particular, every periodic point other than 00 is repelling. (c,d) The density of the (repelling) periodic points in .:J(¢) and the topological transitivity of ¢ on .:J( ¢) = A follow from the corresponding facts for L acting D on {O, l}N (Proposition 5.23(d,e)).
5.6 A Nonarchimedean Montel Theorem In this section we give an important characterization of the Fatou and Julia sets and use it to draw a number of conclusions. The analogous results over C are classical and due mainly to Fatou and Julia. The nonarchimedean results are due to Hsia, whose paper [208] we follow. We assume throughout that K is complete and algebraically closed.
264
5. Dynamics over Local Fields: Bad Reduction
Theorem 5.26. (Hsia [208]) Let
a
f/:
U ¢(D(a,r)) .
Then
I¢(z) - o]
=
II¢- all is constant for all z
Suppose that a i= 0 (the case a decomposition of the set
=
E
D(O, 1).
(5.10)
0 is similar) and consider the following
p(¢(z),¢(w)) ::; /¢(z) - ¢(w)1
I(¢(z) - a) - (¢(w ) - a) 1 from Proposition 5.10, ::; II ¢ - all . [z - wi since ¢ E
This shows that the functions in
I¢(z) - al = ll¢ - all > lal for all z
E
D(O , 1).
(5.11)
Using (5.11) and Lemma 2.3, and then applying (5.10) again, we find that
1¢(z)1 = I(¢(z) - a) + a / = /¢(z) - a / = II¢- all for all z E D(O , 1). (5.12) We use this formula to compute, for z , w E D(O ,I ),
I¢(z) - ¢(w)1 p(¢(z),¢(w )) = max{1 , 1¢(z)I} ' max{l , 1¢(w)l} I(¢(z) - a) - (¢(w) - a)1 max j l , 1¢(z)l} . maxj l , 1¢(w)l }
5.6. A Nonarchimedean Montel Theorem
265
11 4> - all· [z - wi < 2 from (5.12) and Proposition 5.10, - max{1 ,11 4> - a ll} :S [z - wi regardless of the value of 114> - all, = p(z, w) since z, wE D(O , 1). Thus functions in the set
It is now a simple matter to use Theorem 5.26 to prove a nonarchimedean version of Monte1's theorem for rational functions. Theorem 5.27. (Nonarchimedean Montel Theorem, Hsia [208]) Let
U 4>(D(a, r ))
(5.13)
omits two or more points of jp'1 (K) . Then satisfies a uniform Lipschitz inequality on D( a, r), so in particular
By construction, we have
'lj; (z) i- oo
for all 'lj; E \.Ii and all z E D (a,r),
so Proposition 5.8(c) tells us that the functions in \.Ii are holomorphic on D(a, r ) (i.e., they are given by convergent power series). We also know that 'lj; (z) i- 0 for all z E D(a, r), so the functions in \.Ii omit at least one point in K. It follows from Theorem 5.26 that there is a constant C1 such that
p('lj;(z) ,'lj;( w)) :S C1 P(z, w)
for all 'lj; E \.Ii and all z, w E
b i«. r ).
(5.14)
Finally, let A(z) be the linear fractional transformation
The assumption that a and ,6 are distinct implies that A is invertible . Then a very special case of Theorem 2.14 (for the rational map A - I ) says that there is a constant C2 = C2( A) = C2 (a,,6) > 0 such that
266
5. Dynamics over Local Fields: Bad Reduction for all z, w E pI (K).
Further, by construction we have ¢ E q> if and only if A
0
¢
E
(5.15)
W, so for any
z, wE bt«. r), CIP(z, w) ;::: p(A(¢(z)), A(¢(w))) ;::: C:;Ip(¢(Z),¢(w))
from (5.14), from (5.15).
This completes the proof of the nonarchimedean version of Montel theorem.
D
Remark 5.28. The proof of Theorem 5.27 is fairly straightforward. Later, in Section 5.10.3.4, we describe a deeper p-adic version of Montel's theorem on Berkovich space; see Theorem 5.80.
In the classical setting, there are a number of important properties of the Julia set that follow more or less formally from Montel's theorem. We conclude this section with a few instances.
Proposition 5.29. Let ¢ : pI (K) -+ pI (K) be a rational map of degree d ;::: 2 and let U C pI (K) be an open set such that U n J (¢) # 0. In particular, we are assuming that the Julia set of ¢ is nonempty. (a) The set Un :::: O¢n(u) omits at most one point of pI (K). (b) Suppose that the set in (a) does omit a point. Then ¢ is a polynomial function and the omittedpoint is the totally ramifiedfixedpoint. (In other words, there is a change ofvariables f E PGL 2 (K ) such that ¢f (z) E K[z] and the omitted point has been moved to 00.) Proof The set U is covered by disks D(a, r), which are both open and closed, so it suffices to prove the proposition under the assumption that U = D( a, r). If the union omits two or more points ofpl(K), then Montel's theorem (Theorem 5.27) implies that ¢ is equicontinuous on U, contradicting the assumption that U contains a Julia point. This proves (a). If the union omits 0:, then ¢ -1 (0:) = {o}, so 0: is a totally ramified fixed point of ¢. Hence ¢ is a polynomial map by definition (page 17), and after a change of variables it becomes a polynomial in K [z] (Exercise 1.9(c)). D Let ¢ : pI (K) -+ pI (K) be a rational map of degree d ;::: 2. Recall that a subset E of pI (K) is said to be completely invariant for ¢ if it is both forward and backward invariant,
¢-I(E) = E = ¢(E). Proposition 1.24 says that the Fatou set F( ¢) and the Julia set J( ¢) are completely invariant. (The proof in Chapter 1 is over C, but the proof works, mutatis mutandis, for any complete field.) In Chapter 1 we used the Riemann-Hurwitz formula to characterize all finite completely invariant sets (see Theorem 1.6). More precisely, we showed that a finite completely invariant set E has at most two elements. Further, if #E = 1, then after a change of variables, ¢(z) E K[z] is a polynomial and E = {oo}, and if#E = 2,
5.6. A Nonarchimedean Montel Theorem
267
then again after a change of variables, ¢(z) = zd or ¢(z) = z-d and E = {O,oo}. We now show that except for these trivial cases, the Julia set is the smallest closed completely invariant subset of lP'I (K).
Proposition 5.30. Let ¢ : lP'I (K) ---+ lP'I (K) be a rational map of degree d 2: 2, and let E ~ lP'I (K) be a closed completely invariant subsetfor ¢ containing at least three points. Then E is an infinite set and E :2 J (¢).
Proof Theorem 1.6 tells us that a finite completely invariant subset contains at most two points, so our assumption that #E 2: 3 implies that E is infinite. Notice that the complete invariance of the closed set E implies the complete invariance of its complement U, which is an open set. It follows that the union Un>O ¢n(u) omits at least two points, since it in fact omits the infinite set E. Montel's theorem (Theorem 5.27) tells us that U ~ F( ¢). Hence E :2 J( ¢). D Remark 5.31. Proposition 5.30 tells us that if the Julia set J( ¢) is nonempty, then it is the smallest closed completely invariant set containing at least two points. (Notice that the case of exactly two points is ruled out by the fact that if ¢ has a completely invariant subset containing exactly two points, then ¢ is conjugate to either zd or z:", in which case its Julia set is empty.) Corollary 5.32. Let ¢ : lP'I (K) assume that J (¢) -I- 0. (a) J(¢) has empty interior. (b) Let P E J(¢) and let
---+
lP'I (K) be a rational map of degree d 2: 2, and
O;(P) =
U¢-n(p) n2:0
be the backward orbit ofP. The Julia set J( ¢) is equal to the closure of in pI (K). (c) J( ¢) is a perfect set, i.e..for every point P E J( ¢), the closure ofJ( ¢) contains P. (d) J (¢) is an uncountable set.
0; (P) <,
{P}
Proof (a) Let oJ(¢) denote the boundary of the Julia set J(¢). Theorem 1.24 tells us that F( ¢) and oJ (¢) are completely invariant, so the same is true of their union oJ(¢) U F(¢). This union is also closed, since its complement is the interior of J (¢). Proposition 5.20 says that the Fatou set F( ¢) is always nonempty, and since it is open, it must contain infinitely many points. Hence the union oJ (¢) U F( ¢) is an infinite, closed, completely invariant set, so Proposition 5.30 tells us that
J(¢)
~
oJ(¢) U F(¢).
But J(¢) and F(¢) are disjoint by definition, which proves that J(¢) = oJ(¢), i.e., the Julia set has empty interior. (b) We know that J (¢) is completely invariant, so in particular 0; (P) c J (¢) for any point P E J( ¢).
268
5. Dynamics over Local Fields: Bad Reduction
Next let U be any open set with U n .:J( ¢) i- 0. Then Proposition 5.29(a) tells us that Un>O ¢n(u) omits at most one point, and Proposition 5.29(a) says that ifit does omit a point, that point is a totally ramified fixed point, hence is in the Fatou set. In particular, the possible omitted point cannot be P, since P E .:J(¢) by assumption. This proves that P E Un>O ¢n(u), or equivalently, that there is some n 2': 0 such that Un ¢-n(p) i- 0. This proves that every open set U that intersects .:J(¢) nontrivially also intersects (P) nontrivially. Hence .:J (¢) is contained in the closure the backward orbit
0;
ofO;(P). (c) Let Po E .:J(¢ ). We claim that the backward orbit (Po) must contain a nonperiodic point. To see this, suppose instead that (Po) consists entirely of periodic points. Then ¢ -1 (Po) consists of a single point, so Po is a totally ramified periodic point and hence in the Fatou set, contrary to assumption. Therefore we can find a nonperiodic point PI E (Po). The point PI is in .:J(¢ ), since .:J(¢) is completely invariant, so (b) tells us that
0;
0;
0;
Po E closure of On the other hand, Po is not in
0; (Pd.
0; (PI), since otherwise PI would be periodic. Hence
Po E closure of (.:J(¢) ,,{P}). (d) The Baire category theorem [387, §5.l,5.2] implies that a nonempty perfect subset of]p'1 (K) is uncountable. 0
5.7 Periodic Points and the Julia Set Our goal in this section is to show that the Julia set .:J(¢) of a rational map ¢ is contained in the closure of the periodic points of ¢. We begin with an elementary lemma that is obvious in the classical setting by a compactness argument, but which requires a different proof over a non-locally compact field such as Cpo
Lemma 5.33. Let ¢1(Z) and ¢z(z) be power series that converge on D(a, r), and suppose that ¢1 (D(a, r)) n ¢z (D(a, r)) = 0. Then inf
p(¢I(Z), ¢z(z)) > O.
zED(a,r)
Proof Let
M1 =
syP zED(a,r)
1¢I(Z)1
and
Mz =
sup
l¢z(z)l.
zED(a,r)
The maximum modulus principle (Theorem 5.l3(a)) says that there are points ZI,Zz E Di«, r) such that ¢1(zd = M 1 and ¢z(zz) = M z. In particular, M 1 and M z are finite, since ¢1 and ¢z are power series that converge on D(a, r).
269
5.7. Periodic Points and the JuliaSet Let M
= max{M1 , M2 , I}.
Then for any
On the other hand, the function (PI so Theorem 5.13(b) tells us that
-
Z
E
Dto; r) we have
¢2 does not vanish on D( a, r) by assumption,
for all z E D(a, r). Hence
o The next lemma is used in conjunction with Lemma 5.33 to move a varying set of pairs of points {a,,8} to the specific pair {O, I}.
Lemma 5.34. Let A, B c C p be bounded sets that are at a positive distance from one another. In other words, there are constants ~, 0 > 0 such that
sup lal ::; "'EA
~,
sup 1,81 <
~,
and
inf
"'EA, (3EB
(3EB
p(a,,8) = 0
> O.
(5.16)
For each (a,,8) E A x B, define a linear fractional transformation L",,(3(z) Then there is a constant C
= (,8 -
a)z
+ a.
> 0, depending only on ~ and 0, such that
p(L",,(3(z), L"",(3' (z')) ::; C· max{p(a, a'), p(,8, ,8'), p(z, z')} for all a, a' E A, all,8,,8' E B, and all z, z' E ]P'l(C p ) . Remark 5.35. Although Lemma 5.34 appears somewhat technical, it is not saying anything mysterious. The linear fractional transformation L",,(3 is determined by the three conditions L",,(3(O) = a,
The lemma is asserting, roughly, that if we take two nearby (a,,8) values, then the associated transformations are close to one another, where we use the chordal sup norm p(L,L') = sup {p(L(P),L'(P))} PElF" (iC p )
to measure the closeness oftwo maps. Thus Lemma 5.34 is equivalent to the assertion that the map (a,,8, z) ~ L",,(3(z)
is Lipschitz.
= (,8 -
a)z
+a
270
5. Dynamics over Local Fields: Bad Reduction
ProofofLemma 5.34. Toease notation, for x , y E C p we write
Ix, yl = maxj ]»], Iyl}· We also assume (without loss of generality) that fj. any 0:, 0:' E A and (3, (3' E B we have
~
1 and 8 :::; 1. Then for
10: - 0:'1 = p(o: , 0:') , 10:, 11, 10:', 11:::; fj. 2p (0:,o:'),
(5.17)
1(3 - (3' 1= p((3, (3' ) , 1(3,1 1,1(3' ,1 1:::; fj. 2p((3 ,(3' ). Let 0:, 0:' E A, let(3, (3' E B, andlet z, z' E of L o ,{3 and the chordal metric, we have
]P'I(Cp ) . Directly from
the definitions
, 1(((3' - o:')z ' + 0:') - (((3 - o:) z + 0:) I p(L o ,{3 (z ), L o ' ,{3' (z )) = max{ 1((3 _ o:)z + 0:1, I} . max{I((3' - o:') z' + 0:'1, I} . (5.18)
Assuming for the moment that z i:- 00 and z' and estimate it using the triangle inequality:
i:-
00,
we multiply out the numerator
1(((3' - 0:') z' + 0:' ) - (((3 - 0:) Z + 0:) 1 = 1(3' z' - (3z - 0:' Z '
+ o:z + 0:' -
0: /
= 1(3' (z' - z) + ((3' - (3 )z - o:' (z' - z ) + (0: - o:') z
+ (0:' -
0:)1
:::; max{I(3' - (31· lzl, 10:' - o: l· lz, 11, Iz' - zl ' lo:' ,(3' I} :::; max{fj.2p( (3, (3') ·lzl, fj.2p( 0:,o:') ·Iz, 11 , Iz' - zl·lo:' , (3'I} from (5.17),
:::; max{fj.2p(I3,I3' ) ' lzl, fj.2p (0:, o:') ' Iz, 11 , fj. ·I z' - zl} from (5.16), :::; max{fj. 2p(I3,I3') 'lz l, fj. 2p (0:, o:') 'Iz, 11, fj.p (z, z' ) · Iz , II · Jz' , I I} definition of p, :::; fj.2 . max{p(l3, (3'), p(o:, 0:'), p(z , z' )} · Iz, 11·l z' , 11. Substituting this into (5.18) and doinga little bit of algebrayields
max{p(l3, 13'), p(o:, 0:' ), p(z, z') } Iz' , 11 < fj. 2 . _~-,--,-:I---, z '---,1-,--1--,----""7 max{ I(13 - o:)z + 0:1 , I} max{ 1(13' - o:')z'
+ 0:' 1, I} .
(5.19)
We are left to show that the righthand side is bounded in terms of 8 and fj.. By symmetry, it suffices to boundthe first fraction. Weconsidertwo cases. First, if Izi :::; fj. / 8, then we havethe trivial estimate Iz,11
5.7. Periodic Points and the Julia Set
271
Second, suppose that Izl > tl/o. Then the fact (5.16) that 113 - al ~ 0 implies that so
1(13 - a)z + al = 1(13 - a)zl ~
Hence Iz,11 max{l(jJ-a)z+al,1}
=
014
Iz,11 < ~ < tl max{olzl,l} - 0 - O·
Thus tl/o serves as an upper bound in both cases, and substituting this bound into (5.19) yields the estimate
p(L a,(3(z), La/,(3I(z'))
< ~24 max{p(jJ,jJ'),p(a,a'),p(z,z')}.
This completes the proof of Lemma 5.34 with explicit dependence on 0 and tl in the case that z i- 00 and z' i- 00. The remaining cases are similar and are left to the
0
~cr
We next prove a version ofMontel's theorem in which the two omitted points are allowed to move. Lemma 5.34 is the key technical tool that allows us to uniformly replace the two moving points with two particular points, thereby reducing the proof to our earlier result.
Theorem 5.36. (Montel Theorem with Moving Targets, Hsia [208]) Let power series that converge on D( a, r), and suppose that
cPl, cP2
be
Further let be a collection ofrational, or more generally meromorphic, functions on tn« r) such that for all cP E and all z E D(a, r). Then satisfies a uniform Lipschitz inequality on D( a, r), so in particular is an equicontinuousfamily offunctions on D(a, r). Proof The proof is very similar to the proof of Theorem 5.26, but somewhat more elaborate. First we note that since cPl, cP2 : D(a,r) ----> K have disjoint images, there is at least one point a omitted by both of them. Making a linear change of variables z f---+ z - a, we may assume that cPl and cP2 omit the value O. Then Theorem 5.13(b) tells us that
for all z E D(a, r). We are going to apply Lemma 5.34 to the disjoint bounded sets andcP2(D(a,r)). Thus for each point w E D(a,r),ifwelet
cPl (D(a, r))
5. Dynamics over Local Fields: Bad Reduction
272
then Lemma 5.34 says that there is a constant C such that
p(Lw(z), Lu(z')) ::; C max{p(
(5.20)
=
(3' =
C' 2 1 such that
for all w, U E D(a, r) and i
= 1,2.
Substituting this into the inequality (5.20) yields
p(Lw(z),Lu(z')) < CC'max{p(w,u),p(z,z')} for all w, u E
bi«. r) and all z, z' E ]P'1(C p ) . (5.21)
We are now ready to prove Theorem 5.36. The idea is that we know that each
and we consider the family of functions
Each 'l/;¢ can be expressed as a rational function of
'l/;¢(w)
#- 0
and
'l/;¢(w)
#- 1
for all wE D(a, r).
To see why this is true, note that
contradicting the assumption on
p('l/;¢(u),'l/;¢(w)) ::; C"p(u,w)
for all
tu« r).
(5.22)
5.7. Periodic Points and the Julia Set
273
Using this and our earlier estimates, we compute, for ¢ E and u, w E D(a,r ),
p(¢( u), ¢( w)) = p(Lu ( 1/Jq,(u)),L w ( 1/Jq,(w)))
by definition of 1/Jq"
:s cc max{p(u, w),p(1/Jq, (u),1/Jq,(w)) }
from (5.21),
< ee'e"p(u,w)
from (5.22).
This completes the proof that the family of maps is uniformly Lipschitz.
D
We now have the tools to prove the main theorem of this section. Theorem 5.37. (Hsia [208]) Let ¢(z ) E K (z ) be a rational function of degree d with d 2: 2. Then :J(¢) c Per (¢ ),
i.e., the closure ofthe p eriodic points of ¢ contains the Julia set of ¢. Proof We may clearly assume that :J (¢) is not empty. Take any open set U having nontrivial intersection with :J (¢). We must show that U contains a periodic point. The Julia set is a perfect set (Corollary 5.32), so the open set U actually intersects :J (¢) in infinitely many points . In particular, there is a point P E un :J(¢) that is not the image of a ramification point of ¢, since ¢ has at most 2d - 2 ramification points . This implies that there is a neighborhood D (P, r ) C U of P such that
consists of d disjoint open sets with the property that the maps
¢ : V;
---->
D (P, r)
for 1
:s i :S d
are bijective. In particular, they have inverses
¢i : D (P, r ) ~ Vi given by convergent power series. (This is a p-adic version of the one variable inverse function theorem. See Exercise 5.5.) We take i = 1 and i = 2 and consider the maps ¢I and ¢2 and the disjoint sets VI and V2 as illustrated in Figure 5.2. We now examine the effect of applying the iterates ¢n of ¢ to the disk D(P, r). The assumption that D(P, r) contains a point of the Julia set of ¢ means that ¢ is not equicontinuous on D (P, r ), so Theorem 5.36 tells us that the iterates of ¢ cannot avoid both of the moving targets described by the power series ¢ I and ¢2. Hence there exists an iterate ¢n of ¢ and a point Q E D (P, r ) such that either or Applying ¢ to both sides and using the fact that ¢O¢ I and ¢ o¢2 are both the identity map on D (P, r ) yields ¢n+I(Q) = Q, so Q E D(P, r) C U is a periodic point.
D
274
5. Dynamics over Local Fields: Bad Reduction
Figure 5.2: Inversion of ¢ over a critical-point-free neighborhood.
Remark 5.38. In the classical setting over C, one can further show that the Julia set is equal to the closure of the repelling periodic points. This follows from the complex analogue of Theorem 5.37 combined with the fact that a rational function over C has only finitely many nonrepelling periodic points. Unfortunately, a rational function over C p may well have nonempty Julia set and infinitely many nonrepelling periodic points. However, one still hopes that the classical result is true in the nonarchimedean setting.
Conjecture 5.39. (Hsia [208]) Let ¢( z) E K (z) be a rational function of degree d 2: 2. Then the Julia set J (¢) is equal to the closure ofthe repelling periodic points of d: Some evidence for Conjecture 5.39 is provided by the following result of'Bezivin. It says that if the conjecture is false, then there are maps with nonempty Julia set containing no periodic points.
Theorem 5.40. (Bezivin) If a rational function ¢( z) E C p (z) has at least one repelling periodic point, then J (¢) is the closure ofthe repelling periodic points of ¢. In particular, one repelling periodic point implies infinitely many repelling periodic points. Proof See [71] for the first assertion. The second then follows immediately from Corollary 5.32(d), since an uncountable set cannot be the closure of a finite set. D
However, some evidence against Conjecture 5.39 is provided by Benedetto [58, Example 9], who shows that it is possible for a rational function to have a sequence of attracting periodic points whose limit is a repelling periodic point! Further, a slight variation of [57, Example 3] shows that for every n > 0 there is a polynomial ¢( z) E C p [z] of degree p + 2 that has no repelling periodic points of period smaller than n, yet ¢( z) does have repelling periodic points of higher periods. Example 5.41. Consider the polynomial map
¢(z)=zp-z. p
5.7. Periodic Pointsand the Julia Set
275
It is clear that the Julia set of ¢ is contained in 15(0, 1), since if 1001 > 1, then laP I > 1001, so
a Pp -001 1¢(a)l= - =p·laP-al=p·laPI>plal· l
Hence ¢n(a) - t 00, so a E F(¢), since a is attracted to the attracting fixed point at infinity. We also observe that if a E 15(0,1) n Qlp = Zp, then Fermat's little theorem tells us that a P == a (mod p), so ¢(a) E Zp, Thus Zp is a completely invariant subset of ¢. Hsia [206, Example 4.11] (see also [449]) explains how to identify the dynamics of ¢ on Zp with a shift map on p symbols, similar to the example studied in Section 5.5, from which one deduces the following facts:
• .J(¢,Qlp) = Zp, • .J (¢, Qlp) contains all of the periodic points of ¢ (other than (0), so in particular all of the periodic points of ¢ are defined over Qlp, and all except 00 are repelling.
• .J( ¢, Qlp) = .J(¢, C p), since Theorem 5.37 tells us that .J(¢, C p) is contained in the closure of the periodic points of ¢. Thus .J( ¢, C p ) is compact. (See also [73].)
Example 5.42. Let p ~ 5 be a prime, and let ¢(z) = pz3 + az 2 + b with a, b « Z;. We first consider the fixed points of ¢, which are the roots of the equation
pz3 + az 2
-
z + b = 0.
The assumption that a, b E Z; implies that the roots satisfy 10011 = p and 10021 10031 = 1. (Look at the Newton polygon!) We also observe that paf and aai have norm p2, while 001 - b has norm p, so 001 must have the form
a
001 = -- + e p
for some
ewith lei = 1.
This allows us to compute
1¢'(a1)1 = 13pai + 2aa1 - 11 = 1001 (3pa1 +2a) -11 = lad-a + 3pe) -11 = 10011 =p. Thus 001 is a repelling fixed point of ¢. A similar, but more involved, calculation can be used to show that there are repelling periodic points of higher orders. Alternatively, we can invoke Bezivin's Theorem 5.40, which says that the existence of the single repelling fixed point 001 implies that ¢ has infinitely many repelling points whose closure is .J (¢ ). In order to study the periodic points in the Fatou set, we observe that ¢ is nonexpanding on 15(0, 1). To see this, note that ¢ maps the disk 15(0, 1) to itself, so in particular II¢II :::; 1. Applying Proposition 5.10 yields
I¢(z) - ¢(w)1 :::; [z - wi
for all z, wE D(O, 1).
276
5. Dynamics over Local Fields: Bad Reduction
Hence ( z) = iiz 2 + b):
2n 1 2n = ,Az3n + ... + B z2n + 1, + Cz f +'D z - + ...+ E z + F .
coefficie:ts in »z,
T
C E Z;
coefficie~ts in z,
I
Again using the Newton polygon, we see that the polynomial
5.8 Nonarchimedean Wandering Domains We first recall a famous theorem from complex dynamics (see Theorem 1.36). Theorem 5.43. (Sullivan' s No Wandering Domains Theorem [426]) Let m > 0 such that
In other words. the connected component U does not wander, whence the name of the theorem. The first obstacle to translating Sullivan's theorem to the nonarchimedean setting is the fact that Qp and C p are totally disconnected. Thus with the classical definition of connectivity, there is no good way to break up the Fatou set into "connected" components. Various alternatives have been studied, including disk connectivity, rigid analytic connectivity, and the use of Berkovich spaces. In this section we consider disk connectivity, which is the simplest of the three to describe. We state a theorem of Benedetto to the effect that a large class of rational maps in Qp (z) have no wandering "di sk domains" and illustrate the result by proving it under the somewhat stronger hypothesis that there are no critical points in the Julia set of
5.8. Nonarchimedean Wandering Domains
277
5.8.1 Disk Domains and Disk Components The ordinary definition of connectivity is not useful for studying the totally disconnected spaces Q p and C p, nor is the notion of path connectedness helpful. We begin with an abstract notion of connectivity that uses a chain of "disks" in place of a path. Definition. Let X be a topological space and let V be a collection of open subsets of X . (In practice , V will be a base for the toplogy of X.) For convenience , we refer to the sets in V as disks. Let U c X be an open subset and let P E U . The disk component of U containing P (relative to V) is the set of Q E U with the property that there is a finite sequence of disks D 1 , D 2 , . . . , D n C U such that
D i n Di+l =I- 0 for all 1 :::; i < n. In other words, Q is connected to P by a path of disks, as illustrated in Figure 5.3, although as we shall see, in the nonarchimedean world, Figure 5.3 is somewhat misleading. Note that we define disk components only for open subsets of X. It is easy to see that U breaks up into a disjoint union of disk components (Exercise 5.22).
Figure 5.3: A path of disks from P to Q.
Example 5.44. Let X = C and let V be the usual collection of open disks in C. Then the disk components of an open set U C X are the same as the usual pathconnected components. This is clear, since if r is a path from P to Q, then r can be covered by open disks contained in U, and the compactness of r shows that it suffices to take a finite subcover. Thus the definition of disk components and the related notion of disk connectivity (see Exercise 5.23) are reasonable. For example, Sullivan 's theorem may equally well be stated in terms of the disk components of the open set F (¢). For the purposes of this section, we modify slightly our definition of disks in]p'l (Cp ) . The resulting topology is the same, and indeed the disks contained within the unit disk D(O , 1) are the same, but the alternative definition is more convenient for working with disks that may contain the point at infinity. Definition. The standard collection of closed disks in ]p'l (C p ) , denoted by V closed , is given by
278
5. Dynamics over Local Fields: Bad Reduction
Vc10sed
=
all closed disks D(a, r) and { the complement jp>l(Cp ) < Dta, r) of all open disks D(a, r).
Of course, despite the name, all of the disks in Vc10sed are both open and closed sets. Note that Vclosed includes all such disks, not only the disks of rational radius (cf. Remark 5.7). One can show (Exercise 5.24) that the disks of rational radius in Vclos ed are exactly the images of the unit disk D(O, 1) via elements ofPGL 2 (Cp ) . Similarly, the standard collection ofopen disks in jp>l (Cp ) , denoted by V open, is given by all open disks D(a, r) and V
open
= { the complement jp>l(Cp )
"D(a, r) of all closed disks D(a, r).
In the nonarchimedean world, every disk component has a very simple form. More precisely, it is either a disk, the complement of a single point, or all of jp>l (Cp ) .
Proposition 5.45. Let V open and Vc10sed be, respectively, the collections ofstandard open and closed disks in jp>l (Cp ) as defined above. (a) Let D 1 , D 2 E Vc1osed. Then one ofthe following is true: (i) D 1 n D 2
= 0.
(ii) D 1 U D 2 = jp>l(Cp ) .
(iii) D 1
~
D2.
(iv) D 2
~
D 1.
(b) Let U C jp>l (Cp ) be an open set and let V be a disk component of U relative to Vclosed. Then V has one ofthefollowingforms: (iii) V
E Vc10sed U V open.
Proof (a) If D 1 U D 2 = jp>l(Cp ) , we are done. Otherwise, choose any point in the complement of D 1 U D 2 and use a linear fractional transformation to move that point to 00. This reduces us to the case that neither D 1 nor D 2 contains 00, so they have the form D 1 = D(al,rd and D 2 = D(a2,r2). If D 1 and D 2 are disjoint, we are done, so we may assume that there is a point
and switching D 1 and D 2 ifnecessary, we may also assume that rl ::::: r2. Let f3 E D 1 . Then
so f3 E D 2 • Hence D 1 ~ D 2 . (b) If V = jp>l(Cp ) , we are done, so we assume that V -=I- jp>l(Cp ) . Using a linear fractional transformation to move a point not in V to 00, we are reduced to the case that 00 ~ V.
279
5.8. Nonarchimedean Wandering Domains
Let D 1 , • .• , D n E Dclo sed be any path of disks contained in V . Each pair of adjacent disks (D i , Di+d has nonempty intersection, so (a) tells us that one of them is contained within the other. Applying this reasoning to each pair, we see that the union U~=l D, is itself a closed disk, i.e., the union is in D closed. This shows that every disk path in V actually consists of a single disk. Fix some point a E V and let
R = sup{r ~ 0 : D(a, r ) c V }. Note that R > 0, since a E V and V is open. If R done, so we assume that R < 00 . We claim that
= 00, then V = C p
D(a, R ) ~ V ~ D(a, R ).
and we are
(5.23)
The lefthand inclusion is clear from the definition of R. For the righthand inclusion, suppose that b E V and consider a disk path from a to b lying within V. From our previous remarks, this disk path consists of a single disk D( a, s). The definition of R and the fact that D(a, s) ~ V tell us that s :s; R, and then the fact that b E D(a, s) tells us that bE D(a, R ). This gives the other inclusion . But we get even more. If there is even a single b E V satisfying Ib - al = R, then s = Rand D(a,R) = D(b, s ) ~ V, so we find that V = D(a,R) E D closed. On the other hand, if Ib- al < R for every b E V, then V c D ( a, R), so (5.23) tells us that V = D (a, R) E D open • This completes the proof of Proposition 5.45. D
5.8.2 Hyperbolic Maps over Nonarchimedean Fields In this section we prove that the Julia set of a rational map ¢ contains no critical points if and only if ¢ is strictly expanding on its Julia set. This result is used later to prove that such maps, which we call (p-adically) hyperbolic, satisfy a p-adic version of Sullivan 's no wandering domains theorem . On first reading, the reader may wish to omit the somewhat technical proof of the main theorem in this section and proceed directly to the application of the theorem in proving Theorem 5.55 in Section 5.8.3.
Theorem 5.46. (Benedetto [56]) Let K / Qp be ajinite extension ofp-adicjields and let ¢(z ) E K(z) be a rational fun ction ofdegree d ~ 2. Proposition 5.20(c) tells us that F( ¢) =1= 0, so changing variables if necessary, we may assum e that 00 E F( ¢). Then the following are equivalent : (a) There are no critical points in :J( ¢). (b) For every jinite extension L/ K there exists an integer m ~ 1 such that for all ex E :J(¢) n L. In other words, ¢ffi is strictly exp anding on :J (¢)
n L.
Definition. Let K / Qp be a finite extension. A rational function ¢ E K (z ) will be called p-adically hyperbolic if it satisfies the equivalent conditions of Theorem 5.46. See Exercise 5.25 for the relationship with the classical definition ofhyperbolicity.
280
5. Dynamics over Local Fields: Bad Reduction
Remark 5.47. The classical analogue of Theorem 5.46 over C is much weaker. It says that some iterate of
IT
(¢m)'( o:) =
(since ¢'( o:)
= 0).
i =O
Thus the existence of a critical point in :J (¢) implies that (
I
wi
for all z, w E D(o:, Ta ) .
These disks form an open covering of the compact set :J(
The (nonarchimedean) triangle inequality implies that each disk in this covering is contained in one ofthe disk s ofthe previous covering, so we conclude that ¢ stretches by a constant factor on each D ( 0:; , f) . In other words, there are constants S; such that for each 1 :::; i < n ,
I¢(z) - ¢(w)1= S;lz -
wi
for all z, w E D(O:i, f).
The next step is to show that for any fixed a E :J(
n L , the set of derivatives
n = 1,2 ,3, . . . ,
(5.24)
is unbounded. We prove this by contradiction, using the following claim:
Claim 5.48. Let
0:
E :J(
n L and supp ose that there is an R > 1 such that
1(¢n)'(o:)l :::; R for all n 2:: 1. Then
(5.25)
Proofof Claim 5.48. We verify the claim by induction on n. The inclusion (5.25) is clearl y true for n = 0, since R > 1. Suppose the inclusion (5.25) is known for all 0:::; n < N. Our choice of f ensures that for any j3 E :J(¢) n L, the map ¢ stretches by a constant factor on the disk D(j3, f), so in particular each of the map s
281
5.8. Nonarchimedean Wandering Domains
o :::;:n < N , stretches by a constant factor. It follows that
stretches by a constant factor. In other words, there is a constant S such that for all z, ill E D(ex, e]R). Taking ill
=
ill
I
I,
= ex and letting z -+ ex, we see that S = (¢N)' (ex ) and then taking E D(ex , f/ R) arbitrary, we conclude that
ex and z
I¢N(Z) - ¢N( ex) 1= 1(¢N)' (ex)I' lz - ex l
< Rlz :::;: f
exl
since I(¢n)' (ex) I :::;: R for all n by assumption, since z E D(ex, e]R ).
This shows that the inclusion (5.25) is true for n
=
N , hence for all n by induction .
o
Recall that we are in the midst of a proof by contradiction that the derivatives (5.24) are unbounded. Under the assumption that the derivatives are bounded , we have proven the inclusion (5.25), which in turn certainly implies that
U ¢n(D(ex ), f/R) c U D( ¢n(ex, f)) C U n~ l
n 2:1
D({3, f).
{3 EJ(
However, the Julia set is bounded (since we assumed that 00 E F (¢)), so an e-neighborhood of the Julia set is also bounded. In particular, the above union omits at least two points, so by the nonarchimedean Montel theorem (Theorem 5.27), the map ¢ is equicontinuous on D(ex , e] R). This is a contradiction, since ex E J (¢ ) by assumption, which completes the proof that the derivatives (5.24) are unbounded. We now know that for each point ex E J (¢) n L there is some integer m a such that (¢ma )'( ex) 2:: 2. (There is no significance to the number 2; any number strictly larger than I would suffice.) By continuity, there is a disk D(ex, so) such that I(¢ma )' I 2:: 2 at every point in the disk. Taking a finite subcovering, we can cover J( ¢ ) n L by disks
I
I
with the property that there are integers ml , . . . , mt 2:: 1 such that for all 1 :::;: i :::;: t and all z E
o,
b;
For convenience , we may assume that the disks D1 , . . . , are pairw ise disjoint , since if two closed disks have a point in common, then one is contained in the other and may discarded. It remains to show that there is a single iterate ¢m that is expanding on all of the disks. In fact, we show that this is true for all sufficiently large m.
282
5. Dynamics over Local Fields: Bad Reduction
To ease notation, we let 'lj;i = qr i • Given any point a E :J(
Dio ) , (index so that a 1 E Di l ) , (index so that a2 E Di 2 ) ,
ao = a
and
i o = (index so that ao E
a 1 = 'lj;io(ao)
and
i1 =
a2 = 'Ij;i l(a1)
and
i2 =
and In other words, if an is in We note that
b; then a n+1
is obtained by applying 'lj;i =
= 0,1 ,2, ...,
(5.26)
since by construction, the point ain is in Din and 'lj;in = 0 such that for all (3 E :J(
I(
n- 1
II I
for all (3 E :J(
i= O
Let M = max{ m1 , . . . , md and let N be any integer satisfying
2N > 4M J1-
M2
(5.28)
•
We claim that I( 1 on :J(
We continue this process until the exponent N - m comes smaller than M. Notice that this implies that
io - m i l -
. .. - m i k
M>N-m to' -m'1 1 -" ' -m''l.k > N -(k + 1)M , so k
+ 2 > N 1M. We have thus found integers k and r
satisfying
first be-
5.8. Nonarchimedean Wandering Domains
283
(¢N)'(o:) = 'ljJ~o(O:O)1/( (o:d'ljJ~2(O:Z)" ·'ljJ~k(O:k)· (¢T)'(O:k+l) with k
> N / M - 2 and r < M.
Hence 1(¢N)'(o:)1
= 1'ljJ~o(O:O)'ljJ~l (o:d'ljJ~2(O:Z)" ·'ljJ~k(O:k)I·I(¢T)'(O:k+dl ,
v
I
2 Zk from (5.26)
2: 2N / M -
Z
. J-LM
>1
since k
'-----v---"
2
r
M from (5.27)
> N/M - 2 and r < M,
from the choice (5.28) of N.
