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1}
is a fundamental domain for P. This is called a normal polygon. One can show that D is a convex polygon with an even number of sides (subject to the convention that if a side contains a fixed point of an elliptic motion of order
2 from r, then this point is considered as a vertex and the side is divided into two sides). The sides of D can be arranged in pairs of congruent sides so that the sidepairing motions generate the group F. These properties imply the following:
Proposition 2.2. Every Fuchsian group of the first kind has a finite number of generators and a fundamental domain of finite area.
The finiteness of the area of a fundamental domain does not guarantee the compactness (after closure). In fact the most important arithmetic
2.3. Discrete groups  Fbchsian groups
31
groups do not have compact fundamental domains. Suppose that the closed polygon D is not compact. Then D must have a vertex on A, and since the two sides of D which meet at such a vertex are tangent (because they are orthogonal to A), they form a cusp.
For a Fuchsian group of the first kind one can select a fundamental polygon all of whose cuspidal vertices are inequivalent. In such a polygon the two sides joined at a cuspidal vertex are congruent by a motion which fixes the vertex, generates the stability group of the vertex and is parabolic. For this reason a cuspidal vertex is also called a parabolic vertex. Conversely, cusps for r are exactly the fixed points of parabolic motions of F. Therefore we have
Proposition 2.3. A Fuchsian group of the first kind has compact fundamental domain if and only if it has no parabolic elements.
Let F be a Fuchsian group of the first kind. Then the quotient space r \ ]H[ (the set of orbits) is equipped with the topology in which the natural projection 7r : IHI > F \ H is continuous; it is a connected Hausdorff space, and with properly chosen analytic charts it becomes a Riemann surface. Specifically, given a point z E H, consider discs U,z C H sufficiently small so
that for all 7 E F,'y Fz 'yUz = Uz for all 'y c rz.
yU,z fl U,z = O
The stability group Fz is cyclic of finite order, say m > 1. Choose Tz E SL2(C) which maps z to 0 and transforms Uz onto the unit disc
U= {zEC:IzI <1}. Let em : U  U be the power map defined by em,(z) = z"`. Then the charts (7rUz7 emrz7r1) with z ranging over H and Uz (small discs centered at z) form an analytic atlas for IF \ H. If the group F contains only hyperbolic elements and the identity, then F \ H is a compact, smooth surface of genus g > 2. If F has elliptic elements, then the surface F \ H has branch points at the fixed points of the elliptic motions. If F has parabolic motions, then I\ H is not compact. Usually we compactify F \ H by adding cusps with suitable charts. Given a cusp a E A for IF, consider discs Ua C H tangent to A at a sufficiently small so that 7Ua fl U a = o
yUa=Ua
all
forall'yEra.
a
2. Automorphic Forms in General
32
The stability group Pa is cyclic infinite, say generated by rya. There exists Qa E SL2 (R) such that (change rya to ya if necessary)
Qalyaaa= C1 1). We shall call Qa a scaling matrix of the cusp a; it is determined up to (2.15)
Ora00=a,
composition with a translation from the right side. Thus cr sends a to 00 and transforms the disc Ua (cuspidal zone, so to speak) onto the halfplane IEIIY = {z = x + iy : y > Y}. (2.16) If Y is sufficiently large, the cuspidal zones Ua for distinct cusps are disjoint.
Let e : IHI > H be the exponential map e(z) = e2,i,. Then for charts at cusp a we take (irUa, The quotient space r \ H equipped with the above charts at points of H and the cusps is a compact Riemann surface. It is convenient to think of the Riemann surface r \ H as being constructed from a fundamental polygon by gluing pairs of congruent sides at equivalent points. eaa1IF1)
There is a nice construction of a fundamental polygon (due to L.R. Ford) which uses isometric circles (2.6). Suppose r is a Fuchsian group of the first kind which has parabolic motions, so it has a cusp, say a. By conjugation
with the scaling matrix Qa we can assume that a = oo and the stability group r... is generated by the translation z H z + 1 (consider the group Qa 1FQa in place of r). The fundamental domain of r... is the strip
P={z=x+iy:0<x
E = {zEH:Ijy(z)I >1for all yEF,'y
={zeH:Imz>Imyzfor all yeF,yvr,,c}. Thus E is the intersection of exteriors to all isometric circles C,,, ,y E IF, y
Fem. One can show that D = E fl P is a fundamental polygon for F (this is called the standard polygon). Some of the cuspidal vertices of D may be equivalent; nevertheless D has a nice shape which is useful for getting explicit estimates in terms of the group r (see the observations made after Proposition 2.7). If F' C F is a subgroup of finite index, one can construct a fundamental domain for F' out of selected copies of a domain for F. Specifically, letting
yj,1 < j < n, be coset representatives, i.e. F = U F'yj, and D be a fundamental domain for IF, then the union D' = U yjD is a fundamental domain for F' (not necessarily connected). Finally, before giving specific examples of Fuchsian groups, let us mention the following general grouptheoretic result.