This shows that 1(¢N),(o:)1 > 1 for all N > MZlogz(J-L- 1 ) and all which completes the proof of Theorem 5.46.
0:
E :J(¢)
n L, 0
5.8.3 Wandering Disk Domains If U is a disk component of the Fatou set F(¢) of a rational map ¢ E Cp(z), then ¢(U) may not be a full disk component of F(¢). This situation, which does not occur in the complex case, prompts the following definition.
Definition. Let ¢( z) E C p (z) be a rational map defined over a nonarchimedean field, and let
'DC (¢) = {disk components of the Fatou set F (¢ ) } be the collection of disk components of the Fatou set of ¢. Then ¢ induces a map of the set 'DC (¢) to itself according to the rule
'DC (¢) ------; 'DC (¢ ),
U
(disk component of F (¢) containing ¢( U)). (5.29) We say that U E 'DC (¢) is periodic, preperiodic, or wandering according to the behavior of U under iteration of the map (5.29). I----t
Example 5.49. Let p be an odd prime and let
z ¢(z)=z -z p
be the function that we studied in Section 5.5. We proved (Proposition 5.22) that the Julia set of ¢ is contained in the union of two open disks,
:J(¢)
c D(O, 1) U D(I, 1),
and that :J (¢) contains the repelling fixed points 0 and 1. For 0: E C p , let B(o:) denote the disk component of 0: in F(¢). We claim that B ( -1) = D ( -1, 1). To see this, we observe that B ( -1) cannot contain any larger disk, since it does not contain O. On the other hand, D(-I, 1) is in F(¢), since it is disjoint from D(O, 1) U D(I, 1). Hence B( -1) = D( -1,1).
284
5. Dynamics over Local Fields: Bad Reduction
Now consider the image point ¢ (-1 ) disk component
= 2p-1
and the image of the associated
¢ (B( - l) ) = ¢ (D( - l , 1)) = D (2p-l ,p). The disk D(2p-l,p) is contained in F (¢), but it is certainly not the largest disk around 2p-1 contained in F (¢ ).Indeed,
2p-1 E pI (Cp ) " D(O, 1) c F (¢). It is not hard to check that pI (rCp ) containing ¢ (B ( -1 )).
<,
D(O , 1) is the full disk component of F (¢)
Conjecture 5.50. (No Wandering Disk Domains Conjecture) Let K / Q p be afinite extension and let ¢( z) E K (z) be a rational map of degree d :::: 2. Then the Fatou set F( ¢) has no wandering disk components. Benedetto proves Conjecture 5.50 for a large class of rational maps . In order to state his result, we give four definition s (some of which we alread y know): Definition. Let ¢ E Cp (z) be a rational map of degree d :::: 2 and let P E pI (Cp ) . We say that Pis: (i) Julia if P E .J(¢). (ii) critical if e p (¢) :::: 2. (iii) wildly critical if ep (¢ ) == 0 (mod p ). (iv) recurrent if there is a sequence of integers n i ---+ 00 such that ¢ll i( P) i.e., if P is in the closure of the set { ¢ll( P) : n :::: I }.
---+
P,
Theorem 5.51. (Benedetto [54]) Let K /Qp be a finite extension of p-adic fi elds and let ¢( z) E K (z) be a rational map of degree d :::: 2. Assume that ¢ has no wildly critical recurrent Julia p oints (defined over K) . Then the Fatou set F (¢) has no wandering disk components. We make three short observations concerning Theorem 5.51. Remark 5.52. Ifp is odd, then Theorem 5.51 is true for "almost all" rational maps in Cp(z ). This is true because if ¢(z) has a wild critical point P, then in particular it has a point whose ramification index satisfies ep( ¢) :::: p :::: 3. It is not hard to show that all of the critical points of a "generic" rational map of degree d have ramification index equal to 2. (See Exercise 4.23 for a more precise statement.) Remark 5.53. For a fixed degree d, Theorem 5.51 is true for all primes p since p > d rules out the existence of wild critical points.
> d,
Remark 5.54. If a recurrent critical point P is in the Fatou set F (¢), then one can show that P is in fact per iodic (Exerc ise 5.26). On the other hand, if P is critical, then locally around P the map ¢ looks like ¢(z) = ¢( P ) + a(z - z(p ))k + ... for some a E C p and some k :::: 2. Thus if Q is sufficiently close to P, then
5.8. Nonarchimedean Wandering Domains
285
p(¢(Q),¢(P)) = p(p,Q)k, so ¢ is highly contractive near P. And if P is also recurrent, then ¢n(p) gets very close to P infinitely often, so it receives the highly contractive effect of ¢ infinitely often. This should cause points near P to remain near P, and thus force P into the Fatou set. The critical recurrent condition and the Julia condition are thus in opposition to one another, which means that nonperiodic recurrent critical points should be quite rare. On the other hand, Rivera-Letelier [378] has shown that there are maps having wildly critical recurrent points (which are then necessarily in the Julia set) in JP'l(Qp). It is not known whether Rivera-Letelier's examples have wandering disk domains. In the other direction, it is known that there are rational maps defined over C p that have wandering disk domains [63, 59, 62, 378]. In these examples the critical points are in the Fatou set, so Theorem 5.55 implies that the maps cannot be defined over a finite extension ofQp. We now use Theorem 5.46 and a simple compactness argument to prove that the Fatou sets of p-adically hyperbolic maps over finite extensions of Qp have no wandering disk domains. This is not as strong as Theorem 5.51, but still covers an important class of maps. The proof of Theorem 5.51 uses similar ideas, but is more complicated; see [54]. Theorem 5.55. (Benedetto [56]) Let K/Qp be afinite extension ofp-adic fields, let ¢( z) E K (z) be a rational map ofdegree d 2: 2, and assume that the Julia set .:J(¢) contains no critical points of ¢, i.e., assume that ¢ is p-adically hyperbolic. Then the Fatou set F( ¢) has no wandering disk components.
Proof Proposition 5.20 assures us that F( ¢) is nonempty, and it is open, so it contains an algebraic point. (Note that Qp is dense in C p.) Replacing K by a finite extension and changing coordinates, we may assume that 00 E F( ¢), and indeed we may even assume that .:J(¢) c D(O, 1). Equivalently, we may assume that the disk component of 00 contains P! (C p ) -, D(O, 1). We suppose that U C F( ¢) is a wandering disk component of F( ¢) and derive a contradiction. Replacing U with the disk component containing ¢n(u) for a sufficiently large n, we may assume that the orbit of U does not include the disk component at 00. In particular, ¢n(u) C D(O, 1) for all n 2: O. Again taking a finite extension of K if necessary, we can find a point 000 E K and a radius ro such that D(ao,ro) cU. At this stage we fix the field K and we use Theorem 5.46 to find an integer m such that I(¢m)'l > 1 on .:J(¢) n K. Replacing ¢ by ¢m, it suffices to consider the case that WI> Ion .:J(¢) n K. For each n 2: 0, the image ¢n(D(ao,ro)) is a (closed) disk centered at the point an = ¢n(ao), say
In particular, ¢(fJ(a n, rn)) yields
= D(an+l' rn+d, so applying Proposition 5.16(b)
286
5. Dynamics over Local Fields: Bad Reduction
rn l¢/(an) 1< rnH .
(5.30)
We also know that the disks D (an, r n) are disjoint, since D(an , r n) is contained in ¢n(u) , and further, each disk D (a n ,r n ) contains a point of D(O , l )nK. It follows that the radii must satisfy (5.31) lim r « = 0, n- oo
since the set D(O ,I) n K has finite volume, so can contain only finitely many nonempty disjoint disks of any fixed radius . It follows from (5.30) and (5.31) that there are infinitely many n with the property that i.e., since r n of points
---->
0, we must have rn+l
< r n infinitely often. Consider the infinite set
{an: 1¢'(a n)1 < 1,
n
= 1,2,3, ... }.
It is contained in the compact set D(O, 1) n K, so it contains an accumulation point E K. The continuity of ¢' implies that 1¢'((3) 1:::; 1, which shows that (3 E F(¢),
(3
since we used Theorem 5.46 to ensure that WI > Ion .J (¢ ). Let V be the disk component of F (¢) containing (3. Then by construction V contains infinitely many of the iterates an = ¢n(ao). Since the radii of ¢n(u) and ¢n( v) shrink to 0 as n ----> 00, it follows that some iterate ¢n(U) is contained in V and that some (nontrivial) iterate of ¢n (V) is contained in V. Therefore U is not 0 wandering, contradicting our original assumption.
5.8.4 Wandering Disk Domains Exist in c, We have proven that p-adically hyperbolic maps defined over Qp have no wandering disk domains. More generally, Theorem 5.51 shows that rational maps ¢( z) E Qp(z) with wandering disk domains are very rare, if they exist at all. And of course, Sullivan's theorem 5.43 says that rational maps ¢(z) E C(z) defined over the complex numbers never have wandering domains. It is thus somewhat surprising to discover that there are very simple polynomial maps defined over Cp that have wandering disk domains. Theorem 5.56. (Benedetto [59]) For c E C p, let ¢c(z) be thepolynomial
¢c(z) = (1 - c)zpH
+ cz".
Then there exists a valuea E C p such that: (1) .J(¢a) =I- 0, (2) F(¢a) has a wandering disk domain, (3) F( ¢a) contains every criticalpoint of¢a. Proof See [59] for a proof of this specific theorem, and see [63, 62, 373, 378, 380] for generalizations and related results. 0
287
5.9. Green Functions and Local Heights
5.9 Green Functions and Local Heights The canonical height h
h
h
and
=
dh
where the first says that h
1
(v-adic local height of a) = lim d log max{ n-+oo
n
I¢n (a) Iv ,1}.
(5.32)
It is clear that if the limit (5.32) exists, then it transforms canonically, and indeed if ¢(z) is a polynomial, then the limit does exist and everything works quite well (see Exercise 3.24). Unfortunately, for general rational maps the limit (5.32) may not exist. Rather than working directly with a rational map ¢ : JlD1 ----+ JlD1, it turns out to be easier to develop a theory oflocal heights by first lifting ¢ to a map : Pi,. 2 ----+ Pi,. 2 and then constructing a real-valued function 9 on Pi,.2 that satisfies the canonical transformation formula 9 ( (x, y)) = dg(x, y). In this section we construct the Green function g, prove some of its basic properties, and then use 9 to construct local canonical height functions on JlD1 as described in Theorem 3.27. A point in projective space [x, y] E JlD1 is given by homogeneous coordinates, so it is really an equivalence class of pairs (x, y). We make explicit the natural projection map 2 1r: (A " {O,O}) -------+ JlD1, (x,y) f---' [x,y], that sends a point (x, y) E A 2 to its equivalence class [x, y] E JlD1. To ease notation, we write
A;
= Pi,.2"
{O,O}
for the affine plane with the origin removed. Let ¢ : JlD1 ----+ JlD1 be a rational map of degree d. Then ¢ can be written as usual in the form ¢ = [F, G] with homogeneous polynomials F and G of degree d having no common factors. The polynomials F and G then define a map
(x,y)
=
(F(x,y),G(x,y)),
that fits into the commutative diagram A2 &*
A2 &*
JlD1 ~JlD1 We call a lift of ¢. By homogeneity, if = [F, G] is one lift of ¢, then every other lift of ¢ has the form e = reF, eGl for some constant e E K*.
5. Dynamics over Local Fields: Bad Reduction
288
Definition. Let K be a field and let v be an absolute value on K. We denote the absolute value (or sup norm) ofa point (x, y) E A2 (K) by
Similarly, the absolute value (or sup norm) of one or more polynomials is the maximum of the absolute values of their coefficients. (We have already made use of this convention in the proof of Theorem 3.11.) We begin by recalling how a map : A2(K) ofa point.
-t
A2(K) affects the v-adic norm
Proposition 5.57. Let K be afield with an absolute value v. Let = (F, G) : A 2
-t
A2
be given by homogeneous polynomials F, G E K[x, y] ofdegree d 2: 1, and assume that F and G have no common factors in K [x, y]. (a) There are constants Cl, C2 > 0, depending only on and v, such that
< 11(x,Y)llv < II
Cl _
(b)
(x,y)
II d v
-
for all (x,y) EA;(K).
C2
If v
is nonarchimedean and satisfies 11llv explicit constants
IRes (F , G)I v
< 11(x,Y)llv < 1 -
=
(5.33)
1, then (a) is true with the
for all (x,y) E A;(K).
II(x, y)ll~ -
(5.34)
Proof (a) Weproved inequality (5.33) for generalmorphisms pN
- t pM during the course of proving Theorem 3.11. More precisely, see (3.6) on page 92 for the upper bound with an explicit value for C2 ( , v), and see (3.7) on page 93 for the lower bound. (b) By homogeneity, it suffices to prove (5.34) for points satisfying II (x, y) Ilv = 1. Then the upper bound is obvious from the triangle inequality, and the lower bound 0 was proven during the course of proving Theorem 2.14, see (2.5) on page 57.
The next result describes a kind of v-adic canonical height associated to a map : A 2 - t A 2 . The constructionis the same as the one that we used to construct canonicalheights in Section 3.4.
Proposition 5.58. Let K be a field with an absolute value v, let cP : pI morphism ofdegree d 2: 2, and let = (F, G) : A2 - t A2 be a lift of cP. (a) For all (x, y) E (K) the following limit exists:
-t
pI be a
A;
gif>(x, y)
=
.
1
}!..,~ dn loglln(x, y)llv'
We call gif> the Green function of .
(5.35)
5.9. Green Functions and Local Heights
289
(b) The Greenfunction is the uniquefunction A; (K) twoproperties: Q(1)(x,y)) = dQ(x,y) Q(x, y) = log] (x, y) Ilv
+ 0(1)
----+
IR having the following
for all (x, y) E A;(K).
(5.36)
for all (x,y) E A;(K).
(5.37)
(c) Ifv is nonarchimedean and iIl satisfies IliIlll v = 1 and IRes(F, G) Iv = 1, i.e.,
the map ¢ = [F, G] : lP'1 Q(x,y)
=
----+
if
lP'1 has good reduction at v, then
logll(x,y)llv
for all (x,y) E A;(K).
(The converseis also true. See Exercise 5.27.) (d) Forall (x, y) E A; (K) and all c E K*, the Greenfunction Q has thefollowing
homogeneity properties: Q(cx, cy) = Q(x, y)
+ log Icl v.
1 Qc(x, y) = Q(x, y) + -d-log Jelv. -1
(5.38) (5.39)
(e) The Greenfunction Q : A; (K)
----+ IR is continuous. (In fact, Q is Holder continuous, but this is more difficultto prove. See Exercise 5.28.)
Proof We consider the two functions and Proposition 5.57(a) tells us that they satisfy loglliIl(x, y) Ilv
= dlogll (x, y) Ilv + 0(1)
for all (x, y) E A;(K).
(5.40)
This is exactly the situation needed to apply Theorem 3.20, from which we conclude that the limit (5.35) exists and satisfies (5.36) and (5.37). Further, Theorem 3.20 says that Q is the unique function satisfying (5.36) and (5.37). This completes the proof of (a) and (b). (c) The assumptions 111>llv = 1 and IRes(F,G)lv = 1 combine with Proposition 5.57(b) to tell us that II iIl(x, y) Ilv Hence by induction we obtain
= II (x, y) II~
for all points (x, y) E
A; (K).
Then the definition of Q immediately gives Q(x,y) = logll(x,y)llv' which proves (c). dn1>n(x,y). (d) The map iIl n is homogeneous of degree d", so iIln(cx,cy) = c Substituting this into the definition of Q gives (5.38). Similarly, ifwe let iIl c (x, y) = ciIl(x, y), then homogeneity and an easy induction argument show that
290
5. Dynamics over Local Fields: Bad Reduction
Hence
1 9c
10gllcp~ (x,y) 11 v
1(
= J~~ dn
d
n-l)
log llcpn( x , y)llv + d=1 log [c] , 1
= 9
(e) Let 9n(x,y) = d- n 10g llcpll(x , y) Ilv' Then for any fixed value of n, the function 9 n : A; (K ) ----t ~ is continuous, since it is the composition of the continuous maps cpn : A;(K) ----t A;(K) and log II . Ilv : A;(K) ----t R Further, the telescoping sum argument used in Theorem 3.20 to prove the existence of limn->oo 9n(x, y) implies that for all n ~ 1 and all (x, y) E A;(K), where C = C(CP , v) depends only on the 0 (1 ) constant in (5.40). (In the inequality (3.15) on page 98, switch the roles ofm and n and then let m ----t 00.) Hence the sequence of continuous functions 9n(x,y) converges uniform ly to 9
L
he/>(P) =
n v9
for all P
= [x ,y] E JP>l(K).
vEMK
Proof. Let
TJ(x ,y) =
L
n v9
for (x, y) E A; (K ),
v EM K
so a priori the function TJ is a function on A;(K) . However, applying the Green function homogeneity property (5.38) from Proposition 5.58(d) and the product formula (Proposition 3.3), we see that
TJ (CX,cy)=
L v EM K
n v (9
L vE MK
n v9
5.9. Green Functions and Local Heights
291
sc n gives a well-defined function on pi(K). Next we use the transformation property (5.36) from Proposition 5.58(b) to compute
".,(¢(P)) =
2:
n v9<1> ,v( (P )) =
vEMK
2:
n vd9<1> ,v(P ) = d".,(P).
vEM K
Similarly, we use the normalization property (5.37) from Proposition 5.58(b) to estimate
".,(P) =
2:
n v9<1> ,v(P )
vEMK
where h(P) is the usual height of the point P E pi (K) and the Ov(1) are the bounded functions appearing in (5.37). We further observe that Proposition 5.58(c) says that we may take Ov(1) = 0 for all but finitely many v E M K . More precisely, we may take Ov(1) = 0 for all v satisfying
(i) v E M~,
(ii) 11llv
= 1,
and
(iii) IRes(
Iv =
1.
Hence the final sum in (5.41) is a bounded function. We have now proven that n : pi (K) -+ JR satisfies
".,(¢(P)) = d".,(P)
and
".,(P) = h(P) + 0(1).
It follows from the uniqueness of the canonical height (Theorem 3.20) that n is equal ~~. 0
In Section 3.5 we described a decomposition of the canonical height h¢ as a sum of local canonical height functions ~¢,v, but we deferred the proof. The intuition in Section 3.5 was that the local canonical height should measure
~¢,v(P)
=
-log(v-adic distance from P to (0).
More generally, it is convenient to define a local canonical height that measures the v-adic distance from P to a collection of points. In the following theorem we describe a set of points, possibly with multiplicities greater than 1, by specifying the homogeneous polynomial E E K[x, y] at which they vanish. In slightly fancier terminology, we are identifying positive divisors in Div(Pi) with homogeneous polynomials in K[x, y].
Theorem 5.60. Let K be a field with an absolute value v and let ¢ : pi -+ pi be a rational function ofdegree d :::: 2 defined over K. Fix a lift = (F, G) of ¢ and let 9<1> be the associated Green function.
5. Dynamics over Local Fields: Bad Reduction
292
(a) Forany homogeneous polynomial E (x, y) E K [x, y] ofdegree e ~ 1 we define
~¢,E([X,y]) = eY
i- O.
Then ~¢,E is a well-definedfunction on jp'l, i.e., the value of~¢,E(P) does not depend on the choice ofhomogeneous coordinates [x, y]for P. Thefunction ~¢,E is called a local canonical height associated to ¢ and E. In the special case that E(x, y) = y, we drop E from the notation and refer simply to a local canonical height associated to ¢. (b) Forall P E jp'1(K) with E(P) i- 0 and E(il>(P)) i- 0 we have A¢,E (¢(P)) = dA¢,E(P) - log A
A
!(EOil»(P) I E(P)d v .
(Note that the homoegeneity ofE and il> ensures that the ratio (E well-definedfunction on jp'l.) (c) Thefunction IE(P)lv P f------+ A¢,E(P) + log IIPII~ ,
0
il» / Ed is a
A
(5.42)
which a priori is defined only at points P satisfying E(P) i- 0, extends to a bounded continuousfunction on all ofjp'I(K). (d) Given the particular choice of lift il>, the function ~¢,E defined in (a) is the unique real-valuedfunction
satisfying (b) and (c). If cil> is a differentlift, then with the obvious notation,
In particular, any two local canonical heights differ by a constant. Proof The Green function satisfies Y
Icl v ,
while the
eY
~¢,E(¢(P))
= eY
from Proposition 5.58(b),
= d~¢,E(P) + dlog IE(x, y)lv -logIE(il>(x, y))lv'
293
5.9. Green Functions and Local Heights
(c) Directly from the definition of ~1>,E we see that the righthand side of (5.42) is equal to e(yep(P) -log 1IPIIv)' (5.43) The boundedness of(5.43) is exactly (537) in Proposition 5.58(b), Further, we know from Proposition 5,58(e) that yep(x, y) is a continuous function on A;(K), and it is clear that logll(x,y)llv is also a continuous function on A;(K), so the difference (5.43) is a continuous function on A;(K), Further, the difference is invariant under (x, y) f-+ (cx, cy), so it descends to a continuous function on ]P'l (K), (d) Suppose that ~ and ~I both satisfy (b) and (c), and let ~II = ~ - ~I. Writing
~II(P) = (~(P) + log
IE(P)lv) _ 11P11~
(~I(P) + log IE(P)lv) IIPII~
'
we see from (c) that ~II extends to a continuous bounded function on all Of]P'l (K). Let C be an upper bound for I~III. From (b) we find that
~II (¢(P))
= d~1I (P)
for all P E ]P'l (K) with E(P) =I- 0 and E( ¢(P)) =I- 0,
and iterating this relation yields
~II ( ¢n(P)) Hence
= dn~II (P)
provided E (¢i (P)) =I- 0 for all 0 ::; i
1~II(p)1 =
dIn
1~1I(¢n(p))1
::;
~
< n. (5.44)
for all points P E ]P'l (K) satisfying E (¢i (P)) =I- 0 for all 0 ::; i ::; n. But each equation E (¢i (P)) = 0 eliminates only finitely many points, so the inequality (5.44) is true for all but finitely many points Of]P'l (K). Then the continuity of ~II tells us that I~II (P) I < C d- n is true for all P E ]P'l (K). Since n is arbitrary, this proves that ~II (P) = 0, so ~ = ~I. Finally, the effect of replacing
for all P E ]P'l (K) with E(P) =I- O.
294
5. Dynamics over Local Fields: Bad Reduction
Proof We use the definition of X,p,E,v in terms ofthe associated Green function Q
Theorem 5.59 says that the first sum is equal to h,p(P), while the product formula (Proposition 3.3) tells us that the second sum is o. (Note that this is where we use the D assumption that E(P) 1= 0.) Remark 5.62. If v E M~ is nonarchimedean and ¢ has good reduction at v, then the Green function and the local canonical height are given by the simple formulas
,
(
)
Q
(max{lxlv,IYlv})
IE(x,Y)lv
.
Thus it is only for maps with bad reduction that the Green and local canonical height functions are interesting. This should not come as a surprise to the reader, since bad reduction is the situation in which dynamics itself becomes truly interesting . Of course, this is said with the understanding that every rational map over C has "bad reduction," so the dynamics of holomorphic maps on p I(C) are always interesting and complicated. Remark 5.63. For additional material on dynamical Green functions and dynamical local heights, see [21, 88,234,233].
5.10 Dynamics on Berkovich Space We have seen that C p is a natural space over which to study nonarchimedean dynamics, since it is both complete and algebraically closed . However, the field C p has various unpleasant properties : • C p is totally disconnected. • C p is not locally compact, so the unit disk {z E C p line pI (Cp ) are not compact.
:
Izi ::; I} and projective
• The value group IC;I = pQ consists of the rational powers of p, so it is not discrete in lR>o, yet neither is it all oflR >o. This list suggests that it might be better to work in some larger space. There is a general construction, due to Berkovich [64, 67], that solves these problems for C p and other more complicated spaces. The study of dynamics on Berkovich spaces started during the 1990s and is an area of much current research . In this section we briefly describe the Berkovich disk and associated affine and projective lines and
5.10. Dynamics on Berkovich Space
295
discuss some very basic dynamical results. In a final subsection we state without proof some recent results. For further reading, see [26, 51] for an introduction to dynamics on Berkovich space and see [373, 375, 376, 379, 377, 380, 381] for RiveraLetelier's fundamental work in this area.
5.10.1 The Berkovich Disk over C p The unit Berkovich disk lJB is a compact connected metric space that contains the totally disconnected non-locally compact unit disk in Cpo We describe two constructions of lJB, the first an explicit description as the union of four types of points and the second as a set of bounded seminorms on Cp[z]. 5.10.1.1
The Four Types of Berkovich Points
The most concrete description of lJB is as the union of the following four sets of points: Type-I Berkovich Points. Each point a in the standard unit disk lJ(O , 1) of C p is associated to a point of the Berkovich disk, which we denote by ~a ,O
-B
E D.
Type-II Berkovich Points. Each closed disk lJ(a,r) contained in lJ(O , 1) with radius r E IC; I = pQ is associated to a point of the Berkovich disk, which we denote by ~a .T
-B
ED .
Type-III Berkovich Points. Similarly, each closed disk lJ(a,r ) ~ lJ(O, 1) with positive radius r ¢ IC;I = plQ is associated to a point of the Berkovich disk, which we naturally also denote by ~a .T ' Type-IV Berkovich Points. These are the trickiest points in lJB. They are associated to nested sequences of closed disks
with the property that
nbi«;
rn) = 0.
n 2: l
We denote these points by ~a,r , where, as the notation suggests, the vectors a and r are a = (al ' a2 ,"') and r = (r l ' r2, . . .).
Remark 5.64. Note that Berkovich points ~a , T of Types II and III are disks lJ(a, r), so different values of a may yield the same Berkovich point. Indeed, we have ~a , T
= ~b , s
if and only if
r = s and la -
bl :S r ,
296
5. Dynamics over Local Fields: Bad Reduction
since these are the conditions for the disks D (a, r) and D (b, s) to coincide. Similarly, two Berkovich points of type IV are the same if their sequen ces of disks can be suitabl y intertwined. See Exercise 5.40 for details. ~a ,r of Type-I, II, or III corresponds to a disk (possibly of radius 0), so we define the radius of ~a , r to be r, The radii ro, r1, r2, ... of a TypeIV point ~a ,r are non increa sing, so the limit r = lim i---> oo r, exists and is called the
Remark 5.65. A point
radius of~a ,r . We claim that the radiu s of a Type-IV point is strictly positive. To see this, suppose that ~a, r has radiu s O. Then the sequence of points a 1, a2, . . . is a Cauchy sequence in Cp, so it converges to a point a E Cpo Let 6i =
j nf
[z -
al
zED(ai,r;)
be the distance from a to the i th disk. Notice that 0 ~ 61 ~ 62 ~ 63 ~ ... , since the disks form a decreasing sequence. On the other hand, 6i ~ la - ai I -7 0 as i -7 00. Hence 6i = 0 for all i, so a E b i«; ri ) for all i and the intersection is nonempty, contradicting the assumption that ~a,r is a Type-IV point. 5.10.1.2
The Berkovich Disk as a Set of Seminorms
The description of the Berkovich disk DB as a union of points of Types I, II, III, and IV is very concrete, but it can be awkward to apply, since one must deal with four different kinds of points ." There is an alternative description of DB as a collection of seminorms on the ring C p [z] that is sometimes easier to use and that also naturally generalizes to other rings and other spaces. Definition. Fix R
II . IIRon Cp[z] is defined by
> O. The Gauss norm
The maximum modulus principle (Theorem 5.13(a)) tells us that the Gauss norm is equal to the sup norm,
IlfliR=
~up
If(z)l ·
zE D (O,R )
Definition. A (nontrivial)
II . IIR-bounded seminorm on Cp[z] is a nonconstant map
with the following properties: 1.
If I 2: 0 for all f
E
Cp[z].
2. Ifgl = If I·Igl for all I, g E Cp[z] . 40ne can unify the four types of points by defining all of them as equivalence classes of nested sequences of closed disks. See Exercise 5.40.
5.10. Dynamics on Berkovich Space
297
3. If + gl :S max{l f l, Ig l} for all I. s E Cp[z]. 4. If I :S Il fli R for all f E Cp[z]. (This is the boundedness condition.) Thus a seminorm has all of the properties of an absolute value except that there may be nonzero elements f E Cp[z] with If I = o.
Definition. The (closed unit) Berkovich disk fjB is the set fjB
=
{ II . IiI-bounded seminorms on Cp[z]}.
More generally, the closed Berkovich disk ofradius R is the set
fj~
= {II . Ilwbounded seminorms on Cp[z ]}.
The definition of fjB as a set of bounded seminorms is quite unintuitive, but its utility becomes clear when one examines Table 5.2 and sees how each of the four types of Berkovich points naturally defines a seminorm on Cp[z]. A fundamental theorem of Berkovich, whose proof we omit, says that every bounded seminorm on Cp[z ], and more generally on certain power series rings containing Cp[z], comes from one of the four types of Berkovich points. (See [64, page 18] or [26, Proposition 1.1].) Notice that the sem inorm I . 10 1 corresponding to the Berkovich point ~0, 1 is exactl y the Gauss norm Iflo ,1 = Ilf ll; , so following Baker and Rumely, we call ~0,1 the Gauss point.
I Point I Type-I
~a , O
Types-II & III
~a ,r
Type-IV
~a ,r
Seminorm
Ifl a,o = If( a)1 Ifla ,r = sup If(z)1 zE D(a,r) Ifl a,r = lim If lan,rn n~ oo
Table 5.2: Seminorms on
Remark 5.66. It is easy to see that the seminorms associated to points of Type-II, III, and IV are actua lly norms. (See Exercise 5.35.) However, the seminorm assoc iated to a point ~a ,O of Type-I is not a norm, since Ifla,o = 0 if and only if f vanishes at a. For example, Iz - ala,o = O. This explains why the Berkovich disk is defined using seminorms, rather than norms . Remark 5.67 . In order to properly develop function theory on the Berkovich disk and to glue disks together to create larger spaces, it is advantageous to use sets of seminorms on more general rings , such as power series rings . For example, let 11' R be the ring of power series in Cp[z] that converge on the closed disk fj (O , R ) of radius R, i.e.,
lI'R = {f (z ) =
L cnzn E
298
5. Dynamics over Local Fields: Bad Reduction
The ring 11' R is a Banach algebra over C p using the Gauss norm,
which the maximum modulus principle tells us is the same as the sup norm. The Berkovich disk D~ is often defined to be the set of bounded seminorms on the ring 11' R- One can show, using the Weierstrass preparation theorem, that this leads to the same set of points and the same topology as taking the II . Ilwbounded seminorms on the polynomial ring C p [z]; see [26]. The ring 11' R is an example of a Tate algebra. Applying the same construction to more general Tate algebras allows one to construct Berkovich spaces for the associated rigid analytic spaces.
5.10.1.3 Visualizing the Berkovich Disk In order to visualize the Berkovich disk, we place the Gauss point ~O,l at the top of the page and observe that there is a line segment running from any point ~ i- ~O,l up to the Gauss point. If ~ = ~a,r is of Type-I, II, or III, then this line segment is the set of points La,r = {~a,t : r S t S I}. Notice that any two line segments La,r and Lb,s merge with one another at the point ~a,t = ~b,t determined by t
= max{r, s, Ib - al}.
This is the smallest allowable value of t for which the disks D(a, t) and D(b,t) acquire a common point, hence for which they coincide. Thus one can imagine the various line segments continually merging as they run upward toward the Gauss point at the top of the tree. The line segment running up from a Type-IV point ~a,r is slightly more complicated. It is the set of points 00
La,r =
{~a,r}
U
U{~ai,t i=l
00
: ri S t Sri-I} =
{~a,r} U
U
Lai,ri'
(5.45)
i=l
(Note that ro = 1 by definition.) The Type-IV point ~a,r is included in the Berkovich disk precisely to provide an endpoint for the union of line segments ULai,ri' Now imagine starting at the Gauss point and moving downward through the tree. We claim that at any instant, there are three possible scenarios: 1. If you have reached a point ~a,O of Type I or a point ~a,r of Type IV, then you have hit the end of a segment and cannot proceed further. 2. If you have reached a point ~a,r of Type II, then r > 0 is in the value group of Cp and there are countably infinitely many branches along which you can move down the tree. 3. If you have reached a point ~a,r of Type III, then r > 0 is not in the value group of Cp and there is only one direction to move down the tree.
299
5.10. Dynamics on Berkovich Space
The picture for points of Type I is clear. In order to understand Types II and III, suppose that we fix a point ~a,T of Type II or III. Each b E D(a, r) gives a line segment Lb,o that runs up from the Type-I point ~b,O and through the point ~T,a' Two such line segments Lb,o and Lbf,O merge before reaching ~a,T if and only if Ib - b'l < r, so it is really each open disk D(b,r) inside the closed disk D( a, r) that gives a line segment running up to ~a,T' If ~a,T is of Type III, then D(b,r) = D( a, r) for any b E D( a, r), so there is only one segment running downward from ~a,T' The situation is much more interesting, and complicated, if ~a,T is of Type II. In this case D( a, r) is covered by a countable union of open disks D(b,r), so there is a countable set of branches downward from ~a,T' A convenient, although noncanonical, way to describe the branches is as follows. Let IlJ = {z E Cp : Iz I < 1} denote the maximal ideal in the ring of integers of Cp and fix some c E Cp with lei = r. Then the open disks of radius r are in one-to-one correspondence with the residue field JFp via the map {disks D(b,r) inside D(a, r)}
-+
JFp ,
D(b,r)
f------7
(b/c) mod 1lJ.
The surjectivity of this map is clear, and the injectivity follows from the fact that D(b,r) = D(b', r) if and only if Ib - b'l < r = leiIn order to fit the Type-IV points into the picture, let
be any sequence of nested disks and consider what happens as we move down the line segments Lai,Ti' The fact that the disks D(ai,ri) are nested implies that the line segments Lai,Ti extend one another downward as i increases. If the intersection D(ai, ri) is nonempty, then it is equal to D(a,r) for some a E C p and some r :::: 0, so the intersection corresponds to a point ~a,T of Type I, II, or III and the union ULai,Ti together with the endpoint ~a,T forms a closed line segment. However, if the intersection D (ai, ri) is empty, then there is no actual disk D(a,r) sitting at the bottom of the union of the line segments ULai,Ti' Thus as already noted, the points of Type IV exist precisely to remedy this situation and to ensure that every downward path has a termination point. Further, this explains why we defined La,r by (5.45) to be the line segment running from ~a,r to ~O,l' (See Exercise 5.41.) The Berkovich disk DB is "illustrated" in Figure 5.4. Of course, Figure 5.4 is at best a pale imitation of the true glory of the Berkovich disk, since in a complete picture of DB, every line segment contains a countable number of Type-II points, off of each of which there is a countable number of downward branches. Unfortunately, despite advances in modem technology, there are still no computer packages capable of fully rendering a (countably) infinitely branched broomstick.
n
n
5.10.1.4
The Gel'fond Topology on the Berkovich Disk
The next step is to put a topology on the Berkovich disk.
300
5. Dynamics over Local Fields: Bad Reduction
IType-I point I --
~a , O
~O ,O
~b ,O
Figure 5.4: The Berkovich disk
DB.
Definition. The Gel'fond topology on DB is the weakest topology such that for every f E Cp [z ] and every B > 0, the following sets are open:
U(j, B)
= {x E DB : If Ix < B}
and
V(j, B) =
Theorem 5.68. (Berkovich) The Berkovich disk a compact path-conn ected Hausdorffsp ace.
{x E DB: If Ix > B}.
DB with the Gel 'fond topology is
Proof See [64, Theorem 1.2.1] or [29, Appendix D] for the proofthat DB is compact and Hausdorff and [64, Corollary 3.2.3] for the proof that it is path connected. 0 Basic open sets for the Gel'fond topology on 156 , viewed as a tree, can be described by taking (and deleting) branches of the tree as described in the following definition. Definition. The closed branch of DB rooted at (a , r), denoted by Ir. a r' consists of all points ~b,s such that ~a,r is on the line segment L b,s running from ~b,s to ~O , l , together with whatever Type-IV point s are needed to finish off the bottom of any open line segments. Thus Ir.a,r
=
{~b ,s : s
:S r and Ib- al :S r} U {appropriate Type-IV points}.
Ir.:
The open branch rooted at (a, r ), denoted by r is obtained by starting with the closed branch Ir.a ,r ' removing the point ~a,Tl and then taking the connected component that contains ~a ,O' If ~a ,r is of Type III, this is simply the closed branch at ~a,r with the single point ~a ,r removed ; but if ~a , r is of Type II, then there are countably many branches at ~a , r and we select the one that includes the point ~a, O ' It is not hard to see that Ir.: ,r =
{~b,s : s
< rand Ib- al < r } U {appropriate Type-IV points} .
Then a base of open sets on the following three types :
DB for the Gel'fond topology consists of all sets of
5.10. Dynamics on Berkovich Space
301
• Open branches. • Open branches with a finite number of closed subbranches removed. • The entire tree with a finite number of closed branches removed.
Remark 5.69. There is a natural inclusion -
- B
D(O , 1) '-------+ D ,
a
r----+
~a ,O ,
that identifies the unit disk ])(0,1 ) as the set of Type-I points in the Berkovich disk ])B . It is not hard to check that the Gel 'fond toplogy on ])B, restricted to ])(0, 1), is the usual topology induced by the metric on CpoSee Exercise 5.44.
5.10.2 The Berkovich Affine and Projective Lines It is relatively easy to construct the Berkovich affine line A" B and the Berkovich projective line jp'B as topological spaces, which we do in this section. It is more difficult to construct them as ringed spaces with sheaves of functions appropriate for doing analysis. See Remark 5.74 for a brief discussion and references.
5.10.2.1
The Berkovich Affine Line A"B
Recall that the Berkovich disk ])B consists of four types of points . Each may be described in terms of disks that are contained in the closed unit disk ])(0, 1). Equivalently, ])B is the collection of II . Ill-bounded seminorms on Cp[z] . More generally, we define the Berkovich disk])~ of radius R to be the collection of II . ll w bounded seminorms on Cp[z]. It is given the Gel 'fond topology and has its own Gauss point ~O ,R corresponding to the seminorm
It is clear how to define points of Type I, II, III, and IV in ])~ using closed disks contained in ])(0, R),just as we did for ])B. Further, there is an inclusion
This is clear from the definition of ])~ as a set of seminorms. In terms of the picture of D~ as a branched tree, we see that ])~1 is the closed branch of D~2 lying below the Gauss point ~O ,Rl of D~l . Definition. The Berkovich affine line A"B is the union of the increasing collection of Berkovich disks,
A"B =
U D~ ,
R >O
with the topology induced by the direct limit topology on the individual Berkovich disks . It suffices, of course , to take the union over any increasing sequence of radii, for example , over R = pk with k --+ 00 .
S. Dynamics over Local Fields: Bad Reduction
302
Thus every point a E Cp , every disk D(a, r) C Cp , and every nested sequence of disks with empty intersection gives a point in the Berkovich affine line A13, and A 13 is composed of exactly this collection ofpoints. In particular, there is a natural inclusion of C, as the set of Type-I points in A 13,
We also note that A 13 inherits a tree structure from the natural tree structure of the Berkovich disks D~ . However, the tree A13 extends infinitely far upward; there is no Gauss point sitting at the top of A13.
5.10.2.2 The Berkovich Projective Line
jp'13
The easiest way to construct the Berkovich projective line jp'13 as a topological space is to glue together two copies of the Berkovich disk DB along their annuli Ann13
= { ~a , r
lal = 1 }
-13 ED:
using the map We note that the map j is induced from the inversion map j(z) = l i z, since it is easy to check (Exercise 5.37) that ifO rJ. D(a,r), then
j(D (a ,r )) = { Z-l : [z - al:S r} = D(a- 1 ,r/ la I2 ) . In particular, if lal = 1, then j(D(a,r)) = D(a- 1 , r ), so f(~a .r ) = ~a - l . r ' The full Berkovich disk is the disjoint union of the annulus and the open branch containing ~o,o , - 13
D = Ann
13
0
U A\0 ,1 '
Thus when we glue two copies of DB along their annuli , the only parts of the two disks that are not identified are the two open branches A\~ l ' Hence another way to
construct jp'13 is to take one copy of DB and attach one extra copy of A\~ 0 running vertically upward from the Gauss point ~0.1' The result is il1ustrated in Fi~e 5.5. It is natural to denote the extra vertical branch by A\"oo and to label its points using the reciprocals of the points in D(O, 1),
A\"oo = {~a .r : lal > 1 The open and closed branches in 1/. 0
11\0 ,1
-
and
r
< 1} U {~oo .o} .
A\"oo are defined using the natural
identification
1/. 0
.--.. 11\00 '
For example, a basic open neighborhood of the Gauss point ~0 ,1 is obtained by removing from jp'13 a finite number of closed branches, some of which may be in the vertical branch A\"oo at infinity.
5.10. Dynamics on Berkovich Space
303
Extra branch "at infinity"
Figure 5.5: The Berkovich projective line
jp'B.
Remark 5.70. There is a natural embedding of AP into jp'B. However, the inversion map I (z) = 1/z used to glue together the two pieces of jp'B can cause notational confusion regarding the "radius" of points of Types II and III. For example, the point ~p-2 ,p-3 in the branch /r..:a of jp'B would be denoted by ~p-l ,p when viewed as a point in A B . Thus it might be wiser to denote the points in /r..:a using some
alternative notation, for example
i.; but we will not do so.
Remark 5.71. A useful alternative construction oflp>B mimics the construction of the scheme jp'i as the set of homogeneous prime ideals. It starts with the set of bounded seminorms on the two-variable polynomial ring Cp[x, y] that extend the usual absolute value on C p and that do not vanish on the maximal ideal (x, y). Two seminorms I 11 and I . 12 are considered equivalent if there is a constant c > 0 such that I/iI = cdegfl/b for all homogeneous IE Cp[X, y].
Then jp'B is the set of equivalence classes of such seminorms. For details of this construction, see [66]. Let P E jp'I (C p ) . To create a seminorm from P, choose homogeneous coordinates P = [a, b] and set III = I/(a, b) I. Notice that a different choice of homogeneous coordinates for P gives an equivalent seminorm. This embeds jp'I (C p ) into jp'B.
5.10.2.3
Properties of AB and jp'B
As repayment for the effort required to construct them, Berkovich spaces have many nice properties that Cp lacks.
304
5. Dynamics over Local Fields: Bad Reduction
Theorem 5.72. (a) The Berkovich disks D~ are compact, Hausdorff, and uniquely path connected. 5 (b) The Berkovich affine line A B is locally compact, Hausdorff, and uniquely path connected. (c) The Berkovich projective line JlDB is compact, Hausdorff, and uniquely path connected. D
Proof See [26] and [64].
Remark 5.73. As noted earlier, the Berkovich affine line A B contains a copy of Al(C p) = Cp, since each a E C p gives an associated Type-I point ~a,O in D~ provided R :::: [c]. Similarly, the Berkovich projective line JlDB contains a copy of the classical projective line JlDl (C p) via the map
a
f------+
{
~a ,o
E
.0(0,1)
iflal::; 1,
~a,O E
1r..:O ~oo,o E 1r..:O
if1
<
if a =
lal
< 00,
00.
One can show that the restriction of the Gel'fond topology on A B and JlDB to A l (C p) and JlDl (C p), respectively, gives the topology induced by the usual metric on A 1 (C p) and the chordal metric on JlDl (C p). See Exercise 5.44. This explains why the Gel'fond topology is the "right" topology to use on Berkovich spaces. Remark 5.74. We have constructed A B and JlDB purely as topological spaces. It is more difficult, but very important, to refine this construction and make ABand JlDB into ringed spaces with structure sheaves built up naturally from rings of functions. There are two approaches, both due to Berkovich. The first takes unions of open Berkovich disks, which have a natural structure as analytic spaces, and glues them along open annuli; see [26,64]. The second glues affinoids (which are closed) using nets; see [65]. This second construction is less intuitive, but it allows one to functorially attach a Berkovich analytic space to any reasonable rigid analytic space.
5.10.3 Dynamics on Berkovich Space Having constructed the Berkovich spaces study iteration of maps on these spaces.
5.10.3.1
DB, A B,
and JlDB, we are finally ready to
Polynomial and Rational Maps on Berkovich Space
Let ¢(z) E Cp[z] be a polynomial. There is a natural way to extend the map ¢ : Al(C p) ~ Al(C p) to a map on Berkovich affine space ¢: A B ~ A B . In terms of seminorms, the action of ¢ is simply given by composition,
Ifl(o
=
If
0
¢I~·
5Recall that a topological space X is path connected if given any two points XQ, Xl E X there is a continuous map I: [0,1] --+ X with 1(0) = XQ and 1(1) = Xl. Then X is uniquely path connected if any two such paths hand h are homotopic to one another, i.e., h an be continuously deformed to h.
305
5.10. Dynamics on Berkovich Space
However, it is perhaps easier to understand the map ¢ : flo. £3 --+ flo. £3 by looking at the action of ¢ on points of Types I-IV. Recall (Proposition 5.16) that ¢ maps disks to disks, say
¢(D( a, r )) = D( ¢(a),R)
for some R = R(¢,a,r ).
Then for points of Types I, II, and III we define
and for points of Type IV we take the usual limit
Remark 5.75. The maximum modulus principle (Theorem 5.13) allows us to explicitly describe the radius R(¢,a , r ) of the image ¢(D(a,r)). First expand ¢(z) as a polynomial in powers of z - a, say d
¢(z) =
L Ci(¢, a)(z - c)' : i= O
Then
R(¢, a,r ) = max ICi(¢, a)lr i =
sup
1::ot::o d
zE D (a ,r )
I¢ (z) - ¢(a)l.
A rational function ¢(z) E iCp(z) similarly induces a map on the Berkovich projective line Jll'B extending the usual map on Jll'1(iC p ) . If ¢ has no zeros or poles on the disk D (a, r ), then it is relatively easy to describe the value of ¢ (~a , r ) . We know in this situation that ¢ (D(a, r)) is a disk, say equal to D(¢(a), s). Then
¢(~
)=
a,r
{ ~¢(a) ,s
~¢(a ) - l ,s/ I¢(a)12
if l¢(a)1::; 1, if I¢(a)I > 1.
Note that the assumption that ¢ does not vanish on D(a, r) is equivalent to the inequality 1¢(a)1 > s, so the indicated points are in DB. The description of ¢ (~a , r) when ¢ has zeros and/or poles on D(a,r) is more complicated. An explicit description in terms of open annuli is given by RiveraLctelier [373, 375, 376]. (See also [26, Section 2].) A succinct , but less explicit , way to specify the induced map ¢ : Jll'B --+ Jll'B is to use the construction of Jll'B as a space of homogeneous seminorms as described in Remark 5.71. Then for a given seminorm ~ E Jll'B, the seminorm ¢ (~) is determined by writing ¢ = [F, G] using homogeneous polynomial s F and G and setting
If (x , y) l ¢(~) = If (F(x, y), G(x ,Y))I
for all homogeneous f E iCp[x, y].
5. Dynamics over Local Fields: Bad Reduction
306
5.10.3.2 The Julia and Fatou Sets in Berkovich Space A natural way to put a metric on the Berkovich spaces fJB, AB, and lP'B is to use the underlying tree structure and measure distances along line segments. Unfortunately, this path-length metric does not give the Gel'fond topology, and as we have observed, it is the Gel'fond topology that extends the natural metric topologies on fJ (O , 1), A1(Cp ) , and lP'1(Cp ) . (See Exercises 5.42 and 5.44.) It is possible to define a metric that does yield the Gel'fond topology, but the definition of the "Gel' fond" metric is quite indirect. See [26, Corollary 1.3]. So rather than using a metric, we instead characterize the Fatou and Julia sets in Berkovichspace using an abstract topological version of equicontinuity. Definition. Let X and Y be topological spaces and let be a collection of continuous maps X -+ Y. The set is (topologically) equicontinuous at x iffor every point y E Y and every neighborhood V c Y of y there are neighborhoods U c X of x and W C Y of y such that for every rjJ E , the following implication is true:
rjJ( U) n W
=f 0
==}
rjJ(U ) c V.
Intuition: is equicontinuous at x if for each y E Y, whenever rjJ E sends some point close to x to a point that is close to y, then rjJ sends every point close to x to a point that is close to y. One can show that if Y is a compact metric space, then topological equicontinuity agrees with the usual metric definition of equicontinuity. (See [26, Proposition 7.17].) Wesay that is (topologically) equicontinuous on X ifit is topologically equicontinuous at every point of X .
Definition. Let rjJ (z) E Cp(z) be a rational map. The (Berko vich) Fatou set of rjJ is the largest open subset of lP'B on which rjJ is equicontinuous, or more precisely, on which the set of iterates {rjJn } is equicontinuous. The (Berkovich) Julia set ofrjJ is the complementof the BerkovichFatou set. Wedenote these sets by FB (rjJ ) and .JB(rjJ) , respectively. Remark 5.76. Recall that the classical points in lP'B, i.e., the points of Type-I, form
a copy of lP'1 (Cp ) sitting inside lP'B. As noted earlier in Remark 5.73, the restriction of the Gel'fond topology on lP'B to the classical points gives the same topology on lP'1 (Cp ) as that induced by the chordal metric. Using this one can show that equicontinuity at a classical point oflP'B using the Gel'fond topology is equivalent to equicontinuity using the chordal metric. Hence the classical Fatou and Julia sets sit within their Berkovich counterparts: and Remark 5.77. Let rjJ (z) E Cp(z) be a rational map of degree at least 2. Various
authors have shown that there is a unique probability measure Mcf> on lP'B satisfying rjJ* Mcf> = d - Mel>
and
5.10. Dynamics on Berkovich Space
307
(Recall that a probability measure is a nonnegative measure of total mass 1.) We call fJ the canonical measure associated to ¢, since the property ¢* fJ = d . fJ resembles the analogous property h (¢( P)) = d . h (P) of the canonical height. The reader should be aware that other common names for fJ in the literature include Brolin measure, Lyubich measure, and invariant measure. For the construction and applications of fJ
Theorem 5.78. Let ¢(z) E Cp(z) be a rational map ofdegree at least 2. The support ofthe canonical measure fJ is equal to the Julia set :JB (¢). In particular, the Berkovich Julia set :JB (¢) is not empty.
Proof This theorem is an amalgamation ofresults due to Baker, Rumely, and RiveraLetelier. We refer the reader to [26, Section 7.5] for the construction ofthe canonical measure and to [26, Theorem 7.18], [27, Theorems 8.9 and A.7], and [381] for the proof that fJ is supported exactly on :JB (¢). The last part of the theorem is then 0 clear, since the empty set cannot provide the support for a nontrivial measure.
Example 5.79. Let ¢(z) E Cp(z) be a rational map of degree at least 2 and suppose that ¢ has good reduction. We know (Theorem 2.17) that the classical Julia set :J(¢) C jp'1 (Cp ) is empty. Using the construction of the canonical measure, it is not hard to show [26, Example 7.2] that for a map of good reduction, the canonical measure is entirely supported at the Gauss point, i.e., fJ(U)
=1
if ~O,l E U
and
fJ(U)
=0
if ~O,I ~ U.
Thus :JB(¢) = {~O, I }, so the nonempty Julia set guaranteed by Theorem 5.78 is not very interesting, since it consists of a single point. Hence even in Berkovich space, the most interesting dynamical behavior occurs for maps of bad reduction. On the other hand, if the conjugates ¢f of ¢ have bad reduction for every f E PGL 2(Cp ) , then :J(¢) is a perfect set, and hence uncountable. (See Theorem 5.82.)
5.10.3.3
The Map ¢(z) = z2 on Berkovich Space
To conclude our brief foray into Berkovich space, we illustrate Berkovich dynamics by studying the simplest possible map, namely ¢( z) = z2. For any a E C p, we expand
¢(z) - ¢(a) = z2 - a2 = 2a(z - a) + (z - a)2. Assuming henceforth that p 2: 3 and using our convention that all points in jp'B have radius satisfying r ::; 1, we find that with
s = max{12alr, r 2 } = r- maxj ]«], r}.
(5.46)
This explicitly gives the action of ¢ on points of Types I, II, and III in jjB, and the action of ¢ on Type-IV points is given by the appropriate limit.
308
5. Dynamics over Local Fields: Bad Reduction
The formula (5.46) allows us to compute many orbits O (~a , r ) . For example , suppose that lal < 1 and r < 1. Then IcPn (a) I ----+ 0, so the radius of cPn (~a ,r) also goes to O. Thus [o]
<1
and
r<1
lim cPn(~a , r ) = ~o ,o .
=>
n ~ oo
In other words, the open branch It\~ , l is in the attracting basin of the fixed point ~o , o . (We leave it to the reader to make this informal argument rigorous using the Gel 'fond topology on DB.) What are the fixed point s of cP? For a point ~a ,r E DB of Type I, II, or III in the Berkovich disk, we have
r =r . max{[ a[, r } and D(a,r )=D(a2,r) r = 0
and
a = a2 ,
{ max{la[,r}=1
or
and
la -a 2 [::; r.
There are three cases to consider. First, if r = 0, then a = a 2 , so we see that ~o , o and 6 ,0 are the only fixed points of Type I in DB. Second, if r = 1, then ~a , r is equal to the Gauss point ~O, ), which is clearly fixed by cP(z ). Finally, suppose that 0 < r < 1. Then ~a,r is fixed if and only if lal = 1
and
Thus ~a ,r is fixed if and only if a E D (1, r), in which case ~a,r = 6 ,r. This exhausts the Type-I ,-II, and-III fixed points in fJB. A similar analysis on the branch leading up to ~oo,o yields one more fixed point, namely the Type-I point ~oo ,o at the top of the tree. Hence aside from Type-IV points, the fixed-point set of cP on jpB consists of two (attracting) Type-I fixed points ~o,o and ~oo ,o and the line segment running from the Gauss point ~O , l down to the (neutral ) Type-I fixed point ~I ,O ,
Fix(cP, jpB) = {~o ,o, ~oo,o } U {6 ,r : 0 ::; r ::; I }. We leave it as an exercise to show that cP(z ) = z2 has no fixed points of Type IV (Exercise 5.36). We have seen that every point in It\~,o is attracted to ~o , o, and the Gauss point ~o, 1 is fixed. We next show that every other point in DB of Type II or III is preperiodic. Let ~a ,r E DB be such a point , which means that
0 < r <1
and
lal = 1.
Then the disk D (a,r ) has positive radius, so it contains points that are algebraic over lQp (note that Qp is dense in C p). Replacing a with such a point, we mayassume that a is algebraic over lQp. Next we observe that (5.46) combined with the assumption that la l = 1 implies so by iteration
5.10. Dynamics on Berkovich Space
309
Hence
Let K = Qp (a) and noti ce that
U D ((pn (a), r) n K c D (O , 1) n K . n ~O
The disk D(O, 1) n K cannot contain infinitely many disjoint disks of radius r , so there must exist m > n such that
Then 1r (~a, r ) = ¢n(~a ,r), so ~a , ,· is preperiodic. We conclude this section by using the definition of topological equicontinuity to directly demonstrate that the Gau ss point ~O ,l is in the Julia set of ¢. The intuition is that any neighborhood of ~O , l contains points ~O ,r on the line segment connecting ~O ,l to ~o ,o. If r < 1, then the iterates ¢n( ~O ,r ) approach ~o ,o, but the Gauss point ~O ,l is fixed. Hence ¢n (~O, l) doe s not rema in close to ¢n(~O ,r) . We now make this argument rigorous. We suppose that x = ~O , l E ;:6 ( ¢ ) and derive a contradiction. Let y = ~o,o , and let V = !f\~ , 1 /2 be our chosen neighborhood of ~o , o. The definition of equicontinuity says that there are neighborhoods ~O , l E U and ~o , o E W such that for all n ;::: 0, (5.47) The implication (5.47) remains true if we replace U and W by smaller neighborhoods , so we may assume that k
U = JP6
<,
U
with
J.
lI\ a i lr i
°<
r,
< 1 for all i, and
i= l
with 0 < r < 1.
W = !f\°o ,r Choose a value of s satisfying
max r. < s < 1.
l~ i~ k
Then ~o , s E U, since ~o , s is not on any of the closed branches !f\a;,r;' On the other hand , ¢n (~o,s ) = ~o, sn E W = !f\~, r for sufficientl y large n, since we just need to ensure that s"
< r . This proves that
so the assumption that ¢ is equicontinuous at
~O , l
implie s that
310
5. Dynamics over Local Fields: Bad Reduction
But this inclusion is clearly not true, since, for example, ~O ,l E U is fixed by ¢, but ~O,I ~ A\~,1/2' Therefore ¢ is not equicontinuous at ~O ,I, so ~O,l E .JB(¢). A similar case-by-case argument using the explicit description (5.46) of the action of ¢ shows that every other point in pB is in the Fatou set. We leave the details for the reader. 5.10.3.4
Further Results
We briefly describe, without proof, some deeper results on Berkovich dynamics. Our exposition follows [27], and the author is grateful to Baker and Rumely for making their preprint available. Theorem 5.80. (Strong Montel Theorem on pB) Let ¢ E Cp(z) be a rational map ofdegree at least 2, let ~ E pB, let U C pB be an open neighborhood of~, and let V be the union of ¢n (U) fo r all n 2: 1. (a) if pI (Cp ) <, V contains at least 3 points, then ~ E F B(¢). (b) ifpB " (V U PI(C p ) ) isnonempty, then E E F B(¢).
Proof This theorem is due to Baker and Rumely [27, Theorem 7.1] for maps ¢ defined over a finite extension of Qp, and to Rivera-Letelier in the general case ; 0 see [27, Theorem A.I] and [381]. The proof is based on Rivera-Letelier's classification of periodic components in the Fatou set. In order to describe this classification and its applications, we need to define what it means for a periodic point in Berkovich space to be attracting or repelling. For Type-I points, i.e., for points in pI (Cp ) , we use the usual definition. It turns out that all attracting periodic points in pB are Type-I points. (See Exercise 5.45 for an explanation of why this is reasonable.) The definition of repelling periodic points is more complicated. Repelling periodic points are all of Type I or II [376, Proposition 5.5], where we use Rivera-Letelier's definition that a Type-Il periodic point is repelling if its residual degree (see [376, Sect ion 5]) is at least 2. Definition. Let ¢ E C p (z) be a rational map of degree at least 2 and let ~ E pB be an attracting periodic point of period n. The basin ofattraction of ~ is the set
The connected component of this set is called the immediate basin ofattraction of~ . Definition. Recall that a point P is called recurrent for ¢ if it is in the closure of{ ¢n(p ) : n 2: I} . The domain ofquasiperiodicityof ¢ is the interior of the set of points in pB that are recurrent for ¢. We are now ready to state Rivera-Letelier's strong Montel theorem for rational maps on the Berkovich projective line.
5.10. Dynamics on Berkovich Space
311
Theorem 5.81. (Rivera-Letelier) Let Uq, be the set ofpoints ~ E pB with the property that there is a neighborhood U of~ such that pl(C p )
<,
U ¢n(u) n2':1
contains at least 3 points. (a) Every periodic connected component ofUq, is either an immediate basin ofattraction of ¢ or a connected component ofthe domain ofquasiperiodicity of ¢. All such components are in F B (¢). (b) Every wandering component ofUq, is contained in F B (¢). Proof The proof of (a) is given in [27, Theorem A.2] and the proof of (b) is in [27, Corollary A.5]. D As in the classical setting, Montel's theorem has a large number of important consequences, some of which we state here.
Theorem 5.82. Let ¢ E C p (z) be a rational map ofdegree at least 2. (a) (b) (c)
(d)
(e) (f) (g)
If there is some f
E PGL 2(Cp ) such that the conjugate ¢f has good reduction, then the Julia set JB (¢) consists ofa single point. If every conjugate ¢f of ¢ has bad reduction, then the Julia set JB (¢) is a perfect set, and hence in particular it is uncountable. Let ~ E J B(¢), let U E pB be an open neighborhood of~, and let V be the union of ¢n(u) for all n 2': 1. Then (i) V contains pB <, pI (C p ) ; (ii) V contains JB( ¢); and (iii) pI (C p ) <, V consists ofat most two points. The Julia set JB (¢) is either connected or else it has infinitely many connected components. Further, it has empty interior. Let ~ E JB (¢). Then the backward orbit of~ is dense in JB (¢). Let V C pB be a closed completely d-invariant set containing at least 3 points. Then V ~ JB(¢). The Julia set JB (¢) is exactly equal to the closure of the repelling periodic points in pB.
Proof (a) See [27, Lemma 8.1]. (b) See [27, Corollary 8.6]. (c) See [27, Theorem 8.2]. (d) See [27, Corollary 8.3 and Corollary 8.7]. (e) See [27, Corollary 8.5]. (f) See [27, Corollary 8.8]. (g) See [27, Theorem A.7].
D
312
Exercises
Exercises Section 5.1. Absolute Values and Completions
5.1. Prove that up to equivalence , the only nontrivial absolute values on Q are the usual archimedean absolute value and the p-adic absolute values. (This result is known as Ostrowski' s theorem.) Section 5.2. A Primer on Nonarchimedean Analysis
5.2. (a) Let ¢>(z) be a holomorph ic function on D (a, r}. Prove that the Taylor series coefficients of ¢>(z) are uniquely determined by ¢>. (b) Let ¢>( z) be a function that is represented by a convergent Laurent series on the punctured disk D ( a, r ) -, {a}. Prove that the coefficients of its Laurent series are uniquely determined by ¢>. 5.3. (a) Let ¢>(z) and 1jJ(z) be holomorphic functions on D (a,r ). Prove that the product ¢>(z)1jJ (z) is a holomorphic function on D(a, r). (b) Let ¢>(z) and 1jJ(z) be merom orphic functions represented by Laurent series on D(a ,r) . Prove that the product ¢>(z)1jJ(z) is a meromorphic function and is represented by a Laurent series on D (a,r ). 5.4. Let ¢>(z) E Cp[z] be a nonconstant polynomial. (a) Prove directly (i.e., without using Newton polygons) that ¢> sends a closed disk D ( a, r) to a closed disk D(¢>(a), s). (b) Prove that ¢> maps D (a,r ) bijectively to D (¢>(a), s) if and only if
I¢>(z) - cP(a) 1 = ~ I z r
- al
for all z E D(a,r ).
5.5. This exercise outlines a prove of a p-adic inverse function theorem. Let ¢> E C p ( z) be a rational function of degree d 2: 1 and let P E IF'1(C p ) be a point that is not a critical value of cP, i.e., P is not the image of a critical point of ¢>. (a) Prove that there is a disk D (P,r ) centered at P such that ¢>- I (D (P,r ») consists of d disjoint open sets
Show that the Vi are disks. (b) Let ¢>i : Vi ---+ D(P, r) be the restriction of ¢> to Vi for each 1 :::; i :::; d. Prove that ¢>i is bijective. (c) Possibly after reducing r , prove that the inverse maps ¢>; I : D (?, r) ---+ Vi are given by convergent power series. (If 00 E ¢> -1 (P) , either change coordinates or instead use a convergent Laurent series.) 5.6. (a) Let K be an algebraically closed field that is complete with respect to a (nonarchimedean) absolute value and let cP : IF'1(K) ---+ IP'1(K) be a rational map. Prove that ¢> is an open map, i.e., the image of an open set is an open set. (b) Consider the map ¢> : IF'1(Q3) ---+ IF'1 (Q3) defined by ¢>( z ) = Z2 . Prove that ¢> is not an open map by showing that the image of the open disk D(O , 1) in IF'1 (Q3) is not open. Thus rational maps over complete, but not algebraicall y closed, fields need not be open.
Exercises
313
5.7. Let ¢(z) E Cp[z] be analytic on D(a, r) and assume that ¢(a) -I- o. Prove that l/¢(z) is analytic on some closed disk centered at a. More precisely, prove that l/¢(z) is analytic on D(a, t) for any t < rl¢(a)I/II¢II.
Section 5.3. Newton Polygons and the Maximum Modulus Principle 5.8. This exercise generalizes Proposition 5.16(b) by asking you to prove a general Cauchy estimate for p-adic analytic functions. Let ¢(z) E Cp[z] be a power series that converges on oi«, r) and let ¢(D(a, r)) = D(¢(a), s). Prove that for all n (Note that as n increases, the estimate becomes worse, since
;::>:
In!1
1. ---+
0 as n
---+
00.)
Section 5.4. The Nonarchimedean Julia and Fatou Sets 5.9. Prove the implications between uniform Lipschitz, uniform continuity, and equicontinuity stated in Section 5.4. More generally, give a definition of what it means for a family of functions to be equi-Lipschitz and prove the following implications:
I uniformly Lipschitz I
===}
I equi-Lipschitz at every point I
I uniformly continuous I ===} I equicontinuous at every point I 5.10. This exercise develops the abstract theory of residues (on pl) in order to prove Theorem 1.14 in arbitrary characteristic. Let ¢( z) E K (z) and Q E K. The function ¢( z) can be written as a partial Laurent series
a-N ¢ (z ) = (Z-Q )N
a-N+l a-l ( ) + (Z-Q )N-l +...+ -(--) Z-Q + 'IjJ z
and we define the residue of ¢ at
Q
with 'IjJ(z) E K[z - Q],
to be
Res(¢(z)dz) z =o:
=
a-l.
The residue of ¢ at 00 is defined by using the substitution z = l/w; thus Res (¢(z)dz) Z=CXl
= Res(-¢(w- 1)w- 2dw). w=o
(a) Compute all of the nonzero residues of each of the following functions. In each case, check that the sum of the residues is O. 2 Z2 2 2· (i) ¢t(z) = Z : 2 (ii) ¢2(Z) = 4 z - z z - z 5 z(b) If ¢( z) has a simple pole at Q, prove that the residue of ¢ at Q is the value of the function (z - Q) ¢( z) evaluated at z = Q. (c) More generally, for any integer n ;::>: 0, let 'ljJn (z) be the function
2
~
3 2:
n
1 d ( 1/Jn () Z = n! dz n (z - Q)n+l) ¢(z).
Exercises
314
Here we are taking formal derivatives in K(z). Note that in characteristic p, one must be careful to "cancel" the n! before setting p = O. (If you have not seen this kind of computation before, rewrite the expression for 'l/Jn (z) to make it clear that it makes sense in any characteristic.) Suppose that ¢(z) has a pole of exact order n 2: 1 at 0:, i.e., writing ¢(z) = F(z)/G(z) as a ratio of polynomials with no common factors, we have n = ordz=a (G(z)). Prove that Res(¢(z) dz) = 'l/Jn-l(O:). z=o:
(d) Prove that
{p E jp'l(K): Res(¢(z)dz) -# O} P is a finite set. (e) Assume that K is algebraically closed and prove the Cauchy residue formula
L
~s(¢(z)dz)=O.
PElI'l(K)
(This is a hard exercise. Try it first under the assumption that ¢(z) has simple poles and ¢( 00) -# 00, which suffices for (f).) (f) Use (e) to prove that Theorem 1.14 is true in arbitrary characteristic, and use it to deduce that Corollary 5.19 and Proposition 5.20(c) are true. Hence Fatou sets are nonempty for all algebraically closed nonarchimedean fields. 5.11. (a) Let p 2: 3 be a prime and let eE Cp satisfy lei> 1. Prove that the function ¢(z) = z2 + c has exactly one nonrepelling periodic point, namely the totally ramified fixed point at infinity. (b) * Suppose that ¢(z) E Cp(z) has an indifferent periodic point, i.e., a periodic point P whose multiplier satisfies lAp(¢) I = 1. Prove that ¢(z) has infinitely many indifferent periodic points. (This result is due to Rivera-Letelier [372].) (c)
** Do there exist rational maps ¢(z) E Cp(z) that have at least two, but only finitely many, nonrepelling periodic points? (From (b), the nonrepelling periodic points would have to be attracting.)
Section 5.5. The Dynamics of (Z2 - z)/p 5.12. Let S be a finite set, let SM be the space of sequences on S with the symbolic dynamics metric, and let L : SM ----> SM be the left shift map. (See Section 5.5.1.) (a) Let 0:, f3 E SM and suppose that limn~oo p( Ln(o:),L n (f3 )) = O. Prove that there exists an m such that Lm(o:) = L m (f3 ). (b) Prove that the periodic points of L are dense in SM. (c) Prove that there is an element 'Y E SM with the property that the orbit
Ch("() = {C("() : n E N} is dense in SM. One says that L is topologically transitive on SM. (Hint. Create 'Y by first listing all possible blocks oflength I, then all possible blocks oflength 2, and so on.) (d) Let 0: E SM. Prove that the backward orbit Vi (0:)
= Un 2:0 L-n(o:) is dense in SM.
Exercises
315
(e) More precisely, prove that the backward orbit of any following sense: For all (3 E SM and all k 2': 0,
0:
E SM is equidistributed in the
1 (#S)k· (In fact, prove that the limit stabilizes as soon as n. > k.) 5.13. Let ¢(z) = (Z2 - z)/p with p 2': 3. Use Exercise 5.12 and the identification provided by Proposition 5.24 to prove that the backward orbit of a point a E :J(¢) is equidistributed in the following sense: For all b E :J(¢) and all radii r = p-\
5.14. Let p 2': 3 and for any c E C p , let ¢c(z) = (Z2 - cz)/p. (a) If c E Z;, prove that the statement of Corollary 5.25 is true for ¢c. (b) More generally, prove an analogous statement if c is a unit in a finite extension of Zp. (c) What happens if c E pZp? 5.15. Let d
2':
2 be an integer, let p
¢(z)
> d be a prime, and consider the dynamics of the map
= _z(,---z_-_l,--,)(_z_----'2)'---._.·---,-(z_-_d+----,-1) p
over C p . Prove that :J (¢) can be described using symbolic dynamics and use this identification to prove thefollowing generalization of Corollary 5.25: :J(¢) c IQip, the set Per n ( ¢) consists ofthe fixed point at CXJ and dn repelling points, the Julia set :J(¢) is the closure ofthe repelling periodic points, and ¢ is topologically transitive on :J (¢). Section 5.6. A Nonarchimedean Version of Montel's Theorem 5.16. Complete the proof of Theorem 5.26 (see page 265) by writing down the details in the case that 0: = O. 5.17. Let be a collection of power series that converge on D( a, r), and suppose that there is a point 0: E K such that 0: ~ ¢( D(a, r)) for all ¢ E <1>. We proved in Theorem 5.26 that there is a constant C = C ( 0:, a, r) such that
p(¢(z), ¢(w)) S Cp(z, w)
for all ¢ E and all z, wE D(a, r).
Find an explicit value for the Lipschitz constant C in terms of 0:, a, and r, 5.18. Let be a collection of rational, or more generally meromorphic, functions
D(a,r)
-->
jp'l(K),
and suppose that 0:, (3 E jp'l (K) do notlie in ¢ (D(a, r)) for all ¢ E <1>. Our proofofthe nonarchimedean Montel theorem (Theorem 5.27) shows that there is a constant C = C( 0:, (3, a, r) such that
p(¢(z), ¢(w)) S Cp(z, w)
for all ¢ E and all z, wE D(a, r).
Find an explicit value of C in terms of 0:, (3, a, and r.
Exercises
316 Section 5.7. Periodic Points and the Julia Set
5.19. As in the statement of Lemma 5.34, let A, B C C p be bounded sets for which there are constants 0 < 8 < 1 ::; ~ such that sup lal ::;~,
sup 1)31 ::;~,
"EA
(3EB
For each (a,)3) E A x B, let L",(3(z) showed that
p(L",(3(z),L"',(3I(Z')) <
~24
.
=
and
()3 - a)z
inf
"EA, (3EB
p(a, J3)
= 8 > O.
+ a. During the proof of Lemma 5.34 we
max{p(a,a'),p()3,)3'),p(z,z/)}
for all a, a/ E A, all)3,)3' E B, and all z, z' E ]pI (Cp ) . Improve this result by reducing the exponent of ~ and/or 8 in the constant. Try to find the best-possible exponents.
5.20. Let A, B, C C ]pI (Cp ) be three sets that are at a positive distance from each other, i.e., there is a constant 8 > 0 such that inf
"EA,(3EB
p(a,)3) 2: 8,
inf
p(a, 'Y) 2: 8,
"EA,-YEC
inf (3EB,-yEC
p()3, 'Y) 2: 8.
For any triple of points (a,)3, 'Y) E A x B x C, define
to be the unique linear fractional transformation satisfying
Prove that the map
is Lipschitz and find an explicit Lipschitz constant depending only on 8. (This exercise generalizes Lemma 5.34, which is the case that C = {oo} is the single point at infinity.)
5.21. This exercise can be used in place of Lemma 5.34 to prove Montel's theorem with moving targets (Theorem 5.36). (a) Let P, Q E ]pI (K) be distinct points. Prove that there is a linear fractional transformation 11.= Ap,Q E PGL 2 (K ) satisfying
11.(0)
= P,
11.(00) = Q,
IRes(A) I = p(P, Q).
(Hint. If A = (~S) is normalized to satisfy max{lal, Ibl, lei, Idl} = 1, then Exercise 2.8 says that the resultant of A is equal to the determinant ad - be.) (b) Let A, B C C p be bounded sets that are at a positive distance from each other. Prove that the map
(P,Q,R) is Lipschitz.
f-----+
Ap,Q(R),
317
Exercises Section 5.8. Nonarchimedean Wandering Domains
5.22. Let X be a topological space and let D be a collection of open subsets of X that form a base for the toplogy of X. We use D to define disk components as described on page 277. (Note that the sets in D need not be actual disks, since the space X is merely assumed to be a topological space.) (a) Let U C X be an open set, let Pi; P 2 E U be points, and let V l and V2 be the disk components of U containing Hand P2, respectively. Prove that either Vl = V2 or Vl n V2 = 0. (b) Prove that U is a disjoint union of disk components. (c) Prove that the disk components of U are open. 5.23. Let X be a topological space and let D be a collection of open subsets (disks) of X that form a base for the topology of X. An open set U C X is defined to be disk-connected (relative to D) if for every pair of points P, Q E U there is a finite sequence of disks D«, D2, ... , D n C U such that
D, n Di+l
-I 0
for aliI::; i
< n.
(a) Let U, and U2 be disk-connected subsets of X. Prove that either
(b) Let U C X be an open set and let P E U. Prove that there is a maximal disk-connected open subset of U containing P. (c) Prove that the maximal disk-connected open subset of U containing P described in (b) is in fact the disk component of U containing P. This gives an alternative definition of disk component. 5.24. Prove that the standard closed disks with rational radius (cf. Remark 5.7) in jp'l(iCp ) are exactly the images via elements ofPGL 2 (iC p ) ofthe unit disk .0(0,1). Similarly, the standard open disks of rational radius are the images of D(O, 1). 5.25. Let K/Qp be a finite extension and let ¢(z) E K(z) be a rational function of degree d ~ 2 with 00 E F( ¢). Prove that ¢ is hyperbolic, i.e., ¢ satisfies the conditions of Theorem 5.46, if and only if ¢ has the following property:
For every finite extension L/ K there are a set U C L containing .:J( ¢) n L, positive constants b > a > 0, and a continuous function a : U ----> [a, b] such that for all Q E U. This shows that p-adic hyperbolicity and classical (complex) hyperbolicity can both be characterized as saying that a map is everywhere expanding on the Julia set with respect to some reasonable metric. For further information about hyperbolic maps in the classical setting; see, for example [95] or [302]. 5.26. * Let ¢ : jp'l ----> jp'l be a rational map of degree at least 2 and let P E jp'l (iC p ) be a recurrent critical point that is in the Fatou set F( ¢). Prove that P is a periodic point.