2.3. Discrete groups  Fuchsian groups
33
Proposition 2.4 (R. Fricke and F. Klein). A Fuchsian group of the first kind is generated by (primitive) motions A1, ... , A9, B17 ... , Bg, El, ... , Ell P1,. .. , Ph satisfying the relations
[All B1]... [Ag,Bg]EI...EQP1...Ph = 1, E73 = 1
1<j
1
Here A3, Bj are hyperbolic motions, [A;, B3] = A; B., A3 1Bj 1 stands for the commutator, g is the genus of I' \ IEII, E3 are elliptic motions of order m3 >, 2,
Pi are parabolic motions and h is the number of inequivalent cusps.
The symbol (g; ml, ... , ml; h) is group invariant (under conjugation), and is called the signature of F. The area of D = F \ IHI is given by the GaussBonnet formula IDI
(2.17)
= 2g2+E 71
(1 
\
+ h. 7
An interesting class of discrete groups with compact fundamental domains is made from division algebras over Q, called quaternion groups. For example, let us consider the group
r(n,p) _ { 1
fp
(cad
(cad V)_l E SL2(R) : a,b,c,d E Z
I
where p is a prime number which is  1 (mod 4) and n is a positive integer which is not a square modulo p, i.e. (zi) = 1. We shall show that, aside from the identity, F(n, p) has only hyperbolic motions. To this end suppose ry c F(n, p) has ItryJ < 2. Then lal < 1, so a = 0 or ±1. Since a2
 b2n  (c2  d2n)p = 1,
we must have a = ±1 because otherwise b2n  1 (mod p), which violates
the hypothesis that n is not a quadratic residue mod p. For a = ±1 the determinant equation simplifies to b2n = (den  c2)p,
and this is possible only for b = c = d = 0, for the same reason that n is not a square mod p. Therefore we have ry = ±1, proving that I'(n, p) is a hyperbolic group. By J. Nielsen's theorem F(n, p) is discrete, and by Proposition 2.3 it has a fundamental polygon without cusps.
2. Automorphic Forms in General
34
2.4. Congruence groups Let q be a positive integer. Then (2.18)
1 ) (mod q) }
F(q) = y E SL2(Z) : 'Y
is called the principal congruence group of level q. It is a normal subgroup of the modular group I'(1) = SL2(Z) of index
[F(1) : F(q)] = q3 J(1  p2).
(2.19)
pIq
To prove (2.19), consider the exact sequence
1 F(q)  SL2(Z)
(2.20)
SL2(Z/qZ) > 1.
The only question is the surjectivity of the residue class map SL2(Z) > SL2(Z/q7G), which is established by the Chinese Remainder Theorem. By (2.20) we have
[F(1) : r(q)] = ISL2(Z/qZ)I
For any y = (') E SL2 (7G/qZ) we have (c, d, q) = 1, and for a given lower row (c, d) we can find exactly q solutions (a, b) of the congruence ad  be  1 (mod q). Hence we compute that jSL2(Z/qZ)j= q I {(c, d) (mod q) : (c, d, q) = 1}I
L\
= q> u(r)
(q)2
r
=q 3 f[(1 _
p2).
pjq
rjq
Notice that 1
F(q) if q > 2. Moreover, if q > 2 then P(q) has no qv) elliptic elements, and the parabolic elements are ( 1+qt 4u 1qt with u, v, t integers such that uv = t2. All parabolic elements have trace 2. Since r(q) is a subgroup of SL2(Z), the cusps for P(q) are rational numbers and oc. If one writes cusps as rational points in lowest terms, say a = with (a, c) = 1
and a' _
with (a', c') = 1, then they are equivalent, i.e. a' = ya for e F(q), if and only if
f [a] c
[ac]
(mod q).
Hence the number of inequivalent cusps for r(q) is given by
h ={(c, d) (mod q) : (c, d, q) = 1}I
2.4. Congruence groups
35
which yields
h = q2 fJ(1  p2).
(2.21)
plq
Any subgroup of the modular group which contains P(q) is called a congruence subgroup of level q. Here are two examples; (2.22)
Po(q) = S 'Y E SL2(Z) : 'Y =
(* *)
(2.23)
IF, (q) =
(1
1' E SL2(Z) : y =
1
(mod q) }
,
I (mod q) } .