318
Exercises
Section 5.9. Green Functions and Local Heights 5.27. Let K be a field with a nonarchimedean absolute value v, let ¢ : pI -+ pI be a morphism of degree d 2:: 2, let 1> = (F, G) : A? -+ A? be a lift of ¢ satisfying 111> II v = 1, and let Y4> be the associated Green function. If Res(F, G) v =I- 1, or equivalently if ¢ has
I
I
bad reduction at ¢, prove that there exists a point (x, y) E A; (K) such that
This is the converse to Proposition 5.58(c). 5.28. Let K be a field with an absolute value v, let ¢ : pI gree d 2:: 2, and let 1> : A 2 -+ A2 be a lift of ¢. (a) Prove that the map
g(x, y)
= logll1>(x, y) Ilv -
-+
pI be a morphism of de-
dlogll (x, y) Ilv
induces a well-defined function 9 : pI (K) -+ R (b) Prove that 9 is Lipschitz, i.e., prove that there is a constant C
= C( ¢) such that
Ig(p)-g(Q)1 ~Cpv(P,Q) forallP,QEpl(K). (c) If v is nonarchimedean, prove that 9 is locally constant. More precisely, if the lift 1> is chosen to satisfy 111>llv = 1, prove that
g(P)
= g(Q)
forall P,Q E pl(K) with Pv(P,Q)
(d) Define a modified Green function
< IRes (1)) Iv'
94> by
Prove that 94> is a Holder continuous function on pI (K). In other words, prove that Q4> is well-defined on pI (K) and that there are positive constants C and <5 such that for all P, Q E pl(K).
(Hint. Show that Q4> (P) is given by the telescoping series 'L':=o d- n9 (¢n (P) ). For small n, say n < N, estimate the difference I9 (¢n(P)) - 9 (¢n (Q)) I using (b) and the fact that ¢ is Lipschitz, and for large n use an elementary bound. Then make an appropriate choice for N.) 5.29. The definition of the canonical height h¢(P) as the limit of d:" h( ¢n(p)) is not practical for numerical calculations, even for P E pI (Q), because one would need to compute the exact value of points ¢n (P) whose coordinates have O( dn) digits. A better method to compute h¢(P) is as the sum of the Green functions Y4>,v(P) as described in Theorem 5.59. Let ¢ : pI -+ pI be a morphism of degree d 2:: 2 and fix a lift 1> of ¢. If v E MK is a nonarchimedean absolute value such that 111>llv = 1 and IRes(1))Iv = 1, then Proposition 5.58(c) says that Y4>,v(x,y) = logll(x,y)llv. This covers all but finitely many absolute values, so it remains to devise an efficient method to compute Y4>,v (x, y) in the remaining cases.
Exercises
319
Let P = [x, y] E jp'1(K) and use the algorithm described in Figure 5.6 to define a sequence of triples with iu, Xi, Yi E K. This exercise asks you to prove that N iterations ofthe algorithm gives the value of 9<1>, v (x, y) to within O(d- N ) . (a) Prove that IUilv -:; 1 for all i 2: O. Hence the computation of quantities such as F(ui, 1) in the algorithm do not involve excessively large numbers. (b) Prove that for all i 2: 1 we have
(c) As in Exercise 5.28, we let
g(X, Y)
= logll(X,Y)llv
- dlogll(X, Y)t
The homogeneity of shows that 9 is a well-defined function on jp'1(K), and Proposition 5.57 says that 9 is bounded. Prove that
9<1>(x,y)
= logll(x,Y)llv +
:f= ~ig(i-1(X,y)). i=l
(d) Prove that N
9<1>(x,y)
= logll(x,Y)llv +
L ~i
logmax{lxilv, IYilv} + O(d-
N
(5.48)
),
i=l
where the big-O constant depends only on . Deduce that the algorithm described in Figure 5.6 computes 9<1> (x, y) to within O(d- N ) . (e) Find an explicit value for the big-O constant in (5.48) in terms ofthe quantities d, 11llv,
I
and Res(
Iv'
5.30. Implement the algorithm described in Figure 5.6 to compute the Green function 9¢,oo for the archimedean absolute value on jp'l (R). (a) Let ¢(z) = z + l/z and compute 9¢,oo(x, y) to (say) 8 decimal places for each of the points (1, 1), (2, 1), and (5, 2). Check your program by verifying that your values satisfy
Compute the canonical height h¢(I) and compare it with the value that you obtained in Exercise 3.20. (Hint. The map ¢ has good reduction at all primes.) (b) Let ¢(z) = (3z 2 - 1)/(z2 - 1) and note that ¢2(1) = 3 and ¢3(1) = Compute 9¢,oo(x, y) for each of the points (1,1), (3,1), and (13,4). Why does 9¢,oo(3, 1) not equal 49¢,oo(l, I)? What is the difference between these two values?
¥.
5.31. Let ¢ E Q(z) be a rational map of degree at least 2 and let P E jp'l (Q). Write an efficient computer program to compute h¢(P) as a sum of Green functions. (For primes p not dividing Res(
320
Exercises
INITIALIZATION Write ~ as ~ = (F,G) with F,G E K[X, Y] Set N = Desired number of iterations Set Xo = x and Yo = Y and Green = 0 MAIN LOOP: i = 0, 1,2, ... ,N Increment Green By d-qogmax{lxilv, IYilv} I f IXi Iv ::; IYi Iv Set Ui = X;/Yi Compute Xi+l = F(Ui, 1) and Yi+! = G(Ui, 1) Else IYilv < IXilv Set Ui = Y;/Xi Compute xi+l = F(l, Ui) and Yi+l = G(l, Ui) END MAIN LOOP Return the Value Green
Figure 5.6: An algorithm to compute the Green function Q.p(x, y). (b) ¢(z) = Z2 + 1, a = ~. (c) ¢(z) = 3z 2 - 4, a = 1. (d) ¢(z) (e) ¢(z)
1
= z + -, z 3z
2
-
= z2 _
a
1
l'
= 1. a
= 1 and a = 3. Check that h¢(3) ~ 4h¢(1).
5.32. Let K be an algebraically closed field with an absolute value v, let ¢ : pI ~ pI be a morphism of degree d :::: 2, fix a lift of ¢, and for each homogeneous polynomial E E K[x, y], let ).,¢,E be the associated local canonical height function. (See Theorem 5.60.) (a) If E I , E 2 E K[x, y] are homogeneous polynomials, prove that
).,¢,EIE2(P) =
).,¢,EI
(P)
+ ).,¢,E2(P)
at all points such that EI (P) =I 0 and E2 (P) =I O. (b) Let D = ni (QI) + n2(Q2) + ... + nr(Qr) be a divisor on pI, i.e., D is a formal sum of points with nl, ... ,n r E lZ. Use (a) to associate to D a local height function
(c) Prove that there is a rational function
Iv
E K(z) with the property that
at all points where ).,¢,v (¢(P)) and )"¢,v(P) are defined. (d) Let g(z) E K(z) be a rational function and let D g be the divisorofg, which by definition is the formal sum
Exercises
321
o,
=
L
ordQ(g)(Q).
QElI'l(K)
(Here ordQ (g) is the order of vanishing of 9 at the point Q; see Example 2.2.) Prove that the function P f-----+ >"¢.D g (P) + loglg(p) Iv extends to a bounded continuous function on all of jp>1 (K).
Section 5.10. Berkovich Space and Dynamics 5.33. Let ¢(z) E Cp[z] be a polynomial and let a E D(O, 1). Write ¢(z) as a polynomial in z - a, say
Prove that
¢(D(a,r))
=
D(¢(a),s),
where the radius s of the image disk is given by s=r.max{lcII, IC2Ir, IC31r2, ... , ICdl r d}.
(Hint. We already proved that ¢ (D(a, r)) is a disk; see Proposition 5.16. Now use the maximum modulus principle to find the radius.) 5.34. This exercise develops some elementary properties of bounded seminorms as defined on page 296. (a) Let I . Ibe a nonconstant seminorm. Prove that I I = 0 and 111 = 1. (b) Suppose that I . I is an II . Ilwbounded seminorm. Let f(z) = Cbe a constant polynomial. Prove that if I = [c] is the usual absolute value on Cpo (c) Suppose that we replace property (3) of bounded seminorm with the usual triangle inequality If + gl < If I + Igl· Prove that I . I satisfies (3). (d) Suppose that we replace property (4) of bounded seminorm with the weaker statement that there is a constant K such that If I ::; K II f II R for all f E Cp [z]. Prove that (4) is true, i.e., we can take K = 1.
°
5.35. Prove that the seminorms associated to points of TypesII, III, and IV are actually norms. (Hint. For Type IV,use the fact that the limiting radius is positive. See Remark 5.65.) 5.36. Let d 2 2. Prove that the map ¢( z) = zd has no Type-IVfixed points in AB. Does ¢(z) have any Type-IV periodic or preperiodic points? 5.37. Let D(a, r) be a disk with 0 ~ D(a, r) and let f(z) =
f(D(a,r))
= {Z-I: Iz-al
II z. Prove that
::;r} =D(a- I,rllaI 2 ) .
5.38. Let d
¢(z) =
Lcd
E
Cp[z]
be a polynomial with ICil ::; 1 and ICdl = 1.
i=O
The polynomial ¢ induces a map ¢ : DB --+ DB on the Berkovich disk. (a) Prove that the Gauss point ~O,l is fixed by ¢.
(5.49)
322
Exercises
(b) Let a E Cp be a fixed point of c/> : Cp --+ Cpo Prove that ~a,O is fixed by c/>.
°
I
(c) Let a E Cp be a neutral fixed point, i.e., Ic/>'(a) = 1, and let ~ r ~ 1. Prove that ~a,r is fixed by c/>. (In other words, if a is a neutral fixed point, then the entire line segment La,o C DB is fixed by c/>.) (d) Prove that every Berkovich fixed point of c/> of Type I, II, or III is one of the three types listed in (a), (b), (c). (Hint. Use the explicit description of c/>(~a,r) given in Exercise 5.33.) (e) Can c/> have fixed points of Type IV?
5.39. Let c/>(z) = L: CiZi E Cp[z] be a polynomial as in (5.49). Prove directly that the Julia set of c/> is .JB(c/» = {6,0}. (This is a special case of the result described in Example 5.79, since the conditions on c/> imply that it has good reduction.) 5.40. For any nested sequence of closed disks
regardless of whether the intersection is empty, we define a seminorm in the usual way,
n
D(ai, r.) is nonempty and a is a point in the (a) Let r = lim rio If the intersection intersection, prove that Ifla,r = Ifla,r. Thus every point in DB, regardless of whether it is of Type I, II, III, or IV, can be represented by a nested sequence of closed disks. (b) Two nested sequences of closed disks
D(O,l) ::::J D(a1, ri ) ::::J D(a2, r2) ::::J ... D(O,l)::::J D(b1,Sl)::::J D(b2,S2)::::J""
,
are defined to be equivalent if the following two conditions are true: • For every i • For every i
21 thereexistsaj 21 suchthatD(bj,sj) 21 there exists aj 2 1 such that D(aj,rj)
C C
D(ai,ri)' D(bi,Si)'
Prove that two sequences of disks are equivalent if and only if their seminorms I . la,r and I . Ib,. are equal.
5.41. Let ~a,r E DB be a Type-IV point. We defined the "line segment" running from ~a,r to the Gauss point ~0,1 to be the set 00
La ,r
= {~a,r} U U{~ai,t
:
r,
~ t ~ ri-d·
i=l
(Note that ro
= 1 by definition.) Let r = limi--->oo rio Prove that the map h: [r,l]
--->
La,r,
h(t) =
{~a,r
~ai,t
~ft =
r,
if r, ~ t ~ ri-1,
is a homeomorphism, where [r,l] C R has the usual topology and DB has the Gel'fond topology. Thus La,r is indeed a line segment.
323
Exercises
5.42. There is a natural metric on the Berkovich disk DB coming from the tree structure. We first identify the interval [0,1] with each of the line segments
La,o = {~a,r :
°.,:;
r .,:;
-B
I} ED,
so each La,o has length 1. We then define the distance between two points ~a,r and ~b,s to be the length of the shortest path in the tree connecting them. Denote this distance by 1I;(~a,r, ~b,s). (For Type-IV points, take the limit.) (a) If ~a,r and ~b,s both lie on some line segment Lc,o, prove that 1I;(~a,r, ~b,s) = Ir - 81 is simply the distance between them on that line segment. (b) In general, prove that
1I;(~a,r,~b,s)
= max{lr - 81, 21b - al- r - 8}.
(Hint. How far above ~a,r and ~b,s do the line segments La,r and Lb,s merge? Go up one line segment and then down the other.) (c) Prove that the set {
~ E DB : II;(~, ~o,I) < ~}
contains no Type-I points. Prove that every neighborhood of ~O,l in the Gel'fond topology contains infinitely many Type-I points. Deduce that the path metric II; does not define the Gel'fond topology, and indeed that II; is not even continuous in the Gel'fond topology! In Baker and Rumely's terminology, the set DB with the metric II; is called the small mode/. There is also a big model, in which the edges are reparameterized so that if bt«, r) C D(b, 8), then the distance from ~a,r to ~b,s is !log(r/ 8) In particular, points of Type I are at infinite distance from each other and from all points of Types II, III, and IV. It is the big model that is best adapted for doing potential theory on Berkovich space; see [26] for details.
I.
5.43. For points of Type I, II, and III in DB, the Hsia kernel is defined to be
(For points of Type IV, one takes the appropriate limit.) The Hsia kernel is used in studying potential theory on DB; see [26,209]. (a) Prove that 8(~a,r, ~a,r) = r. (b) Prove that 8(~a,o, ~b,O) = la - bl. Thus 8 extends the usual norm on !Cp , where we identify the unit disk in !Cp with the Type-I points in DB. (c) Let ~c,t be the point where the line segments La,r and Lb,s first meet. Prove that 8(~a,r, ~b,s)
= t.
(d) Prove that
8(~,C).,:; max{8(~,e'),8(e',(')} 5.44. There are natural inclusions -
-B
D(O, 1) '---t D , In each case, prove that the restriction of the Gel'fond topology on the Berkovich space yields the usual metric topology on the smaller space, where we use the usual p-adic metric on D(O, 1) and A 1 (!C p ) and the p-adic chordal metric on jp'l (!C p ) .
324
Exercises
5.45. Let ¢ be a rational map of degree at least 2, let P E
jp'B, and
let U C
jp'B be
a neighbor-
hood of P with the property that
¢(U)
~
U
and
n
¢n(U) = {P}.
n;:>O
Prove that P is a Type-I point and that, considered as a point in jp'l(Cp ) , the point P is an attracting fixed point for ¢.
Chapter 6
Dynamics Associated to Algebraic Groups In the forest of untamed rational maps live a select few whose additional structure allows them to be more easily domesticated. They are the power maps, Chebyshev polynom ials, and Lattes maps, whose complex dynamics were briefly discussed in Section 1.6. The underlying structure that they possess comes from an algebrai c group , namel y the multiplicative group for the power maps and Chebyshev polynomial s and elliptic curves for the Lattes maps. Althou gh such maps are special in many ways, they yet provide important examples , testing grounds, and boundary condition s for general results in dynamic s. In this chapter we investigate some of the algebraic and arithmetic propertie s of the rational maps associated to algebraic groups.
6.1
Power Maps and the Multiplicative Group
The simplest rational maps are the power maps given by monic monomials,
where the integer d may be positive or negative. These maps obviously commute with one another under compositi on,
A more intrinsic description of the maps associated to the monic monomial s M d is that they are endomorphisms of the multiplicati ve group ,
More preci sely, there is an isomorphi sm of rings, 325
6. Dynamics Associated to Algebraic Groups
326
The fact that the M d are endomorphisms of the multiplicative group G m makes it quite easy to describe their preperiodic points.
Proposition 6.1. Let d E Z with map Md(Z) = zd. Then
PrePer(Md) = (Gm)tors
Idl 2:
2 and let M d
= {( E Gm : C
=
:
pI
-+
1for some n 2:
p I be the power
1} =
U J.Ln , n~ 1
where recall that J.Ln denotes the group ofnIh roots ofunity. Proof We proved this long ago for any abelian group G and homomorphism z t--+ zd with d 2: 2; see Proposition 0.3. The prooffor d :::;; - 2 is similar and left to the reader. 0 The iterates of Md are given by
so the periodic points of M d are also easy to characterize,
Proposition 6.2. Let Idl 2: 2 and let ( E Per~* (Md ) be a point of exact period n ;::: 2. Then the multiplier of Md at ( is given by
Proof Using M'd(z) = zd" , we can directly compute
In particular, if we are working over C, then every periodic point of Md in C* is repelling. On the other hand, over a p-adic field with p f d, the multiplier d" is a unit, so the periodic points are indifferent. And if p I d, then all of the periodic points are attracting. Of course, there are also the two superattracting fixed points 0 and 00. It is not hard to find all rational maps that commute with the power maps. In particular, we can compute the automorphism group Aut (Md)'
Proposition 6.3. Let K be a fi eld and let .ft!Id(z ) = zd be the d lh -power map for some Idl 2: 2. Further, if K has finit e characteristic p, assume that p f d. (a) The set ofrational maps that commute with Md(z) is given by
{¢(z ) E K (z ) : ¢ 0 Md = Md 0 ¢ } = {cz e : c E J.Ld-1 and e E
Z}.
6.1. PowerMaps and the Multiplicative Group
327
(b) The automorphism group ofM d is
Aut (Md) = {az : a E ltd-I} U {bz- I : bE Itd-d , where we recall that the automorphism group Aut (¢) ofany rational map ¢ is the set of f E PGL 2 (K ) satisfying ¢f = ¢. In particular, Aut (Md) is a dihedral group oforder 2n , where n is the number of (d - 1)51roots ofunity in K*. Proof (a) It is clear by a direct computation that the indicated maps cz" commute with Md(z ), so it suffices to prove that they are the only commuting maps. Suppose that ¢(z) E K (z ) commutes with Md(Z) , so
¢(zd) = ¢(z) d. Let ( E K be a primitive cfh root of unity. (This is where we use the assumption that p f d if K has positive characteristic p.) Then ¢((z)d = ¢(z)d, so for some cfh root of unity ( k .
¢( (z) = (k ¢ ( z)
Consider the function 'l/J (z) = z- k¢(z). It satisfies
'l/J((z)
= ((z)-k¢((z) = (( z)- k(k ¢(z) = z-k¢(z) = 'l/J(z) .
Hence 'l/J is a function of zd. In other words, there is a rational function 'l/JI(z) E K (z) such that 'l/J(z) = 'l/JI (zd), and thus ¢ (z) = zk'l/JI (zd). n More generally, for any n 2: 1 we have ¢ 0 j\;f :l = M:l 0 ¢ , so ¢( zd ) = n ¢(z) d . The above argument then yields an integer 0 :S kn < dn and a rational function 'l/Jn(z) E K (z) such that n ¢(z) = zkn'l/Jn(Zd ). We write 'l/Jn(z) = zj An(Z) for some integer j and some An(Z) E K (z) satisfying An(O ) =f. 0 and An(O ) =f. 00. Then deg(¢ (z))
n
= deg(zkn'l/Jn(zd
))
n n n = deg(zkn+j d An(zd )) ;::: deg(An( Zd
) )
= d" deg(An(z)).
Hence deg(A n) = 0 for sufficiently large n, which proves that ¢(z) has the form ¢(z) = cz" for some c E K* and some e E Z. With this information in hand, it remains only to observe that ¢(zd ) = ¢(z)d if and only if c = cd, i.e., if and only if either c = 0 or c E ltd- I . (b) By definition, Aut(¢ ) is the set of rational maps of degree 1 that commute with ¢ . It follows from (a) that Aut (¢ ) is the set of all CZ± I with c E ltd-I ' In particular, if we let fa(z) = az and g(z ) = Z- I , then
Aut (lVfd ) = {fa : a E Itd-d U {fa 0 9 : a E
ltd-I} '
and the dihedral nature of the group law is evident from the identities f a 0 f b = f ab ,
fa
0
9= 9
0
f a-I.
o
6. Dynamics Associated to Algebraic Groups
328
Example 6.4. We can use the map Md(z) = zd to illustrate the construction of dynamical units in Section 3.11. First we use Theorem 3.66, which says that if 0: has exact order n and gcd(i - j , n ) = 1, then 0:
di
- 0:
dj
d
0:
Taking j
. IS
- 0:
.
a umt.
= 0 and gcd( i , n) = 1, this implies that
:d-l di_ l
1
~1
is a unit for all primitive (dn - 1)st roots of unity
0:.
These are examples of classical cyclotomic units. Similarly, let m and n be positive integers with m n and n m . Then Theorem 3.68 says that if 0: is a primitive (dm - L)" root of unity and if (3 is a primitive (dn - 1)st root of unity, then 0: - (3 is a unit. These are again classical examples of cyclotomic units.
t
t
Example 6.5. Recall that a nontrivial twist of a rational map ¢( z) E K (z ) is a rational map 'ljJ(z ) E K (z) such that 'ljJ(z ) is PGL 2 (K )-conjugate to ¢( z), but 'ljJ(z ) is not PGL 2 (K )-conjugate to ¢(z ). See Section 4.9 for the general theory. Since the power maps Md(z ) = zd have a large automorphism group, they tend to have many twists. For example , the twists associated to the subgroup of Aut (¢ ) are given by (cf. Example 4.81)
K *j (K *)d- l
---+
{cz : c E J.Ld-l}
Twist (Md),
There are also some rather complicated-looking twists associated to the subgroup { Z , Z -l} C
Aut (Md ) .
Each bE K *j (K *)2 leads to a twist
O<~/2 (2~)b'22k /
Ck~ 1)b.z2k+
1
09 <'f-I)/2
•
(6.1)
See Example 4.82 for the derivation of this formula.
6.2 Chebyshev Polynomials The multiplicative group G m has a nontrivial automorphism given by inversion z 1---+ z- l , and the quotient of G m by this automorphism is isomorphic to the affine line A 1 via the map Z ~Z +Z - l.
The inversion automorphism commutes with the ~h_power map Md(z) = zd, so when we take the quotient of G m , we find that Md(Z) descends to give a map on the quotient space A I . This leads to the following definition (cf. Section 1.6.2).
6.2. Chebyshev Polynomials
329
Definition. The rfh Chebyshev polynomial is the polynomial Td(W) E Z [w] satisfying the identity in the field Q(z).
(6.2)
Of course, we need to show that that Td exists. In the next proposition we prove the existence of the Chebyshev polynomials and describe some of their algebraic properties. Proposition 6.6. For each integer d ;::: 0 there exists a unique polynomial T d(w) E
Q [w ]satisfying in thejield Q(z ).
(6.3)
We call T d the dth Chebyshev polynomial. (a) Td(W) is a monicpolynomial ofdegree d in Z[w]. (b) Td(Te(w)) = Tde(w)forall d. e > O. (c) Td(- w ) = (-l)dTd(w ). Thus Td is an oddfunction ifd is odd and it is an even function ifd is even. (d) The Chebyshev polynomials satisfy the recurrence relation
(e) For all d ;::: 1, the dth Chebyshev polynomial is given by the explicitformula
'~ "
T d ( w) =
( -1 )
d (d -k k)wd-2k.
kd _ k
(6.5)
0'5,k'5,d j2 Proof Suppose first that there do exist polynomials Td(w) satisfying (6.3). Then To(z + Z- I) = ZO+ z - o = 2, T 1 (z + Z- I ) = Z + Z- I, T 2(z
+ Z- I) = z2 + z-2 = (z + Z- I )2 -
2,
so we see that To(w) = 2, T 1 (w) = w, and T2(w) = w 2 - 2 are uniquely determined monic polynomials with integer coefficients. Still assuming that the set of Chebyshev polynomials exists, we next observe that they satisfy
(z
+ Z- I )T d+1 (z + Z- I) -
Td(Z + z - l )
= (z + z - l )(zd+l + z - d- l)
_
(zd + z -d)
+ z - d- 2 T d +2 ( Z + z - I ).
= z d+l =
Putting z + Z- 1 = w, this mean s that T d+2(W) = wTd+l(w) - Td(W). Hence if the Chebyshev polynomials exist, they are unique, because they are completely determined by the recurrence
330
6. Dynamics Associated to Algebraic Groups
To(w) =2,
T1(w)=w,
ford~O.
Td+2(W)=wTd+l(W)-Td(W)
and
(6.6) And from this recurrence we see immediately by induction that Td(w) is a monic polynomial of degree d in Z[w]. We now tum this argument around and use the recurrence (6.6) to define a sequence of polynomials Td(w). We claim that Td(w) then satisfies (6.3). This is clear for To and T 1, so we assume that (6.3) is true for To,T 1, ... ,Td+l and use (6.6) with w = z + z-l to compute
Td+ 2(Z + Z-l) = (z + Z-l )Td+l (z + z-l) - Td(Z + Z-l) = (z + Z-l)(zd+l + z-d-l) _ (zd + z-d) = zd+2 + z-d-2. Hence (6.3) is true for Td+2, so by induction we have for all d
~
O.
This proves that the recurrence defines polynomials satisfying (6.3), so Chebyshev polynomials of every degree exist. We have now shown that Chebyshev polynomials exist, are unique, and have the properties in (a) and (d). Next we make repeated use of (6.3) to compute
Td{Te(z+Z-l))
=
Td(ze+Z-e)
=
(ze)d+(ze)-d
=
zde+z-de
=
Tde(z+Z-l).
Hence Td{Te(w)) = Tde(W), which proves (b). To prove (c), we replace z by -z in (6.3) to obtain
Td(-(Z+Z-l)) =Td( -z+(-Z)-l) = (_z)d
+ (_z)-d = (_l)d(zd + z-d) = (-l)dTd(z + Z-l).
Therefore Td(-w) = (-l)dTd(w), which is (c). Next we prove the explicit summation formula (6.5) given in (e). Substituting d = 1 and d = 2 into the formula yields the correct values T 1 ( w) = wand T2(w) = w2 - 2. We now assume that the formula is correct up to Td+ 1 and use the recurrence (6.4) to check it for d + 2. Thus
Td+2(W)
=
wTd+l(w) - Td(w)
=w
""'
r: O~k~(d+l)/2
d+l-k
(d+l-k) wd+l- 2k k
d_(d k
'r:
(_l)k d+l
""' (_l)k_ s: d-k
O~k~d/2
""'
(_l)k d+l
s:
d+l-k
O~k~(d+l)/2
_
""' L...i
(_l)k-l
1~k~d/2+l
(d+l-k)wd+2-2k k
d (d-k+l)wd-2k+2 d-k+l k-l
6.2. Chebyshev Polynomials
T2 = W 2 T3 = w
3
T4 = w
4
T5 = w 5
331
-
2
-
3w
-
4w 2 + 2
-
5w3 + 5w
+ 9w2 - 2 T7 = w 7 - 7w 5 + l4w 3 - 7w T s = w S - 8w6 + 20w4 - l6w 2 + 2 T 9 = w 9 - 9w 7 + 27w5 - 30w3 + 9w T lO = w lO - lOws + 35w6 - 50w4 + 25w2 T ll = Wll - llw 9 + 44w 7 - 77w 5 + 55w3 T6 = w 6
T 12 =
W
6w 4
-
12
-
l2w
10
+ 54w
S
-
112w
6
+ l05w
2
llw 4
-
36w2 + 2
Table 6.1: The first few Chebyshev polynomials.
~ ~
(d + l) (d + 1 - k) + d (d + 1 - k) k
(_l)k
k- 1
d+l-k
O:s;k:S;d/2+l
wd+2-2k '
where we use the standard convention that (;:,) = 0 if n < m or if m < O. A simple algebraic calculation that we leave for the reader (Exercise 6.4) shows that
(d + l)
(d+ k1 - k) + d (d +k _1 -1 k)
=
d+l-k
d + 2 (d + 2 - k) . d+2-k k
Hence
T
() =
d+2 W
~ ~
O:s;k:S;(d+2)/2 which completes the proof of (e).
(_l)k
d
+2 d+2_ k
(d + k2 - k) w d+2-2k , o
Remark 6.7. As mentioned in Section 1.6.2, the classical normalization for the Chebyshev polynomials is
or equivalently,
332
6. Dynamics Associated to Algebraic Groups
Td ( cos B) =
cos(dB)
for all B E R
The two normalizations are related by the simple formula i; (w) = ~ Td(2w ). We have chosen the alternative normalizationbecause it has better arithmeticproperties. In particular, the map Td : pI -> pI has good reduction at all primes. The classical normalization i; has bad reduction at 2. As with the power maps, it is not difficult to describe the periodic points of the Chebyshev polynomials and to compute their multipliers. We state the result and leave the computationas an exercise.
Proposition 6.8. Let T d(w) be the dth Chebyshev polynomial for some d 2: 2. (a) Thefixedpoints ofTd in AI(C) are
27rj ) d+ 1} { ( 27rj ) { 2 cos ( d + 1 : 0 S; j S; -2- U 2 cos d _ 1 : 0 < j
d - 1} . < -2-
(b) The multipliers ofTd at itsfixed points are given by
(2 (}:jl)) =-d AT (2 (d2: d
ATd
cos
d
cos
j
1) )
=
d+l
forO
< j < -2-'
forO
< j < -2-'
d-l
ATd(±2) = d2 . (Note that -2 E Fix(Td) if and only if d is odd.) In general, the periodic points and multipliers ofTd can be derived from the above formulas using T d = Tdn andPern(Td) = Fix(Td).
o
Proof See Exercise 6.5.
We now prove an analogue of Proposition6.3 for Chebyshevpolynomials.
Theorem 6.9. Let K be afield and let Td(W) be the dth Chebyshev polynomial for some d 2: 2. Further, if K has finite characteristic p, assume that p f d. (a) The automorphism group ofTd is given by
Aut(T = d)
{I
J.L2
if d is even, if d is odd.
(b) Assume that K does not have characteristic 2. Let ¢( w) E K (w) be a rational map that commutes with Td(w), i.e., ¢(Td(w)) = Td(¢(w)). Then ¢(w)
= ±Te(w)
The minus sign is allowed stronger result.)
for some e 2: 1.
if and only if d
is odd. (See Theorem 6.79 for a
6.2. Chebyshev Polynomials
333
Proof (a) The assertion that Aut(Td ) C JL2 is an immediate consequence of (b), since (b) implies that any f E Aut(Td) satisfies f (w) = ±T1 (w) = ±w. However, since the proof of (b) is somewhat intricate, we give a direct and elementary proof of (a). Suppose that f E PGL 2 (K ) satisfies Tl = Td . The polynomial Td has a unique totally ramified fixed point at 00 (cf. Exercise 6.8), and Tl similarly has a unique totally ramified fixed point at f- 1(00 ), so the equality Tl = Td tells us that f- I (00) = 00. Hence f (w) = aw + b is an affine transformation. (The same argument applies to any polynomial not of the form awd .) Proposition 6.6(c) says that Td(w) satisfies Td(- w) = (- I )dTd(w), so in particular, T d ( w) = wd + (terms of degree at most d - 2). (6.7) The identity Tl (w)
= Td (w) with f(w) = aw + b can be written as
We evaluate both sides using (6.7) and look at the top degree terms. This gives
adwd + dad- Ibw d- 1 + (terms of degree at most d - 2) = awd
+ (terms of degree at most d -
2).
Hence and By assumption, d¥-O in the field K , so we conclude that ad- 1 = 1 and b = O. In order to pin down the value of a, we use the explicit formula for Td(w) given in Proposition 6.6(e). In fact, we need only the top two terms,
Td(W ) = wd - dwd- 2 + (terms of degree at most d - 4). By assumption we have Td(aw) = aTd(W ), so
Hence ad = a and ad- 2 = a, where we again use the assumption that d¥-O in the field K. It follows that a2 = 1. Further, a = -1 is possible only if (_I)d = -1, so when d is odd. This completes the proof that Aut(Td) is trivial if d is even and is equal to JL2 if d is odd. (b) It is easy to verify that the Chebyshev polynomial Td(W ) cannot be conjugated to a polynomial of the form cw d (Exercise 6.8). It follows that any rational map commuting with Td(w) is necessarily a polynomial, a fact whose proofwe defer until later in this chapter; see Theorem 6.80. We now describe a proof due to Bertram [69] that the only polynomials commuting with T d are ±Te . We begin with two lemmas. The first characterizes the Chebyshev polynomials as the solutions of a nonlinear differential equation, and the second explains how to exploit such equations.
6. Dynamics Associated to Algebraic Groups
334
Lemma 6.10. Assume that K does not have characteristic 2. Let d 2 1 and let F (w) be a polynomialsolution to the differentialequation (6.8)
Then F (w) = ±Td(W). Proof We first check that ±Td(w) are solutions. We differentiate the functional equation (6.2) defining the Chebyshev polynomials to obtain the identity T~ ( z
+ Z- I )(1 -
Z- 2) = dzd-I - dz-d- I,
and then solve for T~,
Putting w
= z + Z-I
as usual and noting that w 2
-
4 = (z d
Z- 1) 2 ,
we compute
_d) 2
(4 - w2)T~(wf = (4 - (z + Z- 1)2)d2 ( Zz ~ ZZ_ l = _d2(zd _ z- d)2 = d2(4_ (zd + z - d)2)
2(4- Td(w f) ·
= d
This proves that ±Td(W) are solutions to (6.8). Next suppose that F (w) is any polynomial solution to (6.8). If F' (w) is identically 0, then (6.8) implies that F (w) = ± 2 = ±To(w), so we are done. We may thus assume that F' (w) f:. O. We differentiate both sides of (6.8) and divide by 2F' (w) to obtain (6.9) (4 - w 2)F"(w) - wF' (w) + d2F (w) = O.
In particular, Td(w) is a solution to (6.9). Suppose now that F is any polynomial of degree k that is a solution to (6.9). We write F (w) = aw k + bw k - I +... with a f:. 0 and substitute into (6.9). The leading term is
so we must have k = d. In other words, we have shown that every nonzero polynomial solution of (6.9) has degree d. But F(w) - aTd(W) is a polynomial of degree strictly less than d that satisfies (6.9); hence F (w) = aTd(w). Finally, substituting w = 2 into (6.8) yields F (2) = ±2, while we know that Td (2) = Td(1 + 1- 1) = I d + I d = 2. Hence F (w) = ±Td(W) , which completes the proof of Lemma 6.10. 0
Lemma6.11. Let A(w) be a polynomial ofdegree r 2 1 and suppose that F (w) is a polynomial ofdegree d 2 2 satisfying
A (w)F' (wr = dr A(F(w)).
(6.10)
335
6.2. Chebyshev Polynomials
Suppose further that G (w) is a polynomial ofdegree e 2: 0 that commutes with F , i.e., F (G(w )) = G(F(w)). Then
A(w)G'(wY = eT A(G(w)). Proof Consider the polynomial
B (w) = A(w)G'(wY - e"A (G(w)). We assume that B (w) i- 0 and derive a contradiction, which will prove the desired result. First we observe that the leading coefficients of A(w )G' (w r and e"A (G(w)) cancel , so (strict inequality). degB < r e Next we use the various definitions and given relations to compute
dTB(F(w)) = dTA(F(w))G'(F(w)f - dT eTA(G(F(w))) = dTA(F(w))G'(F(w)f - dTe TA (F(G(w)))
definition of B, using F oG
= G 0 F,
= A(w )F' (wYG'(F (w)f - e"A(G(w)) F'(G(w)f
using (6.10) twice,
= A(w)( G 0 F )'(wy - e"A( G(w)) F' (G(w)f
chain rule,
G)'(wy - eTA(G(w)) F'( G(w)f = A(w )F' (G(w)f G' (wY - eTA (G(w))F'(G(w)f
using F o G
= A (w)(F
0
= G o F,
chain rule,
= F'(G (w)f [A(w)G'(wr - e"A (G(w))] = F'( G(w)f B (w )
definition of B.
Taking degrees of both sides gives
(deg B)(deg F) = r( deg F - l)(degG) + (deg B) , and the assumption that deg F
2: 2 means that we can solve for
deg B = r (deg G) = reo This contradicts the earlier strict inequality deg B < reoHence B must be the zero polynomial, which completes the proof of Lemma 6.11. 0 We now resume the proof of Theorem 6.9(b). Let ¢(w) be as in the statement (c) with ¢(w) a polynomial and let e = deg(¢). Lemma 6.10 tells us that
(4 - w2) T~(w)2 = d2(4 - Td(w)2). Hence we can apply Lemma 6.11 with A(w) mials ¢ and Td to deduce that
= 4 - w2 and the commuting polyno-
Then another application of Lemma 6.10 implies that ¢( w) = ±Te (w).
0
336
6. Dynamics Associated to Algebraic Groups
Using Theorem 6.9, it is easy to describe all of the twists of the Chebyshev polynomials. Corollary 6.12. Continuing with the notation and assumptions from Theorem 6.9, if d is even, then Td has no nontrivial K I K -twists, and if d is odd, then each a E K * y ields a twist 1 Td,a(W) = ..;aTd( yaw) . Two such twists Td,a and Td,b are K -conjugate
if and only if alb
is a square in K *.
Proof We use the description of Aut (¢ ) from (a). If d is even, then the automorphism group Aut( ¢ ) is trivial, so Proposition 4.73 says that ¢ has no nontrivial twists. For odd d we have Aut (¢) = {±z}, so the desired result follows from Example 4.81 (see also Example 4.75) . D Remark 6.13. Over C, there is a short proof that Aut(Td ) C /L2 using the fact (Exercise 1.31) that the Julia set of Td is J(Td) = [-2,2]. Then the assumption that Tl = Td implies that f maps the interval [- 2, 2] to itself. Since f is bijective on pI (C), it follows in particular that f permutes the endpoints of the interval [-2,2] . Hence f (2) = ±2 and f ( -2) = =f2. Writing f as f (w ) = aw + b, this gives two equations to solve for a and b, yielding b = 0 and a = ±1. Note that this proof does not carry over to characteristic p, since, for example , working over IFp we have Aut(Tp ) = PGL 2 (lFp ) ; see Exercise 6.10.
6.3 A Primer on Elliptic Curves The remainder ofthis chapter is devoted to rational maps associated to elliptic curves . In this section we give some basic definitions and review, without proof, some of the properties of elliptic curves that will be needed later. The reader should also review the summary of elliptic curves over C given in Section 1.6.3. For further reading on elliptic curves and for the proofs omitted in this section, see for example [96, 198, 248,250,254,257,410,412,420].
6.3.1 Elliptic Curves and Weierstrass Equations Definition. An elliptic curve E over a field K (of characteristic different from 2 and 3) is described by a Weierstrass equation, which is an equation of the form 2 E : y = x 3 + ax
+b
(6.11)
with a, b E K and 4a3 + 27b2 =1= O. Of course, we really mean that E is the projective curve obtained by homogenizing equation (6.11), so E has one extra point "at infinity," which we denote by O . If K has characteristic 2 or 3, then equations of the form (6.11) are insufficient, and indeed they are always singular in characteristic 2, so one uses the generalized Weierstrass equation E: y2
+ al XY + a3Y =
x
3
+ a2x2 + a4X + a6.
(6.12)
6.3. A Primer on Elliptic Curves
337
Remark 6.14. Let E / K be an elliptic curve defined over a field K. When we write E, we mean the geometric points of E, i.e., the points in E(K) for some chosen algebraic closure of E. Ifwe want to refer to points defined over K, we always explicitly write E(K), and similarly we write E(L) for the points defined over some extension field L of K. More intrinsically, an elliptic curve is a pair (E, 0) consisting of a smooth algebraic curve E of genus 1 and a point 0 E E. For convenience we often call E an elliptic curve, with the understanding that there is a specified point o. We say that E is defined over afield K if the curve E is given by equations with K -coefficients and the point 0 is in E(K). Using the Riemann-Roch theorem, one can prove that every elliptic curve E / K can be embedded in J!D2 by a cubic equation of the form (6.12) with 0 mapping to the point at infinity. (See [410, III §3].) Then, if the characteristic of K is neither 2 nor 3, we can complete the square on the left and the cube on the right to obtain the simpler Weierstrass equation (6.11). In order to simplify our discussion, we will generally make this assumption. The discriminant t::.(E) and j-invariant j(E) of the elliptic curve E given by (6.11) are defined by the formulas
Proposition 6.15. (a) Let a, b E K and let E be the curve given by the Weierstrass equation (6.11). Then E is nonsingular; and thus is an elliptic curve, if and only
if t::.(E) =J O. (b) Two elliptic curves E and E' are isomorphic over Kifand only ifj(E) = j(E'). More precisely, E and E' are isomorphic if and only if there is a u E K* such that a' = u 4a and b' = u 6b
o
Proof See [410, III §1].
6.3.2 Geometry and the Group Law There is a natural group structure on the points of E that may be described as follows. Let L be any line in J!D2. Then counted with appropriate multiplicities, the cubic curve E and the line L intersect at three points, say
EnL = {P,Q,R}, where P, Q, and R need not be distinct. The group law on E is determined by the requirement that the sum of the points P, Q, R be equal to 0,
P+Q+R= O. The point 0 serves as the identity element of the group. The inverse of a point P, which we denote by - P, is the third point on the intersection of E with the line through P and O.
338
6. Dynamics Associated to Algebraic Groups
Theorem 6.16. Let E / K be an elliptic curve defined over a field K. (a) The addition law described above gives E = E(K) the structure ofan abelian group. (b) The group law is algebraic, in the sense that the addition and inversion maps, E xE
(P,Q)>-+P+Q I
E
E
,
P>-+-P )
E,
are morphisms, i.e., are given by everywhere defined rationalfunctions. (c) The subset E (K) consisting ofpoints ofE that are defined over K is a subgroup ofE(K). Proof See [410, m §§2,3].
D
It is not hard to give explicit formulas for the group law on an elliptic curve, as in the following algorithm.
Proposition 6.17. (Elliptic Curve Group Law Algorithm) Let E be an elliptic curve given by a Weierstrass equation Z
E :y = x
3
+ ax + b,
and let PI = (Xl, Yl) and Pz = (xz, yz) be points on E. If Xl = Xz and Yl = -Yz, then PI + Pz = 0. Otherwise, define quantities
A = yz - Yl Xz - Xl'
Xz -
A = 3XI + a 2Yl '
l/
=
-xi
Xl
+ aXl + 2b
--=------2Yl '
Then Y = AX + v is the line through PI and P z , or tangent to E sum of PI and Pz is given by
if Xl =
Xz·
if PI = P z, and the
As a special case, the duplication formulafor P = (x, y) is x([2]P)
=
4 Z z x - 2ax - 8bx a 4x 3 4ax 4b
+
+
+
D
Proof See [410, m.2.3]
6.3.3 Divisors and Divisor Classes Definition. A divisor on E is a formal sum of points D
=
L np(P), PEE
339
6.3. A Primeron EllipticCurves
with np E Z and all but finitely many np = o. The set of divisors under addition forms the divisor group Div(E). The degree of a divisor d is deg(D)
=
L
n.p:
PEE
The degree map deg : Div(E) ----+ Z is a group homomorphism. There is a natural summation map from Div(E) to E defined by sum: Div(E)
----t
E,
L np(P)
f------t
PEE
L [np](P). PEE
(N.B. The two summation signs mean very different things. The first is a formal sum of points in Div(E). The second is a sum using the complicated addition law on E.) The zeros and poles of a rational function f on E define a divisor div(J) =
L ordp(J)(P), PEE
where ordp(J) is the order of zero of fat P if f(P) = 0, and ordp(J) is negative the order of the pole of fat P if f(P) = 00. A divisor of the form div(J) is called a principal divisor. The principal divisors form a subgroup ofDiv(E), and the quotient group is the Picard group Pic( E). Within Pic( E) is the important subgroup Pico (E) generated by divisors of degree o. The next proposition describes the basic properties of divisors on E.
Proposition 6.18. Let E be an elliptic curve. (a) Every principal divisor on E has degree o. (b) A divisor D E Div(E) is principal if and only sum(D) = O. (c) The summation map induces a group isomorphism sum: Pico(E)
----t
if both
deg(D)
E.
Proof See [410, III.3A and 111.3.5].
6.3.4
o and
D
Isogenies, Endomorphisms, and Automorphisms
Definition. An isogeny between two elliptic curves E 1 and E 2 is a surjective morphism 'ljJ : E 1 ----+ E 2 satisfying 'ljJ( 0) = O. (Note that any nonconstant morphism E 1 ----+ E 2 is automatically finite and surjective.) The curves E 1 and E 2 are said to be isogenous if there is an isogeny between them. Remark 6.19. Every nonconstant morphism 'ljJ : E 1 ----+ E 2 is the composition of an isogeny and a translation (cf. [410, IIIA.7]). To see this, let ¢(P) = 'ljJ(P) - 'ljJ(0). Then the map ¢ : E ----+ E is a morphism, and ¢( 0) = 0, so ¢ is an isogeny. Hence
'ljJ(P) = ¢(P) +'ljJ(0) is the composition of an isogeny and a translation.
340
6. Dynamics Associated to Algebraic Groups
Remark 6.20. We observe that an isogeny is unramified at all points. This follows from the general Riemann-Hurwitz formula (Theorem 1.5) applied to the map
'IjJ : E 1 --+ E 2 , 2g(E 1) - 2 = (deg 'IjJ)(2g(E 2 )
-
2) +
I: (ep ('IjJ ) -
1).
P EEl
The elliptic curves E 1 and E 2 both have genus 1, and the ramification indices satisfy ep ('IjJ ) ::::: 1, so it follows that every ep('IjJ) is equal to 1, so 'IjJ is unramified. Theorem 6.21. An isogeny tb : E 1 --+ E 2 is a homomorphism ofgroups, i.e.,
'IjJ (P
+ Q) = 'IjJ (P ) + 'IjJ(Q)
for all P,Q E E 1(K) .
o
Proof See [410, 111.4.8]
The degree of an isogeny 'IjJ : E 1 --+ E 2 is the number of points in the inverse image 'IjJ - 1(Q) for any point Q E E 2 . This number is independent of the point Q, since, as noted earlier, 'IjJ is an unramified map. It is clear that if deg( 'IjJ ) > 1, then 'IjJ is not invertible, since it is not one-to-one. However, there does exist a dual isogeny that provides a kind of "inverse" for 'IjJ. Theorem 6.22. Let 1/J : E 1
isogeny ,¢ : E 2
--+
'¢('IjJ (P )) = [d] P
--+ E 2 be an isogeny ofdegree d. Then thereis a unique E 1 , called the dual isogeny of 'IjJ, with the propertythat
and
'IjJ( '¢(Q )) = [d] Q
for all PEEl andQ E E 2 .
o
Proof See [410, III §6].
Definition. Let E be an elliptic curve. The endomorphism ring of E, which is denoted by End (E), is the set ofisogenies from E to itself with addition and multiplication given by the rules
(In order to make End(E) into a ring, we also include the constant map that sends every point to 0.) The automorphism group of E, denoted by Aut(E), is the set of endomorphisms that have inverses. Equivalently, Aut(E) = End(E)* is the group of units in the ring End(E). Every integer m gives a multipli cation-by-in morphism in End(E). For m this is defined in the natural way as
>0
m tenus
[m] : E
----+
E,
[m] (P) = 'p + P ; . . . + P.
For m < 0 we set [m](P) = - [-m ](P), and of course [O](P) = O. This gives an embedding of Z into End (E ), and for most elliptic curves (in characteristic 0), there are no other endomorphisms.
6.3. A Primer on Elliptic Curves
341
Definition. An elliptic curve E is said to have complex multiplication if End (E) is strictl y larger than Z . The phrase "complex multipli cation" is often abbreviated byCM. Example 6.23. The elliptic curve E : y2 = x 3 + x has Clvl, since the endomorphism .ljJ : E
----1
E,
'ljJ( x, y) = (-x ,iy),
is not in Z . An easy way to verify this assertion is to note that
'ljJ2(X, y) = (x , -y) = - (x , y) , so tjJ2 = [-1]. This gives an embedding of the Gaussian integers Z [i] into End (E ) via the association m + ni f---+ [m] + [n] 0 'ljJ , and in fact it is not hard to show that End(E ) is isomorphic to Z [i].
Example 6.24. More generally, there are two special families of elliptic curves that have CM, namely those with a = 0 and those with b = O. These are the curves
E a' .. Y2 -_ EI: : y2 =
+ ax , 3 x + b, X
3
j (E~ ) = 1728,
End(E~ )
= Z[i],
Aut (E~ ) = JL4 '
j (EI: ) = 0,
End (EI: )
= Z[p],
Au t (EI: ) = JL6 '
Here p = (- 1+A )/2denote s a cube root of unity and JL n is the group of nIh roots of unity. Of course , all of the E~ are isomorphic over an algebrai cally closed field, since they have the same j -invariant, and similarly for all of the E~' . However, the curves in each family may not be isomorphic over a field K that is not algebraically closed . This is an example of the phenomenon of twisting as described in Section 4.8 (see also [410, X §5]).
Proposition 6.25. Let E / K be an ellip tic curve. Then the endomorp hism ring ofE is one ofthe following three kinds ofrings : (a) End(E) = Z. (b) End (E) is an order in a quadratic imaginaryfi eld F. This means that End (E ) is a subring of F and satisfies End (E ) ® iQl = F. In particular, End (E ) is a subring offinite index in the ring ofintegers ofF. (c) End(E) is a maximal order in a quaternion algebra. (This case can occur only if E is defined over a finite fi eld.)
o
Proof See [410, III §9]. The automorphisms of an elliptic curve are very easy to descr ibe.
Proposition 6.26. Let K be a fi eld whose characteristic is not equal to 2 or 3 and let E / K be an elliptic curve. Then
Au t (E)
=
{
JL2
if j (E ) # 0 and j (E ) #
JL4
ifj (E)
= 1728,
JL6
i/j (E )
=
O.
1728,
342
6. Dynamics Associated to Algebraic Groups
o
Proof See [410, III §1O].
Remark 6.27. It is easy to make the description of Aut(E ) in Proposition 6.26 completel y explicit. Assuming that E is given by a Weierstrass equation (6.11) as usual, for an appropriate choice of n there is an isomorphism
[.J: JL n
---+
Au t (E),
(6.13)
Here we take n = 4 if j(E) = 1728, we take n = 6 if j( E) = 0, and we take = 2 otherwise. Of course, for n = 2 and n = 4 the formula simplifies somewhat to (x, ~y) and (x, ~ - ly), respect ively.
n
6.3.5
Minimal Equations and Reduction Modulo p
Let K be a local field with ring of integers R, maximal ideal p, and residue field k RIp . As in Section 2.3, we write for the reduction of x modulo p.
x
=
Definition. Let ElK be an elliptic curve defined over a local field K. A minimal Weierstrass equation / or E is a Weierstrass equation whose discriminant 6.(E ) has minimal valuation subject to the condition that the coefficients of the Weierstrass equation are all in R.
Example 6.28. If k does not have characteristic 2 or 3, then a Weierstrass equation (6.14) for E is minimal if and only if
a,bE R
and
min{30rd p (a ),2 0rd p (b)}
< 12.
In general, if the residue field k does not have characteristic 2 or 3, then any Weierstrass equation (6.14) can be transformed into a minimal equation by a substitution of the form (x, y) I-t (u 2x , u 3 y) for an appropriate u E K *. If k has characteristic 2 or 3, then a minimal Weierstrass equation may require the general form (6.12). There is an algorithm of Tate [412, IV §9] that transforms a given Weierstrass equation into a minimal one. Definition. Fix a minimal Weierstrass equation for ElK. Then we can reduce the coefficients of E to obtain a (possibly singular) curve E lk. We say that E has good reduction if E is nonsingular, which is equivalent to the condition that 6.(E) E R* . In any case, we obtain a reduction modulo p map on points ,
E (K )
---+
E(k),
Pt-------->P.
Proposition 6.29. If E has good reduction, then the reduction modulo p map E (K) -> E (k ) is a homomorphism.
Proof See [410, VU.2.1].
o
6.3. A Primer on Elliptic Curves
343
Remark 6.30. For elliptic curves defined over a number field K, we say that E has good reduction at a prime p of K if it has a Weierstrass equation whose coefficients are p-adic integers and whose discriminant is a p-adic unit. Note that one is allowed to use different Weierstrass equations for different primes. If there is a single Weierstrass equation that is simultaneously minimal for all primes, then we say that E / K has a global minimal Weierstrass equation. Global minimal equations exist for elliptic curves over (Q, and more generally for elliptic curves over any number field of class number 1, but in general the existence of global minimal equations is somewhat subtle; see [410, VIII §8] and [48]. We discussed a related notion of global minimal models of rational maps in Section 4.11.
6.3.6
Torsion Points and Reduction Modulo p
The kernels of endomorphisms help to determine the arithmetic properties of elliptic curves. Definition. Let E be an elliptic curve. For any endomorphism 'l/J E End(E) we write E['l/J] = Ker('l/J) = {p E E : 'l/J(P) = O}. Of particular importance is the kernel of the multiplication-by-m map,
E[m] = {P E E: [m]P = O}. The group E[m] is called the m-torsion subgroup ofE. The union of all E[m] is the torsion subgroup ofE,
E tors
=
U E[m].
m~l
Theorem 6.31. Let E / K be an elliptic curve and assume that either K has characteristic 0 or else that K has characteristic p > 0 and p f m. Then as an abstract group, E[m] = Z/mZ x Z/mZ.
In other words, E [m] is the product oftwo cyclic groups oforder m.
o
Proof See [410, III.6.4].
The next result gives conditions that ensure that the reduction modulo p map respects the m-torsion points. It may be compared with Theorem 2.21, which tells us what reduction modulo p does to periodic points of a good-reduction rational map. Theorem 6.32. Let K be a local field whose residue field has characteristic p, let E / K be an elliptic curve with good reduction, and let m 2: 1 be an integer with p f m. Let E(K)[m] denote the subgroup of E[m] consisting ofpoints defined over K, i.e., E(K)[m] = E[m] n E(K). Then the reduction map
E(K)[m]
-+
E(k)
is injective. In other words, distinct m-torsion points have distinct reductions modulo p.
344
6. Dynamics Associated to Algebraic Groups D
Proof See [410, VII.3.1].
Let E j K be an elliptic curve defined over the field K . Then the points in E [m ] are algebraic over K, so their coordinates generate algebraic extensions of K. An immediate corollary of the preceding theorem limits the possible ramification of these extensions .
Corollary 6.33. Let K be a local fie ld whose residue fi eld has characteristic p. let E j K be an elliptic curve with good reduction, and let m 2 1 be an integer with p m . Then the field K (E [m]) obtained by adjoining to K the coordinates ofthe m-torsion points ofE is unramified over K .
t
ProofSketch. Let K' = K (E [m ]), let p' be the maximal ideal of the ring of integers of K' , and let k' be the residue field. Suppose that CY E Gal( K ' j K ) is in the inertia group . Then CY fixes everything modulo p', so in particular,
CY(P) == P
(mo d p')
for all P E E[m].
(6.15)
But from Theorem 6.32, the reduction map E[m] --+ E (k' ) is injective, so (6.15) implies that CY(P ) = P for all P E E [m ]. The points in E [m ] generate K' j K, so CY fixes K' . Hence Gal (K' j K ) has trivial inertia group , so K ' j K is unramified . (For further details, see [410, VIlA I].) D
Remark 6.34. The coordinates of the points in E[m] are algebraic over K , so the absolute Galois group G K = Gal(K jK ) acts on E [m ] compatibly with the group structure. In this way we obtain a representation
In order to create a characteristic-O representation , we fix a prime f. and combine all of the f.-power torsion to form the Tate module
(Here Ze denotes the ring of f.-adic integers.) Then the f.-adic representation of E is the homomorphism
These representations are of fundamental importance in the study of the arithmetic properties of elliptic curves .
6.3.7
The Invariant Differential
Definition. Let E : y 2 = x 3 + ax + b be an elliptic curve given by a Weierstrass equation . The invariant differential on E (associated to the given Weierstrass equation) is the differential I-form
dx
WE
dy
= -2y = 3x 2 + a .
6.3. A Primer on Elliptic Curves
345
The next result explains why the invariant differential is so named and shows that it linearizes the group law in a useful way. Theorem 6.35. Let E be an elliptic curve given by a Weierstrass equation and let WE be the associated invariant differential on E. (a) For any given point Q E E, let TQ : E ----t E be the translation-by-Q map defined by TQ (P) = P + Q. The differentialform WE is translation-invariant in the sense that for every Q E E. TQ(WE) = WE
(b) The differential form WE is holomorphic at every point of E. (c) Up to multiplication by a nonzero constant, WE is the only holomorphic translation-invariant 1-form on E. (d) For every m E Z the differential form WE satisfies [m]*WE
= mWE·
o
Proof See [410, III §5].
The invariant differential can also be used to fix an embedding of the endomorphism ring of E into Co Of course, this is of interest only when E has Clvl, since there is only one way to embed Z into
R=
{a E
Thenfor each a E R there is a unique endomorphism [a] E End(E) satisfying [a]*(w) = aw,
(6.16)
and this association defines a unique ring isomorphism
[. ] : R
.s:.. End(E).
(Without the normalization (6.16), the isomorphism R ~ End(E) is unique only up to complex conjugation ofR.) Proof We fix the isomorphism E(
F :
----7
E(
F(z) = (gJ(Z),
~gJ'(Z)) ;
see Section 1.6.3 and [410, VI §3]. Then F*(w)
=
F* (dX) 2y
=
dgJ(z) gJ'(z)
= dz.
346
6. Dynamics Associated to Algebraic Groups
Let 'IjJ E End(E), so using the identification E(C) ~ Cj L, the endomorphism 'IjJ defines a holomorphic map 'IjJ : Cj L -+ Cj L satisfying 'IjJ(O) = O. We claim that the analyticity implies that 'IjJ lifts to a map 1[J : C -+ C of the form 'IjJ (z) = o z for a unique a E C, and hence in particular that 'IjJ is a homomorphism. (Compare with the algebraic statement of this fact given in Theorem 6.21.) To prove this, we first observe that the covering map C -+ Cj L is the universal cover of Cj L, so we can lift 'IjJ to some holomorphic map 1[J : C -+ Co Further, the fact that 1[J lifts 'IjJ means that 1[J satisfies
1[J(z + w) - 1[J(z) E L
for all z E C and all w E L.
Fixing W E L, we find that the map z f--+ 1[J(z + w) - 1[J(z) is a holomorphic map from C to the discrete set L, so it must be constant. Thus for each w E L there is a number c(w) E C such that
1[J(z + w) = 1[J(z) + c(w)
for all z E C.
(6.17)
Writing 1[J(z) 2:aizi as a convergent power series, one easily checks that the relation (6.17) forces 1[J to be linear, say 1[J(z) = o:z+ (J. Then the assumption 'IjJ (O) = tells us that {J E L, so o:z and o:z + (J descend to the same map on Cj L. Hence 'IjJ lifts to a map of the form 1[J(z) = oz. Further, a is unique , since if 'IjJ (z) also lifts to a' z, then the map z f--+ (a - a') z sends C to L , hence must be constant, so a = a' . Finally, we observe that in order for az to descend to Cj L , the complex number a must satisfy al. r;;; L . Thus the association 'IjJ f--+ a gives a map End(E ) -+ R, and it is clear that the map 'I/J (z) = o z satisfies 'I/J*(dz) = adz, so with our identifications , we have 'IjJ *(w) = aw. Next we check that the resulting map End (E) -+ R is a ring homomorphism. Let 'l/Jl ' 'l/J2 E End(E) . Then on Cj L we have 'l/Jl (z) = O:I Z and 'l/J2(Z ) = a2Z, so
o
It remains to check that every a E R comes from some 'IjJ E End(E). By definition any a E R induces a map z f--+ o z on C that descends to a holomorphic homomorphism C] L -+ Cj L. Using the theory of elliptic functions (see, e.g., [410, Theorem VI.4.1D, one can show that every such holomorphic map E(C) -+ E(C) is given by rational functions, which shows that End(R) -+ R is surjective. 0
6.3.8 Maps from E to pI The quotient of E by a finite group of automorphisms gives a map from E to pl . These quotient maps play an important role in dynamics.
Proposition 6.37. Let I' be a nontrivial subgroup of Aut (E ). Then the quotient curve E j f is isomorphic to p I and the projection map 1r : E -+ E j f ~ ]P'1 is given explicitly by
347
6.3. A Primer on Elliptic Curves
X
Jr( X Y) ,
=
iff=JL2
(j (E) arbitrary),
if r if f
(j(E) = 1728 only),
= JL4 = JL3 { 3 X if I' = JL6 X2
y
(j(E) (j(E)
= 0 only), = 0 only).
Proof By definition, the quotient curve Elf is the curve whose function field is the subfield of K(E) = K(x, y) fixed by f. Using the explicit description (6.13) of the action of Aut (E) on the coordinates of E, it is easy to find this subfield. For example, iff = JL2' it consists ofthose functions that are invariant under (x, y) f---* (x, -y), so the fixed field K(Et is K(x, y2) = K(x). As a second example, if f = JL6' then we need functions invariant under (x, y) f---* (px, -y), where p is a primitive cube root of 1. This fixed field is K (El = K (x 3 , y2) = K (x 3 ) , since in this case the elliptic curve is given by an equation of the form y2 = x 3 + b. The other cases are similar. 0 For later use, we prove that the isomorphism class of an elliptic curve E is determined by the critical values of any double cover E ---+ JlDI.
Lemma 6.38. Let E be an elliptic curve defined over a field of characteristic not equal to 2 and let Jr : E ---+ JlDI be a rational map of degree 2. Then Jr has exactly four critical values and they determine the isomorphism class of E.
Proof The Riemann-Hurwitz formula (Theorem 1.5) for the map Jr : E that 2g(E) - 2 = (degJr)(2g(JlD I) - 2) + (ep(¢) -1).
---+
JlDI says
L
PEE
The map Jr has degree 2, the elliptic curve E has genus 1, and JlDI has genus 0, so we find that (ep(¢) -1) = 4.
L
PEE
The ramification indices satisfy 1 :S e p ( ¢) :S deg Jr = 2, so we conclude that there are exactly four critical points, i.e., four points with e p (¢) = 2 and all other points satisfy e p ( ¢) = 1. Further, these four critical points must have distinct images in JlDI , since for any point t E JlDI we have
L
ep(¢)
= deg(Jr).
PE7l'-1(t)
Let iI, t2,t3, t4 E JlDI be the four critical values of n, i.e., the images ofthe critical points, and let f E PGL 2 be the unique linear fractional transformation satisfying
Explicitly,
348
6. Dynamics Associated to Algebraic Groups
(If any oft l , t 2 , t 3 equals 00, take the appropriate limit.) The quantity
is called the cross-ratio oft l , t2, iz, i«; cf. Section 2.7, page 71. We let x = f 0 7r, so x is a rational function of degree 2 on E with critical values 0, 1, 00, and 11,. To ease notation, we let
Taking 0 to be the identity element for the group law on E, we see that div(x)
= 2(To)-2(0),
div(x-l)
= 2(Tr)-2(0),
div(x-11,)
= 2(T,,)-2(0),
so To, TI , T" are in E[2], i.e., they are points of order 2. The sum of the three nontrivial 2-torsion points on any elliptic curve is equal to 0, so Proposition 6.18 tells us that there is a rational function y on E with divisor div(y) = (To)
+ (Tr) + (T,,) - 3(0).
After multiplying y by an appropriate constant, it follows that x and yare Weierstrass coordinates for E (cf. [198, Iv'4.6] or [410, 111.3.1]). More precisely, the rational functions x and y map E isomorphically to the curve with Weierstrass equation
E : y2 =
X (x
- 1)(x -
11,).
It is easy to compute the j-invariant of E in terms of 11, (cf. [410, III. 1.7(b)]). We find
that
'(E) = 28 (11,2 J
- 11, 11,2(11, _
+ 1)3 1)2 '
so in particular j(E) is uniquely determined by t l , t2, t3,t4. Then we apply Proposition 6.15 (or [410, IIL1.4(b)]), which says that the isomorphism class of E is detennined by its j-invariant. This completes the proof of Lemma 6.38. 0
6.3.9
Complex Multiplication
Let ElK be an elliptic curve with complex multiplication defined over a number field. As described in Proposition 6.25, the endomorphism ring of E is isomorphic to a subring of the ring of integers of a quadratic imaginary field. We briefly recall an analytic proof of this important fact and then discuss the relationship between complex multiplication and the ideal class group of the associated quadratic imaginary field. This material is used only in Section 6.6, so may be omitted at first reading. Proposition 6.39. Let E IC be an elliptic curve with complex multiplication, i.e., the endomorphism ring End(E) is strictly larger than Z. Choose a lattice L c C such that E(C) ~ CI L, let
349
6.3. A Primer on Elliptic Curves
R = {a E C : O'L ~ L} , and let
[. J: R ~ End (E) be the isomorphism described in Proposition 6.36. Then R is a subring of the ring ofintegers ofa quadratic imaginary fi eld F , andfor any 0' E R . the degree of [a] is given by deg ]o] = 0'& = NF/Q(O'), where & denotes the complex conjugate of
0'.
Proof To describe the ring R, we choose a basis for L , say L = ZWl have R -I Z by assumption, so there exists an a E R with a (j. Z . Write
aWl = aWl The numbers
WI
+ bW2
and
aW2 =
CWl
+ dW2
with
+ ZW2. We
a, b, e, d E Z .
(6.18)
and W2 are JR.-linearly independent, so the relation
(
a - a
-b) (WI) (0)°
-e a - d
W2
-
implies that the matrix has determinant 0,
a2
-
(a + d)a
+ (ad - be) = 0.
(6.19)
Hence a is an algebraic integer in a quadratic field. Further, we must have a (j. JR., since if a were real, then the relation (a - a )WI = bw2(and a (j. Z) would contradict the JR.-linear independence of Wl and W2 . This proves that every element of R is an algebraic integer in a quadratic imaginary field, and hence R is a subring of the ring of integers of such a field. Finall y, in order to compute the degree of the endomorphism [0'] : E ----> E corresponding to a E R, we observe that deg(a)
=#
Ker
(elL ~ elL) = (L : O'L).
If a E Z, then it is clear that (L : a L ) = a 2 , since L = Z 2 as an abstract group. Suppose now that a (j. Z. Then continuing with the earlier notation, the transformation formulas (6.18) imply that the index of eel. in L is (L : a L) = ad - be. On the other hand , the product a& is the constant term in the minimal equation (6.19) for a 0 over Q, hence also equal to ad - be. The theory of complex multiplication uses elliptic curves to describe the abelian extensions of a quadratic imaginary field F in a manner analogous to the description of abelian extensions of Q using torsion points in G m , i.e., using roots of unity. For a complete introduction to the theory of complex multiplication, see, for example, [257, Part II], [399 , Chapter 5], or [412, Chapter II]. We now describe the tiny piece of the theory that will be needed in Section 6.6 in order to prove Theorem 6.62.
6. Dynamics Associated to Algebraic Groups
350
Let F be a quadratic imaginary field, let RF be the ring of integers of F, and let IF be the group of fractional ideals of F. If we fix an embedding FcC, then each fractional ideal a E IF is a lattice a C C; hence it determines an elliptic curve En whose complex points are (6.20) We observe that En has complex multiplication by RF, since any a E R F has the property that ou C a, hence it induces a holomorphic map
[a] : Cia
1----+
CI a,
Z 1----+
az,
which in turn yields an isogeny [a] : En ----t En. In fact, since R F is the maximal order in F, we have End(En) = R F . We denote by £e£(R F ) the set
£e£(R F ) = {isomorphism classes of elliptic curves E with End(E) ~ R F
} .
We thus have a natural map
a 1----+ (isomorphism class of En).
(6.21)
We also observe that if we multiply a by a principal ideal, then the isomorphism class of En does not change, since for any e E K* there is an obvious isomorphism
Cia ~ C/ea,
Z
1----+
ez.
Hence the map (6.21) induces a natural map from the ideal class group CF to elliptic curves with complex multiplication by RF,
= IF I F*
Proposition 6.40. Let F be a quadratic imaginary field with ring of integers RF and ideal class group CF, and let h F = #C F be the class number ofF. Then with notation as above. the natural map
is a bijection. In particular, there are exactly h F isomorphism classes of elliptic curves whose endomorphism ring is RF.
Proof See [412, 11.1.2].
D
6.4 General Properties of Lattes Maps Chebyshev polynomials arise by restricting the power map z" to the quotient of JlDl by the finite group of automorphisms {z, z-l }. As already briefly described in Section 1.6.3, quotients of elliptic curves lead similarly to rational maps called Lattes maps. In this section we define and discuss general properties of these Lattes maps. For an excellent introduction to Lattes maps over C, including historical remarks and proofs of their basic geometric and analytic properties, see [300].
351
6.4. GeneralProperties of Lattes Maps
Definition. A rational map ¢ : pI ~ pI of degree d 2: 2 is called a Lattes map if there are an elliptic curve E, a morphism 7/J : E ~ E, and a finite separable' covering 7r : E ~ pI such that the following diagram is commutative:
(6.22)
pI
..«: pl.
Example 6.41. Let E : y2 = x 3 + ax + b be an elliptic curve. Then the classical formula for x(2P) (Proposition 6.17) and the isomorphism x : E / {±l} ~ pI yield the Lattes map
2 2 2ax - 8bx + a 4x 3 + 4ax + 4b Here 7/J is the duplication map 7/J(P) = [2]P, and the projection 7r is given by 7r(P) = 7r(x, y) = x. Example 6.42. Let E be the elliptic curve E : y2 = x 3 + ax with j (E) = 1728 and again let 7/J(P) = [2]P be the doubling map. Ifwe take 7r(x, y) = x, then we are in ¢(x)
= x(2P) = x
4
-
the b = 0 case of Example 6.41, and we obtain the Lattes map
¢(x)
x(2P)
=
=
(x 2 _ a)2 4 (2 )' X x +a
However, for this curve we may instead take 7r (x, y) map ¢1. We find a formula for ¢1 using the relation 2
¢l(X) =¢(JX) =
((x_a)2)2 4)X(x+a)
= x 2 . This gives a new Lattes
(x - a)4 16x(x + a)2'
Note that the map 7r(x, y) = x 2 corresponds to taking the quotient of E by its automorphism group Aut(E) ~ 1L4 via the association described in Remark 6.27.
Example 6.43. In a similar manner, the doubling map on the elliptic curve
E: y2
= x3 + 1
with j(E) = 0 and Aut(E) = 1L6 gives various Lattes maps corresponding to taking the quotient of E by the different subgroups of 1L6' Explicitly, the Lattes maps corresponding, respectively, to 1L2' 1L3' and 1L6 are
We leave the verification of these formulas to the reader; see Exercise 6.12. 1The assumption that 1r is separable is relevant only when one is working over a field of characteristic p, in which case it is equivalent to the assumption that 1r does not factor through the p-power Frobenius map.
352
6. Dynamics Associated to Algebraic Groups
We begin with an elementary, but useful, characterization of the preperiodic points of a Lattes map (cf. Proposition 1.42). Proposition 6.44. Let ¢ be a Lattes map associated to an elliptic curve E. Then PrePer(¢)
= Jr(Etors).
Proof Let ( E pi and let PEE be any point satisfying Jr(P) orbits of ( and P. Thus
= (. We consider the
Jr(01jJ(P)) = Jr ((~n(p): n ~ a}) = {Jr~n(p): n ~ o} =
{¢nJr(p): n ~ o} = {¢n((): n ~ o}
=
Oq,(().
The map Jr is finite, so this shows that 01jJ(P) is finite if and only if Oq,(() is finite. Hence PrePer(¢) = Jr(PrePer(~)), and it is left to prove that PrePer( ~) = Etors. We observe that the map ~ : E ~ E has the form ~(P) = ~o(P) + T for some ~o E End(E) and some point TEE. (See [410, III.4.7].) We are going to prove Proposition 6.44 in the case that ~ = ~o E End(E), i.e., assuming that T = O. For the general case, which requires knowing that the point T is a point of finite order, see Exercise 6.14. Suppose first that P E Etors, say [n]P = 0 for some n ~ 1. Consider the images of the iterates ~, ~2 , ~3 , ... in the quotient ring End (E) / n End (E). It follows from the description of End( E) in Proposition 6.25 that this quotient ring is finite, so we can find iterates i > j ~ 1 such that ~i
==
~j
(mod nEnd(E)).
In other words, there is an endomorphism f3 E End(E) such that ~i
= ~j + f3n.
Evaluating both sides at P and using the fact that [n]P = 0 allows us to conclude that ~i(p) = ~j (P). Hence P E PrePer( ~), which proves that Etors C PrePer( ~). Next suppose that P E PrePer(~), say ~i(p) = ~j(P) for some i > j. We rewrite this as (~i - ~j) (P) Theorem 6.22 to obtain
=0
-----
and apply the dual isogeny ~i - ~j described in
[deg(~i - ~j)](P)
= O.
We know that ~i -I ~j, since deg(~) = deg(¢) ~ 2 and i > j, so ~i - ~j has positive degree. This proves that P E E tors, which gives the other inclusion PrePer(~) C Etors. 0 Many dynamical properties of a rational map can be analyzed by studying the behavior of the critical points under iteration of the map. This is certainly true for Lattes maps, whose postcritical orbits have a simple characterization, which we give after setting some notation.
6.4. General Properties of Lattes Maps
353
Definition. Let = {P E C I
:
:
ep(
The set ofcritical values of
00
n =O
n =I
U ) = U CritVal1>n .
PostCrit1> = (See Exercise 6.15.)
Proposition 6.45. Let
----7
pI be a Lattes map that fits into a commutative
CritVal1r
=
PostCrit .
In particular, a Lattes map is postcritically finite. Proof The key to the proof of this proposition is the fact that the map 'Ij; : E ----7 E is unramified , i.e., it has no critical points, see Remark 6.20. (In the language of modem algebraic geometry, the map 'Ij; is etale.) More precisely, the map 'Ij; is the composition of an endomorphism of E and a translation (Remark 6.19), both of which are unramified. For any n 2: 1 we compute
Critval, = CritVal1r ,pn
because 'Ij; is unramified ,
= CritVal1>n
from the commutativity of(6.22),
1r
= CritVal1>n U
1r )
from the definition of critical value,
:2 CritVal1>n . This holds for all n 2: 1, which gives the inclusion 00
CritVal1r
:2
U ) = PostCrit1> . n=O
In order to prove the opposite inclusion , suppose that there exists a point Po E E satisfying Po E CritPt" and 7f( Po) t/: PostCrit1> . (6.23) Consider any point Q E 7j; - I (PO) . Then Q is a critical point of 7f'lj;, since 'Ij; is unramified and 7f is ramified at 7j; (Q) by assumption. But 7f'lj; =
354
6. Dynamics Associated to Algebraic Groups
On the other hand,
so n( Q) is not a critical point for ¢. It follows that Q is a critical point of tt, Further, we claim that no iterate of ¢ is ramified at n (Q). To see this, we use the given fact that n(Po) is not in the postcritical set of ¢ to compute
n(Po) ~ Postcrlt;
===} ===} ===} ===} ===}
n(Po) ~ ¢n(CritPtq,) n(1/J(Q)) ~ ¢n(CritPtq,) ¢(n(Q)) ~ ¢n(CritPtq,) n(Q) ~ ¢n(CritPtq,) n( Q) ~ PostCritq, .
To recapitulate, we have now proven that every Q E
Q E CritPt1r
1/J-l (Po)
for all n
~
1,
for all n
~
1,
for all n
~
1,
for all n
~
0,
satisfies
n( Q) ~ PostCrit,p .
and
In other words, every point Q E 1/J-l (Po) satisfies the same two conditions (6.23) that are satisfied by Po. Hence by induction we find that if there is any point Po satisfying (6.23), then the full backward orbit of 1/J is contained in the set of critical points of n, i.e., 00
CritPt1r =>
U 1/J -n (Po). n=l
But
1/J is unramified and has degree at least 2 (note that deg 1/J =
deg ¢), so
This is a contradiction, since n has only finitely many critical points, so we conclude that there are no points Po satisfying (6.23). Hence
which gives the other inclusion Critval, ~ PostCritq,.
o
As an application of Proposition 6.45, we show that Lattes maps associated to distinct elliptic curves are not conjugate to one another. Theorem 6.46. Let K be an algebraically closed field of characteristic not equal to 2 and let ¢ and ¢' be Lattes maps defined over K that are associated, respectively, to elliptic curves E and E'. Assume further that the projection maps nand n' associated to ¢ and ¢' both have degree 2. If ¢ and ¢' are PGL 2 (K)-conjugate to one another, then E and E' are isomorphic.
355
6.5. Flexible Lattes Maps
Proof Let f E PGL 2 (K ) be a linear fractional transformat ion conjugating ¢' to ¢. Then we have a commutative diagram 1r'
. - - E'
We let n" = f 0 n' , Note that since f is an isomorphism, the map degree 2. This yields the simplified commutative diagram
tt"
still has
showing that ¢ is a Lattes map associated to both elliptic curves E and E' . Applying Proposition 6.45, first to E and then to E', we find that
CritVal1r
= Postcrit, = CritVal
1r "
•
(6.24)
In other words, the degree-2 maps 1r : E --t pI and n" : E' --t p I have the exact same set of critical values. Then Lemma 6.38 tells us that E and E' are isomorphic .
o 6.5 Flexible Lattes Maps A Lattes map is a rational map that is obtained by projecting an elliptic curve endomorphism down to pi . For any integer m ::::: 2, every elliptic curve has a multiplication-by-m map and a projection E --t E / {± 1} ~ pI , so every elliptic curve has a corresponding Lattes map. As E varies, this collection of Lattes maps varies continuously, which prompts the following definition.
Definition. Aflexible Lattes map is a Lattes map ¢ : p I --t pI that fits into a Lattes commutative diagram (6.22) in which the map 'l/J : E --t E has the form 'l/J (P ) = [m](P)
+T
for some m E Z and some TEE
and such that the projection map 1r : E
deg(1r) = 2
and
--t
p I satisfies
1r(P ) = 1r(- P ) for all PEE.
Remark 6.47. The condition that 1r be even, i.e., that it satisfy 1r(- P ) = 1r(P ), is included for convenience. In general, if 1r : E --t p I is any map of degree 2, then there exists a point Po E E such that 1r ( - (P + Po)) = 1r(P + Po) for all PEE. Thus 1r becomes an even function if we use Po as the identity element for the group law on E. See Exercise 6.16.
356
6. Dynamics Associated to Algebraic Groups
Remark 6.48. We show in this section that the Lanes maps of a given degree have
identicalmultiplier spectra. This is one reason that these Lattes maps are called "flexible," since they vary in continuous familieswhose periodic points have identical sets of multipliers. We saw in Section 4.5 that symmetricpolynomials in the multipliers give rational functions on the moduli space M d of rational maps modulo PGL 2 conjugation. Flexible families of rational maps thus cannot be distinguished from one another in M d solely through the values of their multipliers. Example 6.49. We saw in Example 6.41 that the Lattes function associated to the
duplication map 7jJ(P) = [2] (P) on the elliptic curve E : y2 = x 3 by the formula
+ ax + b is given
4 2 2 cPa b(X) = x(2P) = x - 2ax - 8bx + a , 4x 3 + 4ax + 4b It is clear that if a and b vary continuously, subjectto 4a3
+ 27b2 i=- 0, then the Lattes
maps cPa,b vary continuously in the space of rational maps of degree 4. More precisely, the set of maps cPa,b is a two-dimensional algebraic family of points in the space Rat4, given explicitly by
(a, b) f------+ [1,0, -2a, -8b, a2 , 0, 4, 0, 4a, 4b]. Ifwe conjugate by fu(x)
= UX, the Lanes map cPa,b transforms into
Thus assuming (say) that ab i=- 0, we can take U = b/a to transform cPa,b into WIith C
,;.,fb/a,l. 'f'a,b = wc,c
= a 3/b . 2
In other words, the two-dimensional family of Lattes maps {cPa,b} in Rat., becomes the one-dimensionalfamily of dynamical systems (6.25) Of course, it is not clear a priori that the map (6.25) is nonconstant. But if the Lattes maps cPe,e and cPel,e l are PGL 2 -conjugate, then Theorem 6.46 tells us that their associated elliptic curves E and E' are isomorphic. The j-invariant of the elliptic curve E; : y2 = x 3 + CX + c is .()
J E;
8
3
C
= 2 ·3 4c + 27'
so we see that j (E e) = j (Eel) if and only if c = c'. This proves that the map (6.25) is injective, so these flexible Lattes maps do indeed form a one-parameter family of nonconjugaterational maps with identicalmultiplierspectra,i.e., they are a nontrivial isospectra1 family.
357
6.5. Flexible Lattes Maps
Example 6.50. The elliptic curve E : y 2 = x 3 + ax 2 + bx has the 2-torsion point T = (0, 0) . To compute the Lanes function ¢ : pI -+ pI associated to the translatedduplication map ljJ(P ) = [2](P) + T, we first use the classical duplication
formula to compute _ ( x4 2P -
-
2bx 2 4y2
+ b2
x6
+ 2ax 5 + 5bx 4 -
5b2 x
2
-
2ab2 x - b3 )
8y 3
'
.
Then the addition formula and some algebra yield ¢(x) = x(2P
+T ) =
3
2
4b(x + ax x 4 - 2bx 2
+ bx ) .
+ b2
As in the previousexample,these Lattes maps form a one-dimensional familyin M 4 . We begin with a few elementary, but useful, properties of flexible Lattes maps. Proposition 6.51. Let ¢ : pI -+ pI be a flexible Lattes map whose associated map ?j; : E -+ E has the form ?j;(P) = [m]P + T. (a) The map ¢ has degree m 2 . (b) The point T satisfies [2]T = 0. (c) Fix a Weierstrass equation (6.11) for E . Then there is a linear fractional transformation f E PGL 2 such that 1r = f o x. Hence ¢f fits into a commutative diagram
E ~ E (6.26)
p I ~ p l. Proof (a) The commutativity of the diagram (6.22) tells us that
deg(¢) deg(1r ) = deg(1r) deg(?j;). The map e has degree m 2 , since multiplication-by-m has degree m 2 and translationby-T has degree 1. Therefore deg(¢) = m 2 . (b) We are given that the map 1r : E -+ pI has degree 2 and satisfies 1r(P) = 1r( -P). It follows that 1r(P) = 1r(Q) if and only if P = Q or P = -Q. We use the commutativity of (6.22) to compute
1r( -[m]P - T)
= 1r([m]P + T) = 1r(?j;(P)) = ¢(1r(P)) = ¢(1r( -P)) = 1r(ljJ (-P)) = 1r(- [m]P + T ).
Hence for every PEE we have either
-[m]P - T = -[m]P + T
or
- [m]P - T = - (-[m]P
+ T ) = [m]P -
Simplifyingthese expressions, we find that every point PEE satisfieseither [2]T = O
or
[2m]P = O.
T.
358
6. Dynamics Associated to Algebraic Groups
But there are only finitely many points PEE satisfying [2m]P = 0; hence we must have [2]T = O. (c) The map 1f is a rational function on E satisfying 1f( - P) = 1f(P). It follows that 1f is in the subfield K(x) ofthe function field K(E); see [410,111.2.3.1]. In other words, there is a rational function I(z) E K(z) such that 1f = I(x). Equivalently, the map 1f : E ---+ lP'1 factors as
In particular,
2 = deg( 1f)
= deg(J 0 x) = deg(J) deg( x) = 2 deg(J) ,
so we see that deg(J) = 1. Hence 1 is a linear fractional transformation, which proves the first part of (c). Finally, we compute
o
which proves the commutativity of (6.26).
Our next task is to compute the periodic points and multipliers of flexible Lattes maps. For ease of exposition, we do the pure multiplication case, i.e., for maps of the form 'IjJ (P) = [m] (P), and leave the general case for the reader. Proposition 6.52. Let ¢ : lP'1 ---+ lP'1 be ajlexible Lattes map and assume that T = 0, so 'IjJ(P} = [m](P). (See Exercise 6.18for the case T I- 0.) (a) The set ofn-periodic points of ¢ is
(b) Let ( be a periodic point of ¢ ofexact period n. Then
if( E 1f(E[mn if( E 1f(E[mn if( E 1f(E[mn (Notice that
1f
-
1]) and ( rf. 1f(E[2]),
+ 1]) and ( rf. 1f(E[2]), + 1]) n 1f(E[2]).
(E[2]) is the set ofcritical values of1f.)
Proof (a) Let ( E lP'1 be a fixed point of ¢ and choose a point PEE with 1f(P) = (. Note that there are generally two choices for P, so we simply choose either one of them. Then
1f(P) = (
= ¢(() = ¢(1f(P)) = 1f('IjJ(P)) = 1f([m]P).
As noted during the proof of Proposition 6.51(b), we have 1f(P) = 1f(Q) if and only if P = ±Q, so we conclude that either
[m]P=P
or
[m]P =-P.
6.5. Flexible Lattes Maps
359
Conversely, if [m]P = ±P, then
¢(7r(P))
= 7r(W(P)) = 7r([m]P) = 7r(±P) = 7r(P), = 7r(P ) E Fix(¢ ) if and only if
so 7r(P) is fixed by ¢ . This proves that (
[m- I]P = O
or
[m +I]P =O ,
and hence Fix(¢ )
= 7r (E[m - 1])
U 7r(E[m + 1]) .
(6.27)
In order to find the points of period n, we observe that
so ¢n is also a flexible Lattes map. It is associated to the map [rn"] : E ----+ E. (For a generalization of this observation, see Exercise 6.17.) Applying (6.27) to the Lattes map ¢ n yields the desired result,
(b ) The multipliers of ¢ are invariant under PGL 2-conjugation , so we can use Proposition 6.51(c) to replace ¢ by a conjugate satisfying
¢ o x = xow, where x is the x -coordinate on a Weierstrass equation
E : y2 = x 3 + ax
+ b.
In order to compute the multipliers of ¢ , we are going to use the translation-invariant differential form
dx
dy
w = 2y = 3x 2 + a
(6.28)
on E described in Theorem 6.35. The invariant differential satisfies the formula
w*(w)
= [m]*(w) = mw.
(6.29)
Substituting w = dx/2y and doing some algebra yields
w*
(dx) yo W --=m--. y dx
(6.30)
Let ¢ be the Lattes map associated to W , let ( E Fix(¢) with ( :j:. 00, and let t be a coordinate function on pl. Then the multiplier Act> (() of ¢ at ( can be computed using the differential form dt via the equation
¢*( dt) dt
I t =(
=
d¢ (t ) dt
I t=(
= ¢'(() = A¢(( ).
(6.31)
360
6. Dynamics Associated to Algebraic Groups
Using the relation ¢ 0 x = x
x* ( ¢*(dt)) dt
'l/J and formula (6.30), we compute
0
= x*¢*( dt) = x*(dt)
(¢ox)*(dt) x*(dt )
= (x 0 'l/J )*(dt ) = 'l/J*x*(dt ) = 'l/J*(dx) = m y o 'l/J x*( dt)
x*(dt)
dx
(6.32)
y .
We lift ( to a point PEE satisfying x( P) = ( and evaluate both sides of (6.32) at P to obtain
¢*(dt) I = m (y 0 'l/J ) (P) . dt t=( Y
(6.33)
Equating (6.31) and (6.33) gives the useful formula
y o'l/J ) (P) A,p(() = m ( -y-
for ( = x(P ) E Fix(¢) with (
i
00 .
Assume first that [2]P i 0 , which ensures that y(P) i 0 and y(P ) i we can directly evaluate the fraction in (6.34) and conclude that
(6.34)
00.
Then
A (() = m (y 0 'l/J )(P) ¢ y(P) ' We are assuming that ( = x(P ) is a fixed point of ¢, so
x(P) = ¢(x( P)) = x('l/J(P)). Thus 'l/J(P) = ±P, and hence y('l/J(P )) = ± y(P), which proves that A,p(( ) = ± m. More precisely, A
( E Fix(¢)
A (() = { ¢
m -m
if ( E x(E [m -1]) , if( E x (E [m + 1]).
(6.35)
Next suppose that y(P ) = 0, so [2 ]P = 0 , but P i O. We also have [m]P = ±P from the assumption that ( = x(P) E Fix(¢), so m is odd and 'l/J fixes P. The functions y and y 0 'l/J both vanish at P , so we can use l'Hopital's rule to compute
(y ~ 1jJ) (P) = (d(Yd~ 'l/J )) (P) , assuming that the righthand side has a finite value. (Note that this formula is valid algebraically, since what we are really doing is looking at the linear terms in the local expansions of y and y 0 'l/J at P.) We now use the chain rule to compute
6.5. Flexible Lattes Maps
d('lj;*y) dy
361
d('lj;*y) 'lj;*(3x2 + a) 3x2 + a 3x2 + a . dy 'lj;*(3x2 + a) 2+a). 2+a 3x ='lj;* ( dy ). 'lj;*(3x 2 2 3x + a 3x + a dy 2 'lj;*(w) 'lj;*(3x + a) -- . from (6.28), w 3x 2 + a 'lj;*(3x2 + a) =m from (6.29), 3x 2 + a 3(x o'lj;)2+ a - m --'-----,--'---3x 2 + a .
We evaluate both sides at P = ((, 0). Note that the quantity 3(2 + a is nonzero, since otherwise P would be a singular point of E. Also, x('lj;(P)) = x(P) = (. Hence
(
y 0 'lj;) (P) = d('lj;*y) (P) = m 3(x 0 'lj;(p))2 + a = m 3(2 y dy 3x(P)2 + a 3(2
+a +a
= m.
Substituting this value into (6.34) yields the desired result .\1' (() = m 2 . It remains to deal with the case P = O. There are several ways to do this case. First, we could perform an explicit calculation using local coordinates around O. Second, since we are missing only one multiplier, we could use Theorem 1.14, although this would require knowing a priori that>. i- 1. Third, at least for odd m, we could observe that 'lj;(P) = [m]P looks the same locally around each of the points in E[2], and we already computed>' = m 2 for the nonzero points in E[2]. We leave as an exercise for the reader (Exercise 6.20) to complete the proof using whichever argument he or she prefers. Finally, to compute the multiplier of a periodic point ( E Per., (¢), we apply the results that we have just derived to the fixed points of the Lattes map ¢n satisfying ¢n 0 X = X 0 [m n ]. 0 Remark 6.53. Let ¢ : ]lD1 ----+ ]lD1 be a flexible Lattes map associated to 'lj;(P) = [m]P as in Proposition 6.52. If we work over C, then the multiplier of every periodic point ( E Per(¢) satisfies
for some e
= e( () 2':
1.
Hence ( is repelling, so Per(¢) is contained in the Julia set (cf. Exercise 1.27). Choosing a lattice L and a complex uniformization C/ L ----+ E(C) as described in Section 1.6.3, it is clear that E(C)tors is dense in E(C). Therefore Per(¢) = x(E(C)tors) is dense in ]lD1(C) and is contained in .:J(¢). Further, the Julia set is closed. This proves that .:J(¢) = ]lD1(C) and F(¢) = 0, which is Lanes's Theorem 1.43 discussed in Section 1.6.3. On the other hand, if we work over a p-adic field such as Qp or C p , then every periodic point ( E Per(¢) satisfies for some e = e(() 2': 1.
362
6. Dynamics Associated to Algebraic Groups
Thus every periodic point is nonrepelling, so Per (¢ ) c F (¢) from Proposition 5.20. Further, ifplm, then I\ t> (() I = Iml e < 1, so in this case every point in Per (¢ ) is attracting.
Remark 6.54. Recall that the multiplier spectrum of a rational map ¢ : pI -; pI of degree d is the map that associates to each integer n ~ 1 the set
where we treat Per n (¢ ) as the set of dn + 1 (not necessarily distinct) fixed points of ¢n. Two maps with the same multiplier spectrum are called isospectral. (See Section 4.5, page 187.) Proposition 6.52(b) shows that flexible Lattes maps of degree m 2 are isospectral , since their multiplier spectrum depends only on m . A deep theorem of McMullen [294, §2] (Theorem 4.53) says that these are the only isospectral rational maps that vary in a continuous family. Not surprisingly, good reduction of Lattes maps is closely related to good reduction of the associated elliptic curve.
Proposition 6.55. Let K be a local fi eld, let R be the ring of integers of K, and be a flexible Lottes map of degree m 2 associated to an elliptic let ¢ : curve E / K . Suppose that E has good reduction and that m E R*. Then there exists an f E PGL 2 (K ) such that ¢i has good reduction.
Pk -; lPk
Proof Since E has good reduction , we can find a Weierstrass equation for E with coefficients in R and discriminant in R*. We then use Proposition 6.51 to replace ¢ by ¢i so that it fits into a Lattes diagram (6.26). In other words, the Lattes projection map 7f is the x-coordinate function on a minimal Weierstrass equation for E. For any n ~ 1 we can find polynomials Fn(X) ,Gn(X ) E R [X] such that
x([nJ P) = Fn{x(P)) .
c; (x(p))
This is easily proven by writing out the first few polynomials explicitly and then computing the subsequent ones by a recurrence formula. The recurrence also shows that the leading terms of Fn and G n are 2 n G nX=nX ()
and
2
1 -+ ... .
(See [96, page 133], [410, Exercise 111.3.7], or Exercise 6.23.) We note that the roots of Gn(X ) are the x-coordinates of the n -torsion points , and one can check that Gn(X ) factors as
Gn(X ) = n 2
II
(X - x(P) ).
P EE [n] , P ol-V
We are assuming that 'l/J(P ) = [mJ(P ) + T for some fixed integer m and some T E E[2J. For simplicity we prove here the case T = 0 and leave the general case as an exercise. Then
6.5. Flexible Lattes Maps
363
¢(x(P)) = x(1jJ(P)) = x ([m](P)) , We fix an auxiliary prime £ satisfying £ f m and £ E R* and consider the polynomial
H(X) =
II
(Fm(X) - x(Q)Gm(X)).
QEE[R],Qi-O
Notice that H(X) is a monic polynomial, since Fm(X) is monic and deg(Fm ) > deg(G m ) . We also note that all of the x(Q) are integral over R, since they are roots ofGR(X) and £ E R*. It follows that H(X) E R[X]. Further, Proposition 2.13(b) (see also Exercise 2.6) tells us that the resultant of H(X) and Gm(X) is
II II
Res(H(X), Gm(X)) = ±m2degH
H(()
Gm()=O
= ±m2degH
Fm(( / 2- 1
Gm()=O
=
Res(Fm(X),Gm(X))
R2~1
.
Hence in order to show that ¢ has good reduction, it suffices to prove that
Res(H(X), Gm(X)) E R*. Let K' = K (E[m£]) be the field extension obtained by adjoining the coordinates of the points of order m£ to K, let R' be the ring of integers of K', let 1" be the maximal ideal in R', and let k' = R' /1" be the residue field of R'. The extension K' / K is unramified, because we have assumed that E has good reduction and m.e is a unit in R; see Corollary 6.33. Note that H(X) and Gm(X) factor completely in K', and in fact their roots are in R'. This is clear for Gm(X), since its roots are the x-coordinates of the points in E[m] and its leading coefficient is m2, which is a unit in R. We now analyze H(X) more closely.
Claim 6.56. The roots of H(X) are given by {roots of H(X)}
= x(E[m£] "E[m])
C
R'.
ProofofClaim. The roots of H(X) are the solutions to
for some Q E E[£]. Writing a root of H(X) as x(P) for some PEE, this means that
x([m]P)
=
Fm(x(P)) Gm(x(P))
=
x(Q),
6. Dynamics Associated to Algebraic Groups
364
and hence [m]P = ±Q. But Q E E[£], so P E E[m£]. This shows that the roots of H(X) are contained in x(E[m£]). Further, if P E E[m], then Gm(x(P)) = 0 and Fm(x(P)) i- 0, so
e2
H(x(P)) = Fm(x(P)) -
I
i- O.
This gives the inclusion {roots of H(X)}
c x (E[m£] "E[m]).
The other inclusion is clear from the definition of H(X), since
P
E
E[m£] "E[m]
=?
[m]P E E[£]
<,
{O}.
Thus x(P) is a root of Fm(X) - x([m]P)Gm(X), which is one of the factors in the product defining H(X). Finally, we note that K' contains the x-coordinates of the points in E[m£] by construction. Further, these x-coordinates are the roots of the polynomial Fme(X) E R[X] whose leading coefficient is m 2 £2 E R*, so the roots are integral over R, hence are in R'. D We now resume the proof of Proposition 6.55. We assume that the resultant Res (H (X), G m (X)) is not a unit in R and derive a contradiction. This assumption means that H(X) and Gm(X) have a common root modulo p', so we can find Xl, X2 E R' such that and
Xl
==
X2
(mod p').
From our description of the roots of H(X) and Gm(X), this means that we can find points PI E
E[m£] "E[m]
and
P2
E
E[m] ,,{O}
satisfying
PI
== P2 (mod p").
(In principle, we might get PI == - P2 , but if that happens, then just replace P2 by - P2 .) Since clearly PI i- P2 , this proves that the reduction modulo p' map
E(K') ~ E(k') is not injective on E[m£]. This is a contradiction, since Theorem 6.32 tells us that the prime-to-p torsion injects on elliptic curves having good reduction. D
6.6
Rigid Lattes Maps
In general, a Lattes map ¢ : jp'I
--+
jp'I
is defined via the commutativity of a diagram
E~E (6.36)
6.6. Rigid Lattes Maps
365
where 'lj; is a morphism of degree d 2': 2 and 1r is a finite separable map. Every morphism of an elliptic curve to itself is the composition of an endomorphism and a translation (Remark 6.19), so 'lj; has the form 'lj;(P) = a(P) + T for some a E End(E) and some TEE. However, it turns out that the commutativity of (6.36) puts additional constraints on ¢, 'lj;, and 1r. More precisely, it forces the existence of a similar diagram in which 1r has a special form. We state this important result and refer the reader to [300] for the analytic proof. Theorem 6.57. Let K be afield ofcharacteristic 0 and let ¢ be a Lattes map defined over K. Then there exists a commutative diagram of the form (6.36) such that the map 1r has the form 1r :
E
----+
for some nontrivialfinite subgroup I'
c
Elf ~
jp'l
Aut(E).
Proof For a proof over C, see [300, Theorem 3.1]. The general case for character0 istic-Ofields follows by the Lefschetz principle, cf. [410, VI §6]. Definition. Let ¢ be a Lattes map. A reduced Lattes diagram for ¢ is a commutative diagram of the form
E
E (6.37)
E If ~
jp'l
~
jp'l
~ Elf
Theorem 6.57 says that every Lattes map fits into a reduced Lattes diagram. Corollary 6.58. Let ¢ be a Lattes map given by a reduced diagram (6.37). Then the point 'lj;( 0) is fixed by every element of I', so in particular, 'lj;( 0) E E tars . Iffurther j (E) =I- 0 and j (E) =I- 1728, then
f = J.L2'
deg 1r = 2,
and
'lj;( 0) E E[2].
Proof We defer the proof that 'lj;( 0) is fixed by every ~ E I' until Proposition 6.77(b), where we prove it in a much more general setting. (Cf. the proof for flexible Lattes maps in Proposition 6.51(b).) To see that 'lj;(0) is a torsion point, let ~ E T be a nontrivial element off. Then ~('lj;(0)) = 'lj;(0), so applying Theorem 6.22 to the isogeny ~ - 1, we find that
[deg(~ -l)J ('lj;(0)) = ( 0 ) 0 (~-1)('lj;(0)) = O. For the final statement of the corollary, we note that if j(E) is not equal to 0 or 1728, then Proposition 6.26 tells us that Aut(E) = J.L2' Hence f = J.L2 and deg 1r = 2. Further, since 'lj;( 0) is fixed by every element of I', we have [-1]'lj;(0) = 7/;(0), so [2]7/;(0) = O. 0
366
6. Dynamics Associated to Algebraic Groups
Remark 6.59. The proof of Theorem 6.57 in [300] actually shows something a bit stronger. Suppose that ¢ is a Lattes map fitting into the commutative diagram (6.36). It need not be true that the map 7T : E - t p I is of the form E - t Elr, i.e., the given diagram need not be reduced , and indeed the map 7T may have arbitrarily large degree. However, what is true is that there are an elliptic curve E' , an isogeny E - t E' , and a finite subgroup I" c Aut (E' ) such that 7T factors as
E ~ E' ~ E' /r
9'! pl.
Further, this factorization is essentially unique . See [300, Remark 3.3].
Remark 6.60. The proof of Theorem 6.57 is analytic and does not readily generalize to characteristic p. A full description of Lattes maps in characteristic pis still lacking. Aside from the curves having j-invariant 0 or 1728, every Lattes map has and deg n = 2, so after a change of coordinates, the projection 7T : E - t pI is 7T(X, y) = x. For simplicity, we will concentrate on this situation, although we note that the two special cases with Aut(E) = /-L4 and Aut(E) = /-L6 have attracted much attention over the years for their interesting geometric, dynamical, and arithmetic properties. Our next task is to describe the periodic points of (rigid) Lattes maps and to compute their multipliers.
r = /-L2 = Aut(E)
Proposition 6.61. Let ¢ : pi - t pI be a Lanes map and fix a reduced Lottes diagram (6.37) for ¢. We assume that j(E) -:j:. 0 and j (E ) -:j:. 1728. Wefurther assume that 7jJ is an isogeny, i.e., with our usual notation 7jJ (P ) = [a](P ) + T, we are assuming that T = CJ. (See Exercise 6.24 for the other cases.) (a) The set offixed points of ¢ is given by Fi x (¢ ) = 7T(E[a + 1] U E[a -1]).
(6.38)
(b) The intersection satisfies
E[a + 1]
n E[a
- 1] C E[2].
If deg(a-I) is odd, then the intersection is O. (c) Let 7T(P) E Fix(¢). The multiplier of ¢ at 7T(P) is
if P E E[a - 1] and P rf. E[a + 1], if P E E[a + 1] and P rf. E[a - 1], if P E E [a + 1] n E [a - 1].
(6.39)
Proof (a) We have 7T(P ) E Fix(¢ ) ifand only if
7T(P ) = ¢ (7T(P) ) = 7T(1jJ(P)) . Our assumption on j (E ) means that r = Aut(E) = /-L2' so 7T(P ) is fixed by ¢ if and only if 1jJ (P ) = ±P. Since we are also assuming that 1jJ (P ) = [a](P ), this is the desired result.
6.6. Rigid Lattes Maps
367
(b) Let P E E[a: - 1] n E[a: + 1]. Adding [a: - l](P) = 0 to [a: + l](P) = 0 yields [2]P = 0, so P E E[2]. To ease notation, let m = deg(a: - 1). Then using Theorem 6.22, we find that
[m](P) = [c;=-r] 0 [a: - l](P) = [c;=-r](0) = 0, so P E E[m]. Hence P E E[2] n E[m]' so ifm is odd, then P = O. (c) The proof is identical to the proof of Proposition 6.52. The only difference is that'ljJ = [a:] may no longer be multiplication by an integer, but we still have the key formula
'ljJ*(w)
=
[a:]*(w)
=
a:w
giving the effect of'ljJ on the invariant differential w of E. Using this relation in place offormula (6.29) used in proving Proposition 6.52 and tracing through the argument yields the desired result. D We recall from Section 4.5 that O";n) (¢) denotes the i th symmetric polynomial of the multipliers of the points in Per n ( ¢ ), taken with appropriate multiplicities. For d ~ 2 and each N ~ 1, we write (6.40) for the map defined using all of the functions O";n) with 1 :::; n :::; N. McMullen's Theorem 4.53 says that for sufficiently large N, the map a d,N is finite-to-one away from the locus of the flexible Lattes maps. As noted by McMullen in his paper and stated in Theorem 4.54, rigid Lanes maps can be used to prove that U d,N has large degree. For the convenience of the reader, we restate the theorem before giving the proof.
Theorem 6.62. Define the degreeOfO"d,N to be the number ofpoints in O"d,}Y(P)for
a generic point P in the image U d,N(Md). One can show that the degree of U d,N stabilizes as N ----; 00. We write deg(ud)for this value. Thenfor every E > 0 there is a constant OE such that for all d. In particular, the multiplierspectrum ofa rationalfunction ¢ E Rat., determines the conjugacy class of ¢ only up to DE (d~ -E) possibilities. Proof We prove the theorem in the case that d is square free and leave the general case for the reader. Let F = Q( R), let R F be the ring of integers of F, and let 01, ... , 0h be fractional ideals of F representing the distinct ideal classes of R F . Consider the elliptic curves E 1 , ... , E h whose complex points are given by
1:::; i Each E, has End(R i )
9;!
:::; h.
R F (Proposition 6.40), and we normalize an isomorphism
368
6. DynamicsAssociated to Algebraic Groups
as described in Proposition 6.36. We fix a Weierstrass equation for each E, and we define a Lattes map cPi by
cPi 0 X = X 0 [yCd]. Then deg(cPi) = d from Proposition 6.39, and Proposition 6.61 tells us that the multipliers of cPi are given by (6.39). In particular, they are the same for every cPi' i.e., the set of maps {cPl' ... , cPh} is isospectral, so we see that
Next we observe that cPl' ... ,cPh give distinct points in M d , because Proposition 6.40 says that E 1 , ... , Eh are pairwise nonisomorphic, and then Theorem 6.46 tells us that cPl' ... ,cPh are pairwise nonconjugate. This proves that (J"d,N is generically at least h-to-l, where h is the class number of the ring of integers of Q ( yCd). (Note that the cPi are not flexible Lattes maps, and McMullen's Theorem 4.53 tells us that a d,N is finite-to-one away from the flexible Lattes locus.) To complete the proof we need an estimate for this class number. Such an estimate is given by the Brauer-Siegel theorem [258, Chapter XVI], which for quadratic imaginary fields says that lim d---HXJ
d squarefree
log (class number of Q( yCd ))
1
logd
2
(Note that this is where we use the assumption that d is squarefree, since it implies that the discriminant of Q(R) is equal to either d or 4d.) In particular, the class number is larger than d 1j 2 --- E for all sufficiently large squarefree d, which completes the proof of Theorem 6.62 for squarefree d. In the general case, there are two ways to proceed. The first, which is sketched in Exercise 6.25, is to find a quadratic imaginary field F whose discriminant is O(d 1 --- E ) and whose ring of integers contains an element of norm d. The second is to write d = ab2 with a squarefree and use elliptic curves whose endomorphism rings are isomorphic to the order R b = Z + bRF in the field F = Q( v=a). The class number of Rb is equal to hFb times a small correction factor; see [399, Exercise 4.12]. D
6.7
Uniform Bounds for Lattes Maps
A fundamental conjecture in arithmetic dynamics asserts that there is a constant C = C(d, D) such that for all number fields K/Q of degree D and all rational maps cP(z) E K(z) of degree d ~ 2, the number of K-rational preperiodic points of cP satisfies # PrePer(cP, ]P'l(K)) < C(d, D). (See Conjecture 3.15 on page 96.) Aside from monomials and Chebyshev polynomials, the only nontrivial family of rational maps for which Conjecture 3.15 is known
6.7. Uniform Bounds for Lattes Maps
369
is the collection of Lattes maps. The proof uses the following deep theorem, whose demonstration is unfortunately far beyond the scope of this book.
Theorem 6.63. (Mazur-Kamienny-Merel) For all integers D 2: 1 there is a constant B(D) such that for all number fields K/Q ofdegree at most D and all elliptic curves E / K we have
#E(K)tors < B(D). Discussion. This deep result was first proven by Mazur [292] for K = Q, then by Kamienny [225] for [K : Q] = 2, and then was extended to various specific larger degrees before the proof was completed for all degrees by Merel [297]. The proof uses the theory of modular curves and Jacobians, which do have counterparts in arithmetic dynamics (cf. Sections 4.2--4.6). However, the proof also relies in a fundamental way on the fact that E is a group, and hence that there exist a large number of commuting maps E ---> E. This is in marked contrast to the situation for a general rational map rjJ : jp'l ---> jp'l, for which only the iterates of rjJ commute with rjJ. The inclusion Z c End(E) leads to the existence of Heeke correspondences on elliptic modular curves, and these correspondences provide an essential tool in the proof of Theorem 6.63. Unfortunately, there does not appear to be an analogous theory of correspondences for the dynamical modular curves and varieties attached D to non-Lattes maps on jp'l. Corollary 6.64. For all integers n curves E / K we have
#(
>
1, all number fields K/Q, and all elliptic
U E(L)tors)::; B(n[K : Q])3,
(6.41 )
[L:K]:Sn
where B(D) is the constant appearing in Theorem 6.63. Proof To ease notation, we let D = [K : Q]. Every field L appearing in the union in (6.41) satisfies [L : Q] = [L : K][K : Q] < nD, so Theorem 6.63 tells us that #E(L)tors ::; B(nD). In particular, E(L) contains no points of order strictly larger than B(nD). This is true for every such L, so we conclude that
U E(L)tors C
[L:K]:Sn
U
E[b].
l:Sb:SB(nD)
Then using #E[b] = b2 yields
#(
U E(L)tors)::; L [L:K]:'On
l:'Ob:'OB(nD)
#E[b] =
L l:'Ob:'OB(nD)
b2
::;
B(nD)3.
D
6. Dynamics Associated to Algebraic Groups
370
We now use Theorem 6.63 to prove uniform boundedness of preperiodic points for Lattes maps. This bound is in fact independent of the degree of the Lanes map ¢, which may be surprising at first glance. However, it is easily explained by the fact that Lanes maps associated to the same elliptic curve all commute with one another, so they have identical sets of preperiodic points.
Theorem 6.65. Let D 2: 1 be an integer. There is a constant C(D) such that for all numberfields K /Q ofdegree D and all Lottes maps ¢ : pI -+ pI defined over K we have #PrePer(¢,pI(K))::; C(D). Proof Without loss of generality we fix a reduced Lanes diagram (6.37) for ¢. Then Proposition 6.26 says that the projection map 7r : E -+ pI has degree at most 6, and indeed if j (E) i- 0 and j (E) i- 1728, then deg( 7r) = 2. Proposition 6.44 tells us that PrePer(¢,pI)
= 7r(Etors),
so the fact that deg( 7r) ::; 6 yields PrePer(¢,pI(K)) C
U
7r(E(L)tors).
(6.42)
[L:K]::;6
Corollary 6.64 says that the set on the righthand side of (6.42) has size bounded D solely in terms of D, hence the same is true of # PrePer(¢, pl (K)). Example 6.66. The rational map x4
¢a,b(X) =
2ax 2 - 8bx + a 2 4x3 + 4ax + 4b
-
is the Lattes map associated to multiplication-by-2 on the elliptic curve
Ea,b : y2 = x 3 + ax + b. The j-invariant and discriminant of Ea,b are given by the usual formulas
j(Ea ) = 1728 4a 3
4a 3 + 27b2
and
Theorem 6.65 tells us that # PrePer( ¢a,b, pI (K)) is bounded solely in terms of the degree d = [K : Q]. In general, the best known bounds are exponential in d, but if j (Ea,b) is an algebraic integer, then much stronger bounds can be proven as in the following result.
Theorem 6.67. Let K be a number field of degree D 2: 2, let E / K be an elliptic curve whose j-invariant is an algebraic integer, and let ¢ be a Lattes map associated to E. Then there is an absolute constant c such that
6.7. Uniform Bounds for Lattes Maps
371
Proof. The assumption that the elliptic curve E has integral j-invariant means that it
has everywhere potential good reduction. Replacing K by an extension of bounded degree, we may assume that E has everywhere good reduction. (In fact, it suffices to go to the field K(E[3J), a field of degree at most 48 over K.) Then a result of Hindry-Silverman [204] impliesa bound slightly strongerthan
#E(K )tors ~ 22 1 D log D. Finally, we note that as in the proof of Corollary 6.64, a bound for E( K)tors of the form #E(K)tors ~ B ([K : QJ) for all numberfields K impliesa bound of the form
U E(L)tors) < B(n[K : QJ) 3.
#(
[L:K) :=; n
Hence as in the proof of Theorem6.65 we have
U
PrePer(¢,lP'l(K)) C
7r(E(L)tors) ~ c(D log D)3
[L :K ]:=; 6
o
for an absolute constantc.
Theorem 6.65 proves uniformity for rational preperiodic points of Lattes maps. In the other direction,recall that we proved(Theorem3.43) that the orbitsof rational wandering points contain only finitely many integersexcept in a few preciselyspecified situations. In particular, Lattes orbits contain only finitely many integer points, since Lattes maps are not polynomial maps. Using deep results from the theory of elliptic curves, it is possible to obtain strong uniformityestimates for the number of integerpoints lying in Lattes orbits. For simplicity we state results over Q and Z, but we note that an appropriately formulated versionis true for rings of S-integers in numberfields. Definition. Let E /Q be an elliptic curve. Recall that a global minimal Weierstrass equation for E is a Weierstrass equationthat is simultaneously minimalat all primes (Remark 6.30). We define a quasiminimal Weierstrass equation for E /Q to be an equation of the form E : y2
= x 3 + ax + b,
a,b E Z,
(6.43)
such that 14a3 + 27b2 j is as small as possible. Equivalently, the equation (6.43) is quasiminimal ifthere are no primes p such that p4ja and p6lb. Remark 6.68. Givenan arbitrary Weierstrass equation
E : y 2 = x3
+ ax + b,
a.b E Z,
it is easy to create a quasiminimal equation. Simply let u be the largest integer such that U 12 divides gcd(a3 , b2 ) , and then
6. Dynamics Associated to Algebraic Groups
372
is a quasiminimal equation for E /Q. Elliptic curves over Q have global minimal Weierstrass equations, see Remark 6.30, and it is not hard to show that a quasiminimal equation is minimal at every prime p ::::: 5, and that it is almost minimal at 2 and 3; see Exercise 6.26. The following theorem is a conditional resolution of a conjecture of Lang [254, page 140].
Theorem 6.69. (Hindry-Silverman) Let E /Q be an elliptic curve given by a quasiminimal Weierstrass equation and let E(Z) be the set of points in E(Q) having integer coordinates. Also let v(E) be the number ofprimes dividing the denominator ofthe j-invariant of E. (a) There is an absolute constant C such that for any subgroup r c E(Q),
# (r n E(Z)) < cv(E)+rankr. (b)
If the "ABC conjecture" is true.' then there is an absolute constant C such that for any subgroup r c E(Q), # (r n E(Z)) < c
rankr.
Proof The proof of (a) is given in [407] and the proof of (b) is in [202].
D
We can use Theorem 6.69 to prove a uniform bound for integer points in orbits of flexible Lattes maps (cf. Conjecture 3.47).
Theorem 6.70. Let E / Q be an elliptic curve given by a Weierstrass equation with integer coefficients, let m ::::: 2 be an integer, and let ¢(z) E Q(z) be the Lattes map satisfying for all PEE. ¢(x(P)) = x([m]P) Assume further that ¢ is affine minimal in the sense that
Res(¢) =
min
Res(¢!).
(6.44)
!EPGL2(iQ) !(z)=az+b
(See page 112 for the definition ofthe resultant Res(¢) ofa rational map.) Thus the assumption (6.44) says that we cannot reduce the resultant of ¢ by conjugation by an affine linear transformation j(z) = az + b. Let ( E Q and consider the orbit O¢(() of( by ¢. (a) There is an absolute constant C such that
where v( E) is the number ofprimes dividing the denominator ofthe j-invariant
ofE. 2The ABC conjecture of Masser and Oesterle says that if A, B, C integers satisfying A + B = C, then C «€ TIplABC pH€.
> 0 are pairwise relatively prime
6.7. Uniform Bounds for Lattes Maps
(b)
373
If the ABC conjecture
is true, then the number of integer po ints in 0 4> (() is bounded by an absolute constant independent ofE and (.
Proof Write the given Weierstrass equation for E as with a , b « Z . We begin by showing that the minimality assumption (6.44) implies that there are no primes p with p21a and p3lb. The rational function ¢ (x ) = F (x )/G (x ) is associated to the rnultiplication-by-rn map, so it is given by polynomials
F (a,b;x ),G(a,b ;x ) E Z [a , b,x] that are weighted homogeneous in the sense that
For example, if m = 2, then
¢ (x ) = x
4
2 2 2ax - 8bx + a 4x 3 + 4ax + 4b
-
Hence ifp2 a andp3 lb, then conjugating ¢ (x ) by f (x ) = px yields 1
¢ f (x ) =p_1F(a,b;px ) G(a, b;px )
= F (p- 2a ,p- 3b;X) . G(p- 2a, p- 3b;x )
The assumption that p2ja and p31b implies that these polynomials have integer coefficients, and then homogeneity yields
Res(¢f ) = Res(F(p- 2a , p- 3b;x), G(p-2 a , p- 3b;x )) 2
2_1
2
2
= p_ m (m
) Res(F(a ,b;x) ,G(a,b ;x))
= p_m (m _ 1) Res (¢ ). This contradicts (6.44), so we have proven that there are no primes p satisfying p 2 1a andp 3 lb. We would like to apply Theorem 6.69 to the rank-I subgroup generated by a point P = ((, TJ) of E lying above (. Unfortunately, although ( E Q, there is no reason that TJ need be rational. So it is necessary to move to a twist of E. We are given a point ( E Q and we choose a point P = (( , TJ ) E E lying above (. We do not assume that TJ = (3 + a( + b is rational. We write
J
then we factor with u 1 squarefree, and consider the elliptic curve
6. Dynamics Associated to Algebraic Groups
374
(6.45) (In the terminology of Section 4.7, E' is a twist of E; cf. Example 4.71.) Notice that the point P{ = (UIXI,uiv l) E E'(iQ!) is a point of E' having integer coordinates. We claim that the Weierstrass equation (6.45) for E' is quasiminimal. To prove this claim, let p be any prime. We showed earlier that either or Since
UI
ordp(b) < 3.
is squarefree by construction, it follows that either or
which shows that the Weierstrass equation (6.45) is quasiminimal. This means that we can apply Theorem 6.69 to E' and the rank-l subgroup generated by P' to conclude that
#{ n 2:
1 : [n]P' E E'(Z)} :S
ClI(E')+I.
(6.46)
Further, if the ABC conjecture is true, then the upper bound may be replaced by C. The two elliptic curves E and E' are isomorphic, although the isomorphism is defined only over iQ! (JUl). This isomorphism, which we denote by F, is given explicitly by
E : y2 = x 3 + ax (x,y)
+b
In particular, j(E) = j(E'), so v(E) = v(E'). In order to relate integers in Oq'>(() to integer points in E' (iQ!), we write and
P~ = [n]P{ = (x~, y~).
Since the isomorphism F respects multiplication by n, we have
In particular, since P{ E E' (iQ!), itfollows that[n]P{ = P~ E E'(iQ!), so F maps the multiples of PI, which, note, are not in E(iQ!), to points in E'(iQ!). Further, if X n E Z, then it is clear from the definition of F that x~ = UIX n E Z, and hence y~ is also in Z, since we just showed that y~ is in iQ! and the equation of E' shows that y~ is the square root of an integer. To summarize, we have proven that Xn
E
Z
=}
P~ =
F(Pn ) E E'(Z).
(6.47)
6.8. Affine Morphisms and Commuting Families
375
= x([mk]p ) = Xmk, and hence
By con struction we have qi (()
#(()
from (6.47),
:s: #{n ? 0 : P~ E E' (Z )} < cv (E' )+l
from (6.46),
= cv (E )+l
since E and E' are isomorphic.
Further, if the ABC conjecture is true, then we may replace the upper bound by C.
o Remark 6.71. We note that something like the affine minimality of ¢ is necessary in the statement of Theorem 6.70 . Indeed, without some kind ofminimality condition, we saw in Proposition 3.46 that we can make # (() (() n Z ) arbitrarily large by replacing ¢(z) with B¢(B- l z) . This conjugation has the effect of multiplying every point in the orbit by B, hence allow s us to clear an arbitrary number of denominators. Remark 6.72 . Continuing with notation from the statement of Theorem 6.70, we note that there is a cutoff value ko such that
¢k(() E Z
for
O :S: k
:s: k o
and
¢k(( ) ¢:c Z
for
k
> k o.
This reflects the more general fact that if x( [n] P) E Z and if r ln, then x( [r]P) E Z ([4 10, Exerc ise 9.12]). Note that no such cutoff statement holds for general rational maps that are not non-Lattes map s.
6.8
Affine Morphisms, Algebraic Groups, and Commuting Families of Rational Maps
Power maps and Chebyshev maps are attached to endomorphisms of the multiplicative group lG m and its quotient lG m I {z = z - l }, and similarly Lanes maps are attached to maps of quotients of elliptic curves. In this section we put these constructions into a general context and state a classical theorem on commutativity of one-variable rational maps.
Definition. Let G be a commutative algebraic group. An affine morphism of G is the composition of a finite endomorphism of degree at least 2 and a translation.
Remark 6.73. The reason for this terminology is as follows. Let G Ie be a connected commutative algebraic group of dimension g. Then its universal cover is e 9 and every affine morphism 'ljJ : G --+ G lifts to an affine map 9 --+ 9 , i.e., there are a matrix A and vector a such that the following diagram commutes:
e
e9
z ....... A z + a )
e9
1
1
G
G
e
376
6. Dynamics Associated to Algebraic Groups
Example 6.74. Every affine morphism of the multiplicative group G m has the form 'ljJ(z) = az d for some nonzero a and some dE Z. More generally, for any commutative group G, any a E G, and any dE Z there is an affine morphism 'ljJ(z) = az", Notice that it is easy to compute the iterates of this map,
Proposition 6.75. Let ib : G ----7 G be an affine morphism ofan algebraic group G, so 'ljJ has theform 'ljJ(z) = a· a(z)for some a E End(G) and some a E G. (a) The endomorphism a and translation a are uniquely determined by 'ljJ. (b) Let a and a be as in (a). Then the iterates of i] have the form
Proof The definition of affine morphism tells us that there are an element a E G and an endomorphism a of G such that the map 'ljJ has the form 'ljJ(z) = aa(z). Evaluating at the identity element e E G yields 'ljJ(e) = aa(e) = a, so a is uniquely determined by 'ljJ. Then a(z) = a-I'ljJ(z) is also uniquely determined by 'ljJ. This proves (a). The proof of (b) is an easy induction, using the commutativity of G and 0 the fact that a is a homomorphism.
Definition. A self-morphism of an algebraic variety ¢ : V ----7 V is dynamically affine if it is a finite quotient of an affine morphism. What we mean by this is that there are a connected commutative algebraic group G, an affine morphism 'ljJ : G ----7 G, a finite subgroup I' c Aut (G), and a morphism G If ----7 V that identifies G If with a Zariski dense open subset of V (possibly all of V) such that the following diagram is commutative: G
7jJ ------7
1 cit
1
1
------7
1
V
G
ott
(6.48)
1 1
q, ------7
V
Example 6.76. Examples of dynamically affine rational maps ¢ : pI ----7 pI include the power maps ¢( z) = z" with G = Gm and I' = {I}, the Chebyshev polynomials Tn(z) with G = G m and I' = {z, Z-I}, and Lattes maps with G an elliptic curve E and f a nontrivial subgroup of Aut(E).
Proposition 6.77. Let ¢ : V
----7 V be a dynamically affine map and let 'ljJ : G ----7 G and I' c Aut( G) be the associated quantities fitting into the commutative diagram (6.48). (a) For every ~ E I' there exists a unique E I' with the property that 'ljJo~ = o'ljJ. (b) Write'ljJ(z) = a . a(z) with a E G and a E End(G) as in Proposition 6.75. Then ~(a) = afor every ~ E f.
e
e
6.8. Affine Morphisms and Commuting Families
377
(c) Assume that #f ::::: 2 and that C is simple. (An algebraic group is simple if its only connected algebraic subgroups are {I} and C.) Then a E C tars, i.e., the translation used to define 'Ij; is translation by a point offinite order. Proof (a) The uniqueness is clear, since if'lj; 0 ~ = 6 o'lj; = 6 0 sb, then 6 = 6 because the finite map 'Ij; : C ----+ C is surjective. We now prove the existence. The commutativity of (6.48) tells us that for all z E C and all ~ E I',
(n
0
'Ij; 0 O(z) = (¢ 0 n
0
~)(z) =
(¢ 0 n)(z) = (n 0 'Ij;)(z).
Thus ('Ij;o~)(z) and 'Ij;(z) have the same imagefor the projection map n : C ----+ C If, so there is an automorphism E f satisfying
e
'Ij;(~(z))
= (('Ij;(z)).
e,
We claim that the automorphism which a priori might depend on both ~ and z, is in fact independent of z. To see this we fix ~ and write 'Ij; (~( z)) = ~~ ( 'Ij; (z)) to indicate the possible dependence of on z. In this way we obtain a map (of sets)
e
C
------t
I',
Since I' is finite, there exists some C E f such that ~~ = C for a Zariski dense subset of z E C. (Note that a variety cannot be a finite union of Zariski closed proper subsets.) It follows that 'Ij; 0 ~ is equal to Co 'Ij; on a Zariski dense subset of C, and hence they are equal on all of C. (b) From (a) we see that there is a permutation off defined by the rule T :
I'
------t
I',
Evaluating both sides of'lj; 0 ~ = T(~) o'lj; at the identity element 1 E C and using the fact that ~(1) = 1 and 'Ij;(1) = a· a(l) = a, we find that
a = 'Ij;(~(1)) = T(O('Ij;(l)) = T(O(a). But T is a permutation of I', so as ~ runs over I', so does T (0. Hence a is fixed by every element off. (c) From (b) and the assumption that #f ::::: 2, there exists a nontrivial ~ E I' with ~(a) = a. It follows that a is in the kernel of the endomorphism
C
------t
C,
The kernel is not all of C, since ~ is not the identity map, so the simplicity of C tells 0 us that the kernel is a finite subgroup of C. Hence a has finite order. Remark 6.78. In this book we are primarily interested in dynamically affine maps of pI, but higher-dimensional analogues, especially of Lanes maps, have also been studied. See for example [68, 134, 145,439].
6. Dynamics Associated to Algebraic Groups
378
The commutativity of (6.48) implies that deg(¢) = deg(7jJ). It follows that all dynamically affine maps for the additive group G a have degree 1, since every affine morphism ofG a has the form 7jJ(z) = az + b. Hence nonlinear dynamically affine maps on pI are attached to either the multiplicative group G m or to an elliptic curve, since these are the only other algebraic groups of dimension 1. We note that over a field of characteristic 0, the endomorphism ring End(G) of a one-dimensional algebraic group G is commutative.' More precisely, the multiplicative group has endomorphism ring End(G m ) = Z, and the endomorphism ring End(E) of an elliptic curve E is either Z or an order in a quadratic imaginary field. The commutativity of End( G) means that dynamically affine maps commute with many other maps. An appropriately formulated converse of this statement is a classical theorem of Ritt. Theorem 6.79. (Ritt and Eremenko) Let ¢, 7jJ E q z) be rational maps ofdegree at least 2 with the property that ¢ 0 7jJ = 7jJ 0 ¢. Then one ofthe following two conditions is true: (a) There are integers m, n 2: 1 such that ¢n = ib": (b) Both ¢ and 7jJ are dynamically affine maps, hence they are either power maps, Chebyshev polynomials, or Lottes maps. In all cases, the commuting maps ¢ and 7jJ satisfy
F(¢)
= F(7jJ),
:J(¢) = :J(7jJ),
and
PrePer(¢) = PrePer( 7jJ).
Proof The first part of the theorem, in somewhat different language, is due to Ritt [371]. See Eremenko's paper [152] for a proof of both parts of the theorem and some additional geometric dynamical properties shared by commuting ¢ and 7jJ. A higher-dimensional analogue is discussed in [135]. We remark that the equality PrePer(¢) = PrePer( 7jJ) is a formal consequence of the commutativity of ¢ and 7jJ and the fact that the preperiodic points of a nonlinear rational map are isolated; see Exercise 1.15. D
Although we do not give a proof of Ritt's theorem, we conclude this section by proving the easier statement that only polynomial maps can commute with polynomial maps. This result was used in our description of the rational maps commuting with the Chebyshev polynomials (Theorem 6.9). Theorem 6.80. Let K be afield, let ¢(z) E K[z] be a polynomial ofdegree d 2: 2, and let 7jJ(z) E K(z) be a nonconstant rational map. We assume that both ¢ and 7jJ are separable, i.e., neither ofthe derivatives ¢' (z) and 7jJ' (z) is identically O. Suppose further that ¢ and 7jJ commute under composition, ¢ 0 7jJ = 7jJ 0 ¢. Then one of the following is true: (a) 7jJ(z) E K[z], i.e., 7jJ is also apolynomial. 3Evenin characteristicp, most elliptic curves have commutative endomorphism ring. However, there are a finitenumberof ellipticcurveswhoseendomorphism ring is a maximalorder in a quatemionalgebra. These supersingular curves are all definedover IFp2. See [410, V §3].
379
6.8. Affine Morphisms and Commuting Families
(b) After simultaneous conjugation by an affine map f (z) z + (3. the polynomial ¢(z) has the form ¢(z) = azd and the rational map 'ljJ(z) has the form 'ljJ(z) = bz" for some r < O.
Proof The proof is an application of ramification theory and the Riemann-Hurwitz formula (Theorem 1.1). By assumption, the map ¢ is a polynomial, so 00 is a totally ramified fixed point of ¢. Suppose that 'ljJ(z) is not a polynomial. This means that we can find a point a E 'ljJ-l(00) with a=/=- 00. We use the commutativity of ¢ and 'ljJ to compute ea(¢n
0
'ljJ)
= ea('ljJ)
n-l rr eqhj;(a)(¢) = ea('ljJ)eoo(¢t = ea('ljJ)dn i=O
(6.49)
II ea('ljJ Hence
0
¢n) =
Cg
e¢i(a)(¢)) e¢n(a) ('ljJ).
nrr-l e¢i(a)(¢) __ i=O
d
ea('ljJ) > _1_ e¢n (a) ('ljJ) - deg 'ljJ
for all n :::: 1.
(6.50)
Every ramification index e¢i(a) (¢) is an integer between 1 and d, so letting n --+ 00, we see that e¢i(a)(¢) = d for all sufficiently large i. On the other hand, ¢ is a polynomial and a =/=- 00, so ¢i (a) =/=- 00 for all i. Hence there is at least one point (3 =/=- 00 with e{3 (¢) = d. The Riemann-Hurwitz formula then implies that (3 and 00 are the only two points at which ¢ is ramified. It follows that ¢i (a) = (3 for all sufficiently large i, which implies that ¢((3) = (3. In other words, (3 and 00 are both totally ramified fixed points of ¢, i.e., (6.51 )
and
and ¢ has no other ramification points. In particular, since by construction we have ¢i(a) = (3 for some i, it follows that a = (3. But a =/=-00 was an arbitrary point in 'ljJ-l(00), so we have also proven that 'ljJ-l(00) C {(3, oo}. Next let, E 'ljJ-l((3). We use the fact that ¢i'ljJ(r) = (3 and ee (¢) = d to repeat the calculation (6.49) with a replaced by v. This again leads to the inequality (6.50), but with, in place of a, and hence to the conclusion that ¢ is totally ramified at some iterate ¢i (,). It follows that ¢i (,) E {(3, 00} for some i, and hence from (6.51) that, E {(3,00}. We have now proven that
'ljJ-l({(3,OO}) C {(3,00}. Thus 'ljJ is totally ramified at (3 and 00, and since 'ljJ((3) =/=- 00 by assumption, the map 'ljJ must switch (3 and 00. Since we also know that (3 and 00 are totally ramified fixed points of ¢, it follows that ¢ and 'ljJ have the form
¢(z) = (3 + a(z - (3)d Then conjugation by f(z)
and
'ljJ(z) = (3 + b(z - (3y
= z + (3 puts them into the desired
for some r form.
< O. D
380
Exercises
Exercises Section 6.1. Power Maps and the Multiplicative Group
6.1. (a) (b) (c)
Let K be a field of positivecharacteristic p. Let Mp(z) = z", Prove that the automorphism group of Mp over K equals PGL 2(lFp ) . More generally, if q is a power of p, prove that Aut(Mq ) = PGL2(lFq). Again let q be a power of p, and let d be an integerwith p f d. Describe Aut(Mq d ) .
6.2. Let K be an algebraically closed field, let d E Z, and let a E K*. Further, if K has positive characteristic p, assume that p f d. Describe all rational functions ¢(z) E K (z) that commutewith az d under composition. 6.3. ** Let Md(z) = zd be a power map for some Idl 2: 2, and if K has positivecharacteristic p, assume that p f d. Example6.5 describestwo types of twists of Md(Z). The first type has the form ¢a(Z) = az d and the second type 'l/Jb(Z) is given by the complicatedformula (6.1). Does Md(z) have any other twists? If so, describe all of the twists of Md(z). Section 6.2. Chebyshev Polynomials
6.4. Completethe proof of Proposition6.6(e) by verifyingthe identity 1 [( d+ 1) (d + 1 - k) + d(d + 1 - k)] = d+ 2 (d + 2 - k) . d+l-k k k-l d+2-k k
6.5. Let Td(w) be the dth Chebyshevpolynomial for some d 2: 2. (a) Prove that the fixed points ofTd( w) are as describedin Proposition6.8(a). (b) Prove that the multipliers of Td(W) at its fixed points are as described in Proposition 6.8(b). (Hint. For (a) use the trigonometric identity cos(A) _ cos(B) = sin ( B ; A) sin ( B ; A) , and for (b) differentiate Td(z
+ z ~ 1) = Zd + Z-d /
Zd - Z-d
-1
Td(z
to obtain the identity
+Z )=d z _
Z-1 .)
6.6. Proposition 6.8 describes the multipliers of the fixed points of the Chebyshev polynomial Td(w). Prove directly that the multipliers satisfy the summation formula described in Theorem 1.14,
L
(EFix(Td)
= 1.
1 1-
ATd (()
6.7. Let K be a field of characteristic p 2: 3, let n 2: 1 be an integer with p f n, and let I-t n C K* be the nth roots of unity. There is no "cosine function" for tile field K, but we can define a set of cosine values by Cos.,
=
a + a2 {
We also let 2 Cos; = {2( : ( E Cosn } . Let d 2: 2 be an integer with p f d(d2 mial.
-
1
: a E I-t n
}
.
1) and let Td(w) be the a;h Chebyshevpolyno-
381
Exercises (a) Prove that Fix(Td)
= 2 COSd+l U2 COSd-1 .
Also compute the intersection 2 COSd+1 n2 COSd-l. (b) Prove that the multipliers of Td at its fixed points are given by if ( E 2 COSd+1 and ( if ( E 2 COSd-1 and (
f ±2, f ±2,
if( = ±2. (c) Give a similar description of the fixed points and their multipliers in the case that d == ±1 (modp). 6.8. We stated during the proof of Theorem 6.9 that for d ~ 2, the Chebyshev polynomial Td(W) is not equivalent to a monomial, i.e., no conjugate (i-I 0 Td 0 f)(w) has the form cu/". Prove this assertion. 6.9. Prove that the (formal) derivatives of the Chebyshev polynomials satisfy the following identities:
+ dwTd(W) = 2dTd-dw). (4 - w2)T~f(W) - wT~(w) + d2Td(W) = O.
(a) (4 - w2)T~(w)
(b)
6.10. Let K be a field of positive characteristic p. (a) Prove that the pth Chebyshev polynomial Tp(w) is equal to w P in K[w]. q) (b) In general, if q is a power ofp, prove that Tqd(W) = Td(wF = Td(w for all d ~ 1. (c) Again letting q be a power of p, deduce that Aut(Tq) = PGL2 (IF q). (Cf. Exercise 6.1.) 6.11. Let K be a field of characteristic 2 and let d ~ 1 be an odd integer. Prove that wT~ (w) = Td(w). What is the derivative T~ (w) if d is an even integer?
Section 6.4. tattes Maps - General Properties 6.12. Let E be the elliptic curve E : y2 = x 3 + 1 with j(E) = 0, so Aut(E) = 1-t6 is cyclic of order 6. Let 1/J(P) = [2]P be the doubling map. (a) Let 7f : E --> E / 1-t2 ~ pl. Prove that we can take 7f( X, y) = x and that the Lattes map corresponding to 1/J is
=
tih(z)
Z(Z3 - 8b) 4(z3 + b) .
(b) Let 7f : E --> E / 1-t3 ~ pl. Prove that we can take corresponding to 1/J is A,
(
)
=
Z4
,+,2 Z
7f (
+ 18bz2 8z 3
x, y) = y and that the Lattes map
2 27b
(c) Let 7f : E --> E/l-t6 ~ pl. Prove that we can take 7f(x,y) = x 3 and that the Lattes map corresponding to 1/J is
z(z - 8b)3 cP3(Z)
= 64(z + b)3
Compute the conjugate cP3 (z - b) + b of cP3 (z), compare it to cP2 (z), and explain. 6.13. Let cP be a Lattes map. Prove that there does not exist a linear fractional transformation f E PGL 2 such that the conjugate cP f is a polynomial. (Cf. Exercise 6.8.)
382
Exercises
6.14. Complete the proof of Proposition 6.44 in the general case that 'l/J(P ) = a( P ) + T with a E End (E) and T EE not necessarily equal to O . However, you may assume that T E Etars, i.e., T is a point of finite order. 6.15. Let ¢ : C --.... C be a nonconstant rational map from a smooth curve to itself. Recall that Critval, denotes the set of critical values of ¢. Prove that 00
00
n =O
n= l
Section 6.5. Elliptic Curves and Flexible Lattes Maps 6.16. Let E be an elliptic curve and let Jr : E --.... p I be a map of degree 2. (a) Let R be any point on E . Show that we can define a new group law (call it *) on E by the rule
P*Q = P+Q-R. Show that R is the identity element for the group (E, *). (b) Prove that there exists a point Po E E such that tt ( - (P + Po)) = tt (P + Po) for all PEE. (c) Conclude that after choosing a new identity element for E, the map Jr is even, i.e., satisfies Jr (P ) = Jr ( - P ) for all PEE. 6.17. Fix an elliptic curve E and and a degree-2 map Jr : E --.... p I satisfying Jr (P) = Jr(- P ). For any integer m and any point T E E [2], let ¢m,T : pI --.... p I be the flexible Lattes map associated to the map 'l/J(P ) = [m](P ) + T as in the commutative diagram (6.22). (a) Prove that ¢m,TO¢m' ,T' = ¢mm',mT'+T . In particular, the maps ¢m,O commute under composition. (b) Prove that cP':n.,T is either cPmn,T or cPmn, O. More precisely, if m is odd and n is even, prove that ¢':n. ,T = ¢m n,O, and prove in all other cases that ¢':n.,T = ¢mn,T . (Of course, ifT = 0, the cases are all the same.) (c) It follows from (a) that the collection of maps { ¢m,T : m ~ 1, T E E[2]} is closed under composition. Prove that cP I,O is the identity element and that the associative law holds. Thus this set of flexible Lattes maps for E is a noncommutative monoid. 6.18. Let ¢ : pI --.... pI be a flexible Lattes map associated to 'l/J (P ) point T E E[2] is not necessarily equal to O. (a) Prove that the set of fixed points of ¢ is given by Fix(¢)
= x([m-
= [m]P + T, where the
lr I(T)) u x([m + l r I(T)).
(b) Compute the multiplier of ¢ at each point in Fix(¢). (c) Use the results from (a) and (b) and the formula for the composition of Lattes maps in Exercise 6. I7 to describe the periodic points of ¢ and to compute their multipliers. (Hint. Mimic the proof of Proposition 6.52, which dealt with the case T = 0 .) 6.19. Proposition 6.52 describes the multipliers of a flexible Lanes map. Using these values, verify directly that the formula
from Theorem 1.14 is true for flexible Lattes maps.
Exercises
383
6.20. Complete the proof of Proposition 6.52(b) by computing the multiplier A¢(00) at the fixed point 00 = x(O). (Hint. Move 0 to (0,0) using the change of variables z = x/y and w = l/y. Then write the invariant differential in terms of z and wand mimic the proof in the text.) 6.21. Let K be an algebraically closed field and let ¢ and ¢' be flexible Lattes maps defined over K that are associated, respectively, to elliptic curves E and E'. Suppose that ¢ and ¢' are PGL 2(K)-conjugate to one another. We proved (Theorem 6.46) that if the characteristic of K is not equal to 2, then E and E' are isomorphic. What can be said in the case that K has characteristic 2? (Note that in characteristic 2 it is necessary to use a generalized Weierstrass equation (6.12) to define E.) 6.22. We proved Proposition 6.55 in the case that 'IjJ( P) = [m](P). (a) Prove Proposition 6.55 for general flexible Lattes maps, i.e., Lattes maps associated to maps oftheform 'IjJ(P) = [m](P) + Twith T E E[2]. (b) Formulate and prove a version of Proposition 6.55 for rigid Lattes maps. (c) ** To what extent is the converse of Proposition 6.55 true? More precisely, if ¢ is a Lattes map fitting into a reduced Lattes diagram (6.37) and if ¢f has bad reduction for every f E PGL 2(K), does the elliptic curve E necessarily also have bad reduction? 6.23. Let E be an elliptic curve given by a Weierstrass equation
E : y2 = x
3
+ ax + b.
Let m ~ 1 be an integer and write x ([m] p) as a quotient of polynomials
_ Fm(x(P)) x ( [mJP) c.; ( x(P) ).
(6.52)
(a) Prove that Fm and G,« can be taken to be polynomials in x, a, and b. More precisely, prove that there are polynomials Fs«, G,« E Z[a, b, x] satisfying (6.52) and that they are uniquely determined by the requirement that F m be monic in the variable x. (b) Prove that deg(Fm ) = m 2 and deg(G m ) = m 2 - 1 and that their leading terms m2 + ... and Gm(x) = m 2xm2- 1 + .... are Fm(x) = x (c) If m is odd, prove that there is a polynomial 'ljJm(X) E Z[a, b,x] such that Gm(x) = 'ljJm(X)2. Similarly, ifm is even, prove that there is a polynomial 'ljJm(X, y) E Z[a, b,x, y] such that G m (x) = 'ljJm (x, y?, where in the computation we replace y2 by x 3 + ax + b. The polynomial 'ljJm is called the m" division polynomial for E, since its roots are the nontrivial points of order m. (d) Prove that Fm and G m satisfy 2
m Fm(a,b;x) Fm(t 2 a,t 3 b;tx)=t
and
2 G m (2 ) t a,t 3 b;tx ) =t m _ 1 G m ( a,bjx.
Thus Fm and G m are homogeneous if x, a, and b are respectively assigned weights 2, 4, and 6. (e) Let !::>.(E) = -16(4a 3 + 27b2 ) . Prove that the resultant of F m and G m with respect to the variable x is given by
384
Exercises
Section 6.6. Elliptic Curves and Rigid t.attes Maps 6.24. This exercise extends Proposition 6.61. Let ¢ : jp'l --> jp'l be a Lattes map and fix a reduced Lattes diagram (6.37) for ¢. Write 'IjJ(P) = [o:](P) + T as usual, where we use the standard normalization described in Proposition 6.36 to identify End(E) with a subring
of C. (a) Prove that the fixed points of ¢ are given by Fix(¢)
= U{1r(P) : [0: - ~](P) = -T}.
(6.53)
~Er
(b) Let 1r(P) E Fix(¢). Prove that P is a critical point for 1r if and only if P is fixed by a nontrivial element of~. More generally, prove that the ramification index is given by
ep(1r) = {~E I": [~]P =
pl.
In particular, if P is not a critical point, then there is a unique ~ E I' that fixes P. (c) Assume that T = O. Let 1r(P) E Fix(¢) and choose some automorphism ~ E f such that 'IjJ(P) = [WP). Compute the multiplier of ¢ at 1r(P) as in the following table (we have given you the first four values):
ifep(1r) = 1, iff
= JL2 and ep(1r) = 2,
iff = JL3 and ep(1r) = 3, iff = JL4 and ep(1r) = 2, iff = JL4 and ep(1r) = 4, iff
= 1-L6 and ep(1r) = 2,
iff = 1-L6 and ep(1r) = 3. 6.25. In the text we proved Theorem 6.62 under the assumption that d is squarefree. This exercise sketches an argument to eliminate the squarefree hypothesis. We set the notation S(b) for the squarefree part of the integer b 2 1. (a) For each integer d 2 2, let Dd 2 1 be an integer with the property that d is a norm from the ring Z [vi- D d] down to Z. In other words, there are integers u and v such that
u
2
+ D d v 2 = d.
Then with notation as in the statement of Theorem 6.62, prove that for every is a constant CE such that
E
> 0 there
for all d.
(Hint. Use elliptic curves with CM by the ring Z [vi- Dd] and Lattes maps associated to the endomorphism (b) Prove that for every E
[u + vvl- D d ]
and follow the proof of Theorem 6.62.)
> 0 there is a constant C~ > 0 such that max S(d - u 2 )
2
C~dl-E
for all d.
O:O;u<.Jd
(Hint. It suffices to prove that for sufficiently large d the average satisfies
Exercises
385 ~ c:
va1
2
logS(d-u):::: (l-E)logd.
O"Ou
l
Write this as two sums using ord, (S( b)) = ord, (b) - 2 ~ ord, (b)J and show that the first sum is asymptotic to log (d) and the second is bounded as d ---> 00.) (c) Combine (a) and (b) to complete the proof of Theorem 6.62. Section 6.7. Uniform Boundedness for t.attes Maps
6.26. Let E /Q be given by a quasiminimal Weierstrass equation
E : y2
= x
3
+ ax + b,
i.e., the discriminant 116(4a3 + 27b2 ) I is minimized subject to the condition that a and bare integers. (a) Show that the equation for E is minimal at every prime p :::: 5. (b) Let 113(E) be the discriminant of a general Weierstrass equation (6.12) for E that is minimal at 3. Prove that
0::; ord 3(4a 3
+ 27b2 ) -
ord, ( 113(E))
< 6.
(c) Let 112(E) be the discriminant of a general Weierstrass equation (6.12) for E that is minimal at 2. Prove that
6.27. Theorem 6.70 suggests that there should be an absolute upper bound for the number of integer points in the orbits of affine minimal Lattes maps defined over Q. (a) Let E be the elliptic curve
E : y2 = x 3
-
48907
+ 8481094
and let ¢(x) be the Lattes map associated to multiplication-by-2, i.e., fox = x 0 [2]. Verify that the orbit 0(2363) contains five integer points, but that ¢5(2363) tf- 2::. Also verify that there is a point in E(Q) with x-coordinate 2363. (b) Let E be the elliptic curve
E : y2 = x 3
-
40467
+ 4120274
and again let ¢(x) be the Lattes map associated to multiplication-by-2. Verify that the orbit 0(193) contains five integer points, but that ¢5(193) tf- 2::. In this case E(Q) does not contain a point with x-coordinate equal to 193. (c) ** Find an affine minimal Lattes map ¢ and an initial point ( E 2:: such that ¢5 (() E 2::, or prove that none exist. (This will force ¢k (() E 2:: for all 0 ::; k ::; 5; see Remark 6.72.) 6.28. Prove a version of Theorem 6.70 for a Lattes map satisfying
¢(x(P))
=
x ([m]P + T)
for all PEE,
where T is a fixed 2-torsion point of E. (You may need a more general version of Theorem 6.69; see [202,407].)
Chapter 7
Dynamics in Dimension Greater Than One Up to this point our primary focus has been on arithmetic dynamics of rational maps on JlDI . In this chapter we take a look at dynamics in higher dimension s. Even over
387
388
7. Dynamics in Dimension Greater Than One
7.1 Dynamics of Rational Maps on Projective Space Recall that a rational map
Example 7.1. The rational map (7.1) is not a morphism, since it is not defined at the point [0,1,0]. Notice that if we discard [0,1,0], then
°
=
[a 2 , ob, a2 ]
=
[a, b, a].
Thus
h(
7.1.1
Affine Morphisms and the Locus of Indeterminacy
In this section we study rational maps lP'N - t lP'N with the property that they induce morphisms of affine space Pi" N - t Pi" N. Concretely, an affine morphism
is a map of the form
7.1. Dynamics of Rational Maps on Projective Space
389
To avoid trivial cases, we generally assume that at least one of the F; is not the zero polynomial. Definition. The degree of a polynomial
il, ... ,iN
is defined to be
In other words, the degree of F is the largest total degree of the monomials that appear in F. (By convention the zero polynomial is assigned degree -00.) The degree ofa morphism ¢ = (F l , ... , F N) : AN ---+ AN is defined to be deg ¢
= max{ deg F l , ... , deg FN}.
Homogenization of the coordinates of an affine morphism ¢ : AN ---+ AN of degree d yields a rational map ¢ : lP'N ---+ lP'N of degree d. For each coordinate function F; of ¢, we let
Notice that each Pi is a homogeneous polynomial of degree d (or the zero polynomial), so the map dN N ¢=[XO , F l , F 2 , .•. ,FNJ:lP' ---+lP' is a rational map of degree d. We call map need not be everywhere defined. Definition. Let ¢ : AN
---+
¢ the rational
map induced by ¢. A rational
AN be an affine morphism of degree d and let
be the rational map that it induces. The locus ofindeterminacy of ¢ is the set
(To ease notation, we write ¢ and Z (¢) instead of ¢ and Z (¢).) This is the set of points at which ¢ is not defined. Notice that Z(¢) lies in the hyperplane Hi, {X o = O} at infinity, since ¢ is well-defined on AN. The polynomials PI, ... ,PN can be used to define a morphism
The map is called a lift of ¢. If we let
7r
be the natural projection map,
7. Dynamics in Dimension GreaterThan One
390
then
11" ,
cI>, and
¢ fit together into the commutative diagram
Example 7.2. The map
induces the rational map
and has indeterminacy locus Z (¢)
7.1.2
= {[O,0, I]} consisting of a single point.
Affine Automorphisms
Of particular interest are affine morphisms that admit an inverse.
Definition. An affine morphism ¢ : AN -; AN is an automorphism if it has an inverse morphism. In other words, ¢ is an affine automorphism if there is an affine morphism ¢- I : A N -; AN such that
Somewhat surprisingly, ¢ and ¢-I need not have the same degree , nor does deg(¢n ) have to equal (deg¢) n .
Example 7.3. Consider the map ¢(x, y ) = (x, y + x 2 ) . It has degree 2 and is an automorphi sm, since it has the inverse ¢- I (x , y ) = (x, y - x 2 ) . The composition ¢2 IS
¢2(x l y)
= ¢(x , y + x 2 ) = (x, Y + 2x 2 ) ,
so deg(¢2) = 2 = deg(¢). More generally, ¢n( x, y) = (x ,y + nx 2 ) has degree 2, so the degree of ¢n does not grow. This contrasts sharply with what happens for morphism s of jp'N.
Example 7.4. Let a E K * and let f (y ) E K [y] be a polynomial of degree d
~
2.
The map
¢(x, y) = (y , ax
+ f(y )),
is called a Henan map. It is an automorphism of A 2 , since one easily checks that it has an inverse ¢ - I given by
7.1. Dynamics of Rational Maps on Projective Space
391
Henon maps, especially those with deg(f) = 2, have been extensively studied since Henon [200] introduced them as examples of maps R'' -7 ]R2 having strange attractors. There are many open questions regarding the real and complex dynamics of Henon maps; see, for example, [132, §2.9] or [211], as well as [212,413] for a compactification of the Henon map. The rational maps J!D2 -7 J!D2 induced by ¢ and ¢-l are
¢([Xo, Xl, X 2]) = [xg, Xg- lX 2, aXg-lXl + f(XO,X2)], ¢-l ([X o, Xl, X 2 ]) = [xg, a-I xg- l X 2 - a-I f(x o, Xd, xg- l Xd, where we write f(u, v) = ud f (v/ u) for the homogenization of f. It is easy to see that the loci of indeterminacy of ¢ and ¢-l are
Z(¢)
= {[a, 1, oj}
and
Z(¢-l) = {[O,O, 1J}.
In particular, the locus of indeterminacy of ¢ is disjoint from the locus of indeterminacy of ¢ -1. Maps with this property are called regular; see Section 7.1.3. Example 7.5. Consider the very simple Henon map
¢(x, y) = (y, -x + y2). The extension ¢ = [xg, X OX2 , - XOXI + Xi] of ¢ to J!D2 has degree 2, but it is not a morphism, since it is not defined at the point [0, 1,0]. And just as in Example 7.2, there is no height estimate of the form h (¢( P)) = 2h(P) + 0(1) for ¢. We can see this by noting that
¢-( [b, a, b] )
=
[b 2,b2,-ab + b2] = [b, b, -a + b],
so if a,b, E Z with gcd(a, b) = 1 and b > a > 0, then [b, a, bJ and ¢([b,a, bJ) have the same height. Hence for every E > 0 even the weaker statement
h(¢(P)) ::::: (1 + E)h(P)
+ 0(1)
for all P = (x, y) E A 2(Q)
is false. It turns out that ¢ has only finitely many Q-rational periodic points (Theorem 7.19), but the proof does not follow directly from a simple height argument. Example 7.6. More generally, if ¢ : AN -7 AN is an affine automorphism, then it is not possible to have simultaneous estimates of the form
h(¢(P)) ::::: (1
+ E)h(P) + 0(1),
h(¢-l(p)) ::::: (1 + E)h(P)
+ 0(1),
(7.2)
for some E > 0 and all P E AN (K). To see this, suppose that (7.2) were true. Then we would have for all P E AN (K),
392
7. Dynamics in Dimension Greater Than One
Thus h(P ) would be bounded, leading to the untenable conclusion that AN (K ) is finite. So it is too much to require that both ¢(P) and ¢- l(p ) have heights larger than the height of P. However, as we shall see, it is often possible to show that some combination of h( ¢(P )) and h( ¢-I (P )) is large, which is then sufficient to prove that P er ( ¢) is a set of bounded height. We conclude this section with two useful geometric lemmas . The first relates the locus of indeterminacy of an affine automorphism and its inverse, and the second characterizes when the degree of a composition is smaller than the product of the degrees .
Lemma 7.7. Let ¢ : AN ----; AN be an affine automorphism ofdegree at least 2 and denote the hyperplane at infinity by Hi, = {X o = O} = IF'N <, AN, Then
Proof Let d
-
-
-
cI> = (Xo,Fl ,Fz, ... , FN)
and
cI>
-1
e
-
-
-
= (XO ,Gl ,GZ, .. . , G N )
be the lifts of ([> and ([>- 1, respectively. The fact that ¢ and ¢ - l are inverses of one another implies that there is a homogeneous polynomial f of degree de - I with the property that
xge, so we see that f = xge- Thus (xge, xge- l Xl , xge- l X l , .. . ,xge- l X N ), 1
But the first coordinate of the composition is (cI>- 1 0 cI» (Xo, . . , , X N ) =
.
or equivalently, -
d
-
-
Gj(XO' F l , · .. , F N )
= X od e-l x,
for aliI 5:. j 5:. N.
Now let P = [0, Xl, .. " XN] E Hi, -, Z(¢), so ([>(P) with at least one Fi(P) # O. From (7.3) we see that
=
(7.3)
[0, F\ (P), ... , FN(P )]
Hence
so ([>- 1 is not defined at ([>( P) . Therefore ([>(P) E Z (¢ - l ).
o
Lemma 7.8. Let ¢ : AN ----; AN and 1fJ : AN ----; AN be affine morphisms, and let H o = {X o = O} = IF'N -, AN be the usual hyperplane at infinity. Then
deg(1fJ 0 ¢) < deg(1fJ) deg(¢) if and only if ([> (H o " Z (¢ )) C Z (1fJ ).
7.1. Dynamics of Rational Maps on Projective Space
393
Proof Let d = deg( 1», let e = deg( 'ljJ), and let
if;,
respectively. We write
The composition \fJ
0
where £1, ... ,£ N are homogeneous polynomials of degree de. The degree of'ljJ 01> will be strictly less than de if and only if there is some cancellation in the coordinate polynomials of \fJ 0
0
1» < deg('ljJ) deg( 1»
{::::::}
X o divides £j for every 1 ::; j ::; N.
Suppose now that Xol£j for every j and let P Since ¢ is defined at P, some coordinate of
=
[0, Xl, ... , XN] E H o <, Z(1)).
is nonzero. On the other hand, the assumption that Xol£j implies that
(\fJ
0
(0, £1 (
£2(
Hence if; is not defined at ¢(P), so ¢(P) E Z( 'ljJ). This completes the proof that ifdeg('ljJ 0 1» < de, then ¢(Ho " Z(1))) C Z('ljJ). For the other direction, suppose that ¢(Ho " Z(1))) c Z('ljJ). This implies that for (almost all) points of the form (O,XI, ... ,XN), the map if; is not defined at the point ¢([O, Xl, ... , XN]). Hence
so £j(O, Xl, X 2 , ... , X N)
°
= for all j. Therefore Xol£j for all j.
D
Example 7.9. Let 1> be the map 1>(x, y) = (x, y+x 2 ) that we studied in Example 7.3. Dehomogenizing 1> yields
so the locus of indeterminacy for
1> is Z (1»
= {[ 0, 0, I]}. Notice that
Hence ¢(Hi, <, Z( 1») = Z( 1», so Lemma 7.8 tells us that deg( 1>2) < deg( 1»2. This is in agreement with Example 7.3, where we computed that deg( 1>2) = 2.
394
7. Dynamics in Dimension Greater Than One
7.1.3
The Geometry of Regular Automorphisms of AN
In this section we briefly discuss the geometric properties of an important class of affine automorphisms.
Definition. An affine automorphism ¢ : AN ----t AN is said to be regular if the indeterminacy loci of ¢ and ¢- 1 have no points in common,
The following theorem summarizes some of the geometric properties enjoyed by regular automorphisms of AN. We sketch the proof of (a) and refer the reader to [401] for (b) and (c).
Theorem 7.10. Let ¢ : AN (a) For all n 2:: 1,
----t
A N be a regular affine automo rphism.
¢n is regular,
and
(b) Let
d1
= deg e, d2 = deg ¢- l ,
£1
= dimZ(¢)+l ,
£2
= dimZ(¢-I )+1.
Then
£1 + £2 = N
and
d~l = dI2 •
(c) Forall n 2': 1 thesetofn -periodicpointsPer n (¢) is a discrete subset ofAN (C). Counted with appropriate multiplicities,
Proof (a) We first prove by induction on n that
This is trivally true for n = 1, so assume now that it is true for n - 1. Let P E Z (¢n), so in particular P E H«. Suppose that P ~ Z(¢). The induction hypothesis tells us that P ~ z(¢n-1), so applying Lemma 7.7 to the map ¢n-1, we deduce that
(For the last equality we have again used the induction hypothesis.) On the other hand, we have that ¢n- 1 is defined at P and ¢n is not defined at P, which implies that ¢n-1(p) E Z (¢ ). This proves that ¢n- 1(p) is in both Z (¢-l ) and Z (¢) , contradicting the assumption that ¢ is regular. Hence P E Z (¢ ), which completes the proof that z (¢n) c Z (¢ ). Similarly, we find that z (¢- n) C Z (¢-l ). Having shown that z (¢n) C Z (¢) and z (¢-n) C Z (¢- l ), we see that the regularity of ¢ implies that
7.1. Dynamics of Rational Maps on Projective Space
SO
395
¢n is also regular.
Next suppose that deg(¢n) < deg(¢)n for some n 2 2. We take n to be the smallest value for which this is true, so in particular deg( ¢n-1) = deg( ¢ )n-1, and hence We apply Lemma 7.8 with 7/J = ¢n-1 to conclude that
where the last inclusion was proven earlier. On the other hand, Lemma 7.7 says that ¢(Ho" Z(¢)) C Z(¢-l). Hence
¢(Ho" Z(¢))
C
Z(¢) n Z(¢-l)
=
0.
This is a contradiction, which completes the proof that deg( ¢n) It remains to show that Z (¢) C Z (¢n). Let
= deg( ¢ )n.
be a lift of ¢, so
By a slight abuse of notation, we say that P E Z (¢) if and only if cI>( P) = O. (To be precise, we should lift P to AN+1 .) We proved that deg( ¢n) = deg( ¢), which implies that the coordinate functions of cI>n have no common factor. Thus ¢n can be computed by evaluating cI>n and mapping down to lP'N. Hence just as above we have P E Z (¢n) if and only if cI>n (P) = O. Therefore
This proves that Z(¢) C Z( ¢n) and completes the proof of (a). (b) See [401, Proposition 2.3.2]. (c) See [401, Theorem 2.3.4].
o
Remark 7.11. If ¢ : A2 ---.. A2 is a regular automorphism of the affine plane, then Theorem 7.10(b) tells us that = = 1 (which is clear anyway since the indeterminacy locus of a rational map has codimension at least 2) and that d 1 = d2 . Thus planar regular automorphisms satisfy deg( ¢) = deg( ¢ -1). In the opposite direction, if d 1 = d2 , then Theorem 7.10(b) says that = and hence that N = + is even. In other words, a regular automorphism ¢ : AN ---.. AN with N odd always satisfies deg( ¢) i= deg( ¢ -1).
"1 "2
"1 "2,
"1 "2
7. Dynamics in Dimension Greater Than One
396 Example 7.12. Let ¢ : A 3
-.
A 3 be given by
One can check that the inverse of ¢ is
Homogenizing x = XI! X o, y = X 2 / X o, z = X 3 / X o, we have the formulas
from which it is easy to check that Z(¢) = {Xo = X 2 = X 3 = O] = {[O, 1,0, O]}, Z(¢-l) = {X o = Xl = O} = {[O,O,u,v]}. Thus Z (¢) consists of a single point, while Z (¢ -1) is a line. In the notation of Theorem 7.10, we have N = 3 and
d1 = deg e = 2,
£1 = dimZ(¢)
d2
+ 1 = 1,
The map ¢ is regular, since Z(¢)
= deg¢-l = 4,
£2 = dimZ(¢-l) = 1 = 2.
n Z(¢-l) = 0.
Remark 7.13. Let ¢ : AN -. AN be an affine morphism and let : A N+1 be a lift of ¢. The map ¢ is called algebraically stable if
n ({ X o = O}) -1= {O}
-.
A N+ 1
for all n 2: 1.
In other words, ¢ is algebraically stable if for every n 2: 1, some coordinate of n(xo, ... ,XN ) is not divisible by X o. Since the first coordinate of n is a power of X o, this implies that there can be no cancellation among the coordinates, so an algebraically stable map ¢ satisfies
Further, an adaptation of the proof of Theorem 7.1o(a) shows that
Regular automorphisms are algebraically stable, but there are algebraically stable automorphisms that are not regular. For a discussion of the complex dynamics of algebraically stable maps, see [174, 187, 401].
7.1. Dynamics of Rational Maps on Projective Space
397
Remark 7.14. For arbitrary rational maps ¢ : JP>N ----> JP>N, the dynamical degree of ¢ is defined to be the quantity
and its logarithm log dyndeg (¢) is called the algebraic entropy of ¢. (One can show that the dynamical degree is in fact the infimum of deg(¢ n)l / n.) The dynamical degree provides a coarse measure of the stable complexity of the map ¢ , and presumably it has a major impact on the arithmetic properties of ¢ . See [10, 199,290] for an indication of this effect in certain cases. The dynamical degree need not be an integer, or even a rational number; see Exercise 7.4 for an example . However, Bellon and Viallet [49] have conjectured that it is always an algebraic integer. The dynamical degree , and more generally the sequence of integers
dn = deg( ¢n),
n = 0, 1,2, ... ,
can be quite difficult to describe. See [10, 45, 46, 49, 77, 133, 199] for work on this problem . In many cases the sequence ( dn)n~o satisfies a linear recurrence with rational coefficients , or equivalently, the generating function L n>o dnT n is in ((Jl(T). However, see [46] for an example of a birational map ¢ : JP>N ~ JP>N whose degree generating function is not in ((Jl(T).
7.1.4 A Height Bound for Jointly Regular Affine Morphisms In this section we prove a nontrivial lower bound for the height of points under regular affine automorphisms. The theorem is an amalgamation of results due to Denis [131], Kawaguchi [230, 231], Marcello [287, 288, 289, 290], and Silverman [413,418]. Before stating the theorem , we need to define what is meant by the height of a point in affine space. Definition. The height h(P ) of a point P = (X l , ... , X N) E AN (Q) in affine space is defined to be the height of the associated point in projective space using the natural embedding AN ----> JP>N,
Eventually we will apply the following height estimate to a regular affine automorphism ¢ and its inverse ¢ - l , but it is no harder to prove the result for any pair ofjointly regular maps , and working in a general setting helps clarify the underlying structure of the proof. Theorem 7.15. Let ¢ l : AN the property that
---->
AN and ¢ 2 : AN
(We say that ¢l and ¢ 2 are jointly regular.) Let
---->
AN be affine morphisms with
398
7. Dynamics in Dimension Greater Than One and
There is a constant C
= C((P1, ¢2) such that for all P
AN (Q),
E
(7.4)
Remark 7.16. We recall that the upper bound
h(1j;(P)) ::; (deg1j;)h(P)
+ 0(1)
(7.5)
is valid even for rational maps 1j; : lP'N ~ lP'N (see Theorem 3.11), since the proof of (7.5) uses only the triangle inequality. Thus Theorem 7.15 may be viewed as providing a nontrivial lower bound complementary to the elementary upper bound
ProofofTheorem 7.15. Write the rational functions and ¢2 as
lP'N ~ lP'N
induced by ¢l
and where the Pi are homogeneous polynomials of degree dl and the Gi are homogeneous polynomials of degree d2 • The loci of indeterminacy of ¢l and ¢2 are given by
Z(¢d Z(¢2)
= =
{Xu {Xu
=
PI =
=
PN
=
Gl
=
GN
We define a rational map 1j; : lP'2N ~
lP'2N
=
= =
a}, a}.
of degree d l d2 by
The locus of indeterminacy of 1j; is the set
since by assumption Z (¢d and Z (¢2) are disjoint. Hence 1j; is a morphism, so we can apply the fundamental height estimate for morphisms (Theorem 3.11) to deduce that for all P E lP'2N (Q). (7.6) The following lemma will give us an upper bound for the height of 1j; (P).
Lemma 7.17. Let u, al, ... , aN, bi, ... , bN E Q with u i=-
a.
Then
399
7.1. Dynamics of Rational Mapson Projective Space Proof Let ai = aiJu and 13i we have the trivial estimate max{ 1, lall v , "
"
= bi/u for 1 :S i :S
N. Then for any absolute value v
laNlv, l13llv,"" I13Nlv} :S max{ 1, lall v , ... , laNlv} . max{ 1, l13llv, ... , I13Nlv}.
Raising to an appropriate power, multiplying over all absolute values, and taking logarithms yields
This is the desired result, since the height does not depend on the choice of homogeneous coordinates of a point. 0 We apply Lemma 7.17 to the point
with P E AN (Q), which ensures that Xo(P)
-I- O. The lemma tells us that
h('IjJ(P)) :S h ([XO(p)d 1d2 , FHP)d2 , . . . ,PN(p)d2 ])
+ h ([XO(p)d
1d2
c, (p)d
,
1
, •••
,GN(p)d 1 ])
= d2h ([ XO(p)d 1 , PI (P), ... ,PN(P)])
+ dlh ([XO(p)d Gl(P), ... ,GN(p)]) = d2h(r/>I(P)) + dlh(¢2(P)), 2
,
We combine this with (7.6) to obtain
Dividing both sides by d ld2 completes the proof of Theorem 7.15.
o
For regular affine automorphisms, it is conjectured that the height inequality (7.4) in Theorem 7.15 may be replaced by a stronger estimate.
Conjecture 7.18. Let ¢ : AN ----+ AN be a regular affine automorphism. Then there is a constant C = C(¢) such that for all P E AN (Q),
1
d h(¢(P)) l
1
+ d h(¢-l(p)) 2
~
(1 ) + 1
d ld 2
h(P) ~ C.
(7.7)
Kawaguchi [230] proves Conjecture 7.18 in dimension 2, i.e., for regular affine automorphisms ¢ : A 2 ----+ A 2 ; see also [413]. However, for general jointly regular affine morphisms, it is easy to see that (7.4) cannot be improved; see Exercise 7.8. Kawaguchi also constructs canonical heights for maps that satisfy (7.7); see [230] and Exercises 7.17-7.22.
400
7.1.5
7. Dynamics in Dimension Greater Than One
Boundedness of Periodic Points for Regular Automorphisms of AN
Theorem 7.15 applied to a regular affine automorphism ¢ and its inverse implies that at least one of ¢(P) and ¢-l(P) has reasonably large height. This suffices to prove that the periodic points of ¢ form a set of bounded height, a result first demonstrated by Marcello [287, 288] (see also [131,418]) using a height bound slightly weaker than the one in Theorem 7.15.
Theorem 7.19. (Marcello) Let ¢ : AN ----7 AN be a regular affine automorphism of degree at least 2 defined over Q. Then Per(¢) is a set ofbounded height in AN (Q). In particular,
Per(¢) nAN (K) isfinitefor all numberfields K. Proof Let
d 1 = deg¢
d2 = deg¢-l.
and
Applying Theorem 7.15 with ¢1 = ¢ and ¢2 = ¢-1 yields the basic inequality (7.8) where C is a constant depending on ¢, but not on P E Pi:. N (Q). We prove the theorem initially under the assumption that d1 d2 function
f(P)
1 1 ( 1 ) d h(P) - ad h ¢- (P) -
=
1
C
0.-
2
where the real number a > 1 will be specified later. Then
f(¢(P)) - af(P)
1
(~h(P) _ _ 1 h(¢-l(p)) d1
ad 2
= (:l h(¢(P))
2':
+ Lh(¢-l(P)))
(1 - ~d -_1_) h(P) ad 1
l'
4. Define a (7.9)
satisfies
~d h(P) - ~) 0.2 a-I
1
= (d h(¢(P)) _ a
f
>
-
_
~) a-I
(~ + a~Jh(P)+C
from (7.8).
2
Hence if we take
then
a 1 1 -d - -ad - --0, 1
and our assumption that d 1d2 conclude that
>
2
4 ensures that a
>
1, so for this choice of a we
401
7.1. Dynamics of Rational Mapson Projective Space f( ¢(P)) ? o:f( P)
for all P E AN (Q).
Appl ying this estimate to the points P, ¢(P ), ql (p) , . . . , ¢n - l (P), we obtain the fundamental inequality for all P E AN (Q) and all n
? O.
(7.10)
Similarly, we define
l I e g(P) = - h(P ) - - h( ¢(P)) - d2 (3d l (3 - 1
(7.11)
and take
(3
= d 1d2 + J (d 1d2 ) 2 - 4d 1d2 . 2d l
Then an analogous calculation, which we leave to the reader, shows that g satisfies for all P E AN (Q), from whi ch we deduce that for all P E A N (Q) and all n
?
O.
(7.1 2)
We compute
a - nf( ¢n+l( p))
+ (3-n g (¢- n-l(p )) ? f (¢(P )) + g(¢-l (p )) from (7.10) and (7.12), l
I
e
= ( d h( ¢(P )) - ad h(P ) - a - 1 l
)
2
+ (~h (¢-I (P)) ~
_ _ 1 h(P ) _ (3~
~) (3 - 1
from the definition (7.9) and (7. 11) of f and g,
> (1 -
-
_1ad
2
1_) h(P ) _ (1 (3d 1
+
_1_ + _1_) a1
(3 - 1
C
from (7.8).
Using the definition of f and g and rearranging the tenus, we have proven the inequality
(7.13 ) Now suppose that P E A N (Q) is a periodic point for ¢. Then h (¢k (P) ) is bounded independently of k, so lettin g n -+ 00 in (7.13) yields
402
7. Dynamics in Dimension Greater ThanOne (a(3 - 1)0 > (1 _ _ 1 __ 1 ) h(P) (a - 1)((3 - 1) ad 2 (3d 1 '
where we are using the fact that a also ensures that
1 - _1 ad 2
>
1 and (3
1__ (3d 1 -
> 1. Our
VI _d 4d
assumption that di d2
>
4
>0
1 2
'
so the height of P is bounded by a constant depending only on ¢. This completes the proof of the first assertion of Theorem 7.19 under the assumption that d 1d2 > 4, and the second is immediate from Theorem 7.29(£), which says that for any given number field, JlDN (K) contains only finitely many points of bounded height. In order to deal with the case d 1 d2 ::; 4, i.e., d 1 = d2 = 2, we use Theorem 7.10, which tells us that ¢2 is regular and has degree di. Similarly, deg( ¢-2) = d§. Hence from what we have already proven, the periodic points of ¢2 form a set of bounded height, and since it is easy to see that Per(¢) = Per(¢2), this completes the proof in 0 all cases. Remark 7.20. We observe that Theorem 7.19 applies only to regular maps. It cannot be true for all affine automorphisms, since there are affine automorphisms whose fixed (or periodic) points include components of positive dimension. For example, the affine automorphism ¢(x,y) = (x,y + f(x)) fixes all points of the form (a,b) satisfying f(a) = O. Of course, this map ¢ is not regular, since one easily checks that
Z(¢)
= Z(¢-l) = {[O, 1,0]}.
Definition. Let ¢ : V -- V be a morphism of a (not necessarily projective) variety V. A point P E Per(¢) is isolated if P is not in the closure ofPern(¢) <, {P} for all n ~ O. In particular, if Per., (¢) is finite for all n, then every periodic point is isolated. Conjecture 7.21. Let ¢ : AN __ AN be an affine automorphism ofdegree at least 2 defined over Q. Then the set ofisolatedperiodic points of¢ is a set ofbounded height in AN(Q). A classification theorem of Friedland and Milnor [176] says that every automorphism ¢ : .A2 -- .A2 of the affine plane is conjugate to a composition of elementary maps and Henon maps. Using this classification, Denis [131] proved Conjecture 7.21 in dimension 2. (See also [287,288].)
7.2 Primer on Algebraic Geometry In this section we summarize basic material from algebraic geometry, primarily having to do with the theory of divisors, linear equivalence, and the divisor class group (Picard group). This theory is used to describe the geometry of algebraic varieties and the geometry of the maps between them. We assume that the reader is familiar
403
7.2. Primer on Algebraic Geometry
with basic material on algebra ic varieties as may be found in any standard textbook , such as [186, 197, 198,205]. This section deals with geometry, so we work over an algebraically closed field. Let
K = an algebraically closed field, V = a nonsingular irreducible projective variety defined over K , K {V ) = the field of rational functions on V.
7.2.1
Divisors, Linear Equivalence, and the Picard Group
In this section we recall the theory of divisors, linear equivalence, and the divisor class group (Picard group). Definition. A prime divisor on V is an irreducible subvariety W C V of codimension 1. The divisor group of V , denoted by Div{V), is the free abelian group generated by the prime divisors on V. Thus Div{V) consists of all formal sums
where the sum is over prime divisors W C V , the coefficients nw are integers , and only finitely many nw are nonzero . The support of a divisor D = L nw W is
[D[ =
U W. Wwith nw?"O
If W is a prime divisor of V, then the local ring at W is the ring
Ov,w = {J E K {V ) : f is defined at some point of W} . It is a discrete valuation ring whose fraction field is K (V ). Normal izing the valuation so that ord w (K (V) *) = Il, we say that
ordw(f ) = order of vanishing of f along W . Then
f
vanishes on W if ordw (f ) 2: 1, and
Definition. Let the divisor
f
E
f
has a pole on W if ordw (f ) :::; -1.
K{V )* be a nonzero rational function on V . The divisor of f is (f) =
L ordw (f )W E Div(V ). w
A principal divisor is a divisor of the form (f ) for some f E K (V ). The principal divisors form a subgroup ofDiv(V ). The divisor class group (or Picard group) of V is the quotient group p ' (V ) = Div (V) ic (principal divisors)
404
7. Dynamics in Dimension Greater Than One
Two divisors D 1 , D 2 E Div(V) are linearly equivalent if they differ by a principal divisor, D 1 = D 2 + (J), i.e., if their difference is in the kernel of the natural map Div(V) We write D 1
'"
-----t
Pic(V).
D 2 to denote linear equivalence.
The next proposition follows directly from the definitions and the fact that every nonconstant function on a projective variety V has nontrivial zeros and poles. Proposition 7.22. There is an exact sequence
1 -----t K*
-----t
K(V)*
~
Div(V)
-----t
Pic(V)
-----t
O.
Remark 7.23. The exact sequence in Proposition 7.22 is analogous to the fundamental exact sequence in algebraic number theory, 1
-----t
(units) umts -----t (mUltiPlicative) group
(fractional) Ideals
-----t.
-----t
(ideal class) group
-----t
1.
Definition. Let ¢ : V ~ V' be a morphism of nonsingular projective varieties and let W' c V' be a prime divisor such that ¢(V) is not contained in W'. Then ¢ -1 (W') breaks up into a disjoint union of prime divisors, say
Letf E K(V') be a unifonnizer at W', i.e.,ordw,(j) by ¢ is defined to be the divisor
=
1. ThenthepullbackofW'
T
¢*W'
= Lordw,(J 0 ¢)Wi
E Div(V).
i=1
More generally, if D' =
L
nw, W' E Div(V'), the pullback of D' is the divisor ¢* D'
= L nw'¢*(W'), w'
provided that all of the terms with nw, if and only if¢(V) ID'I·
rt
i- 0 are well-defined. Thus ¢* D' is defined
There is also a way to push divisors forward. Definition. Let ¢ : V ~ V' be a morphism of nonsingular projective varieties, let W c V be a prime divisor, and let W' = ¢(W). If dim W' = dim W, then the function field K(W) is a finite extension of the function field K(W') via the inclusion ¢* : K(W') '--------+ K(W), ¢*(j) = f 0 ¢, and we define the pushforward of W by ¢ to be the divisor
405
7.2. Primer on Algebraic Geometry 4>, W = [K(W) : K (W' )]W' E Div(V' ).
If dim W' < dim W , we define 4>, W = O. And in general, for an arbitrary divisor D = L n w W E Div (V ), the pushforward of D is 4>, D =
l: nw4>.(W) . w
Example 7.24. If 4> : V
----t
V ' is a finite map, then
4>, 4>' D' = deg(4))D'
for all D' E Div (V ' ).
Proposition 7.25. Let 4> : V ----t V' be a morphism of nonsingular proj ective varieties. (a) Every D' E Div(V' ) is linearly equivalent to a divisor D" E Div (V' ) satisfying 4>(v ) ~ ID"I· (b) If D' and D" are linearly equivalent divisors on V' such that 4>' D' and 4>' D" are both defined, then 4>' D' and 4>' D" are linearly equivalent. (c) Using (a) and (b), the map
4>' : {D' E Div(V' ) : 4>(V) ~ ID'I } - - - t Div(V) extends uniquely to a homomorphism
4>' : Pic(V' )
Pic(V).
---t
Example 7.26. A prime divisor W of ]P' N is the zero set of an irreducible homogeneous polynomial F E K [X o, ... , XN]. Wedefinethe degree ofW to be the degree of the polynomial F and extend this to obtain a homomorphism deg(l: n w W) = l: n wdeg(W ).
w
w
It is not hard to see that a divisor on ]P'N is principal if and only if it has degree 0, so the degree map gives an isomorphism
deg : p ic(]P'N)
~
) Z.
Any hyperplane He ]P'N is a generator ofPic(]P'N ). Example 7.27. A prime divisor oflP'N X ]P'M is the zero set of an irreducible bihomogeneous polynomial F E K[ X o, .. . , X N , Yo, ... , Y M]. We say that F and W have bidegree (d, e) if F satisfies
F (oXo, . .. , aXN ,,GYO, • •• ,,GYM) = ad,GeF (Xo, .. . , X N,Yo, ··· ,Y M). The bidegree map can be extended linearly to give an isomorphism bideg : Pic(lP'N x ]P'Af) ~ Z x Z. Let PI : lP'N x ]P'M ----t ]P'N and P2 : ]P'N x ]P'Af ----t ]P'Af be the two projections and let HI be a hyperplane in lP'N and H 2 a hyperplane in ]P'Af. Then p ic(lP'N x lP'M) is generated by the divisors piHl = H I
X
lP'M
and
P'2H2 = lP'N
X
H 2.
406
7. Dynamics in Dimension Greater Than One
7.2.2 Ample Divisors and Effective Divisors Definition. A divisor D = LnwW is said to be effective (or positive) ifnw ~ 0 for all W. We write D ~ 0 to indicate that D is effective. The base locus of a divisor D, denoted by Base(D), is the intersection of the support of all of the effective divisors in the divisor class of D,
Base(D) =
n lEI·
E~D
E ?O
Notice that any divisor is a difference of effective divisors ,
L
D=
nw W -
w with
L
(-nw)W.
W with nw
nw>O
Definition. Let D E Div(V). Associated to D is the finite-dimensional K-vector space £(D) = {J E K(V) : (I) + D ~ o} U {O}. We write £(D ) = dim £(D) for the dimension of £(D). Let D E Div(V) be a divisor with £(D) ~ 1. We choose a basis for £(D ) and use it to define a rational map ¢D =
[ft ,... ,fi (D)] : V
---+
fl ,··· , f i (D )
p i(D)-I.
The map ¢D is well-defined up to a linear change of coordinates on p i (D)-I, i.e., up to composition by an element ofPGLi(D)(K). Further, if D and D' are linearly equivalent, then ¢ D and ¢ D' differ by a change of coordinates. Conversely, let i : V <.......> fil'N be a morphism (or even a rational map) and let H C p N be a hyperplane with i (V ) H . Then i is equal to the composition of ¢i' H with a change of coordinates and a projection.
rt
Definition. A divisor D E Div(V) is very ample if the map ¢D : V -4 fil'i (D)-l is an embedding, i.e., an isomorphism onto its image. A divisor D is ample if some multiple nD with n ~ 1 is very ample . Ampleness and very ampleness are properties of the divisor class of D . Notice that ifi : V <.......> fil'N is an embedding and H E Div(fil'N) is a hyperplane, then ¢* H is a very ample divisor on V . Example 7.28. Let p! HI and P2H2 be the generators of Pic(fil'N x pM) described in Example 7.27. Then p!H1 + P2H2 is a very ample divisor on pN x fil'M. The associated embedding is called the Segre embedding. It is given explicitly by the formula fil'N X pAJ ---+ fil'N M+N+M ([Xo, . .. , X N ], [Yo , ... ,YM
])
f----+
[X oYO, X OY1 , .. . , X ) 'j , .. . , X NYM
Now let V be a subvariety ofpN x pM, say ¢ : V
¢*(prHI
+ p;H2) = (P I 0
is a very ample divisor on V.
<.......>
¢)* HI
pN
X
+ (P2 0
pM . Then
¢)* H 2
].
7.3. The WeilHeight Machine
407
7.3 The Weil Height Machine The theory of height functions that we developed in Sections 3.1-3.5 provides a powerful tool for studying the arithmetic of morphisms ¢ : ]p'N - t ]p'N of projective space. For example, if ¢ has degree d ~ 2 and is defined over a number field K, then the fundamental estimate (Theorem 3.11)
h(¢(P )) = d· h(P ) + 0 (1)
for all P E ]p'N (Q)
(7.14)
and the fact (Theorem 3.7) that there are only finitely many points in ]p'N (K) of bounded height lead immediately to a proof of Northcott's theorem (Theorem 3.12) stating that ¢ has only finitely many K -rational preperiodic points. Recall that the height h(P ) of a point P E ]p'N(Q) is a measure of the arithmetic complexity of P. Similarly, the degree of a finite morphism ¢ measures the geometric complexity of ¢. Thus an enlightening interpretation of (7.14) is that it translates the geometric statement "¢ has degree d" into the arithmetic statement "h(¢(P)) is approximately equal to dh(P )." A natural way to define a height function on an arbitrary projective variety is to fix an embedding ¢ : V '---'> ]p'N and define hv (P) to equal h(¢(P )) . Unfortunately, different projective embeddings yield different height functions . But letting H denote a hyperplane in ]p'N , one can show that if the divisors ¢* Hand 1jJ* H are linearly equivalent, then the height functions attached to ¢ : V '---'> ]p'N and 1jJ : V '---'> ]p'M differ by a bounded amount. More intrinsically, the projective embedding ¢ determines the divisor class of the very ample divisor ¢* H . This suggests assigning a height function to every divisor on V . The Weil height machine provides such a construction. It is a powerful tool that translates geometric facts described by divisor class relations into arithmetic facts described by height relations. As such , the Weil height machine is of fundamental importance in the study of arithmetic geometry and arithmetic dynamics on algebraic varieties of dimension greater than 1.
Theorem 7.29. (Weil Height Machine) For every nonsingular variety V /Q there exists a map
h v : Div(V) ---. {functions V(Q)
-t
JR},
D
f--t
hV,D,
with the following properties: (a) (Normalization) Let H C ]p'N be a hyperplane and let h : ]p'N (Q) - t JR be the absolute logarithmic height fun ction on projective space defined in Section 3.1. Then for all P E ]p'N (Q). (b) (Functoriality) Let ¢ : V - t V' be a morphism ofnonsingular varieties defined over Qand let DE Div (V' ). Then
hV,q,"D(P ) = hV1,D(¢(P )) + 0 (1)
for all P E V(Q).
408
7. Dynamics in Dimension GreaterThan One
(c) (Additi vity) Let D , E E Div (V ). Then
hV,D+E(P ) = hV,D (P ) + hV,E(P ) + 0 (1)
for all P E V(Q).
(d) (Linear Equivalence) Let D, E E Div(V) with D linearly equivalent to E. Then
hv,o(P) = hV,E(P ) + 0 (1 )
for all P E V (Q).
(e) (Pos itivity) Let D E Div (V ) be an effective divisor. Then
bv.o 'P) ;::: 0 (1)
for all P E V(Q) " Base(D ).
That is, hV,D is bounded below fo r all points not in the base locus of D . (f) (Finiteness) Let D E Div (V ) be ample. Thenfor all constants A and B , the set
{p E V(Q) : [Q(P) : Q] ::; A is finite. In particular, finite extension, then
if V
and hV,D(P)
< B}
is defin ed over a number fi eld K and
{p E V( L)
if L/ K
is a
: hV,D (P ) ::; B}
is a fin ite set. (g) (Uniqueness) The height functions hV,D are determined, up to 0 (1), by the properties of (a) normalization, (b) functoriality, and (c) additivity. (It suffices to assume functoriality for projective embeddings V '-+ jp'N.) Proof See [76, Chapter 2], [205, Theorem B.3.2], or [256, Chapter 4].
0
Remark 7.30. All ofthe 0 (1) constants appearing in the Weil height machine (Theorem 7.29) depend on the various varieties, divisors , and morphisms. The key fact is that the 0 (1 ) constants are independent of the points on the varieties. More precisely, Theorem 7.29 says that it is possible to choose functions hV,D, one for each smooth projective variety V and each divisor D E Div (V ), such that certain prop erties hold, where those properties involve constants that depend on the particular choice of functions hV,D. In principle, one can write down particular functions hV,D and determine specific values for the associated 0 (1) constants, so the Weil height machine is effective. In practice, the constants often depend on making the Nullstellensatz effecti ve, so they tend to be rather large. Remark 7.31. Many of the properties of the Weil height machine may be succinctly summarized by the statement that there is a unique homomorphism
hV such that if ¢ : V
:
· V P 1c( ) -----+
'-+ jp'N
{functions V( K) -7 R} {bounded functions V (K) -7 R}
-;-,-------'-.,..---:-....,---- -,------'--:'-::-:-::=::---=-----=-o-
is a projective embedding, then bv,« H = h + 0(1).
7.3. The Weil HeightMachine
409
Example 7.32. Let ¢ : jp'N - t jp'N be a morphism of degree d and let H E Div(jp'N ) be a hyperplane. Then ¢* H '" dH, so Theorem 7.29 allows us to compute
hpN,H(¢(P )) = hpN,
= hPN,H1 (PI¢(P )) = h(x ),
hv,