The first one, called the Hecke congruence group (it is not a normal subgroup of SL2 (Z)), is most often encountered in arithmetic. Therefore we are going to examine I'o (q) in great detail.
Proposition 2.5. A set of representatives for 17o(q) \ Po(i) is given by (2.24)
(*
v ) E I'o(1)
with vI q and u (mod q/v).
Hence the index is (2.25)
vq = [FO(1) : I'o(q)] = gfl(1
+p1).
piq
Proof. We have
(y S) (c d)
(u v)'
= (rya+Sc yb+6d) = say. Hence (d, q) is invariant under multiplication in Po(1) by Po(q) on the left side. In fact one can bring any (a d) E Po (1) by multiplication with a suitable (a E Po (q) to a matrix (u v) with v = yb + 6d = (d, q), i.e. v1q. All solutions to the linear equation y'b+8'd = v form a oneparameter
family y' = y + dt, 6' = 6  bt with t ranging freely over integers t  0 (mod q/v) (this congruence ensures that y' = 0 (mod q)). These solutions move the other entry u into u' = u + t, so one can change u freely modulo q/v. Having fixed u, v in the prescribed range, there is no room left for additional changes. This proves the first part of Proposition 2.5. Hence the formula for the index is derived as follows: q) : (u,v) = 1} [Po(1) : I'o(q)] = E {u(mod vjq
(Pa + pa1 _ pa2 p°Ilq
+Pa2 _ pa3 +...) = q n(1 + p1). plq
2. Automorphic Forms in General
36
Proposition 2.6. A set of inequivalent cusps for ro(q) is given by the following fractions: u
with vIq, (u, v) = 1, u (mod (v, q/v)).
(2.26)
Hence the number of inequivalent cusps is
h = E cp((v,w))
(2.27)
vw=q
Proof. All cusps are equivalent to rational numbers since ro(q) C ro(1). We transform the representatives T =
of of type (2.24) into (1 1) T1 =
*) and evaluate these at oo, getting the rational points v with /vjq and V (u, v) = 1. Suppose two such points are equivalent, say u' v'
a
Q
u
ry
b
v
for (ry a) E ro(q). Hence v' = ryu + bv, vIv', v'I v, v' = v, b  1(mod q/v) and u' = au +,iv  au  bu u mod (v, q/v), as claimed. Our selection of inequivalent cusps of type (2.26) for r = ro(q) will be convenient in future computations. Recall that all cusps for IF = ro (q) are equivalent to oo under the modular group action. We choose Ta = ( uV
(2.28)
ESL2(Z),
so Taoo=a.
Then Ta 1ryaTa C rte, because any element of oalraaa fixes oo. Since both groups are cyclic, we must have Ta 1'faTa =
{
f (1
1 bTn,
)
:bcZ}
for some positive integer ma. To interpret ma, observe that the fundamental domain for r = ro(q) can be filled by vq copies of the fundamental domain
for ro(1); the number of such copies which touch the cusp a must be the same for equivalent cusps and is exactly equal to ma. To compute ma, consider the relation
I'a = Tar.TaI n ro(q)
(u _ = {
(
M)
1 muv mv2
mug
1 + muv) : my
E ro(q) 2
=
(
0 mod
q) }
2.5. Double coset decomposition
37
Here m ranges over all integers divisible by q/(q, v2), and this shows that q
M =
(2.29)
.
(q, v2)
a
Now the index formula (2.25) can be written in another way:
vq > ma = E
(2.30)
(q, 2)
P((v, q/v))
vq
The integer ma is called the width of the cusp a. The scaling matrix Qa reduces the width to 1, so it is given by Qa = Tap,,, where
( ma
(2.31)
Pa
2.5. Double coset decomposition Throughout r is a Fuchsian group of the first kind which contains parabolic motions. Select a complete set of inequivalent cusps for I', say a, b, ..., and the corresponding scaling matrices u,,, ub,... in SL2(IR). Recall that Qa la = Qb 1b = ... = oo and Ca 1Faaa = Ub 1rbab = = B, where
B={fl
i ):nEZ}
1
is the group of translations by integers. For any pair of cusps a, b (not necessarily distinct), we shall partition the set ca 1Fab into double cosets with respect to B. First let us examine the subset
'.rub:woo =o0} which consists of the uppertriangular matrices in a 'Fab. Suppose Q,,,, is not empty, say w = ca 1'ycb (for some y in F) belongs to Q. Evaluating y at b, we get yb = aawvy 1b = vaoc = a, showing that the cusps a, b are equivalent; hence a = b, y E ]Pa, w E B. Therefore 1l = B if a = b, and it is the empty set otherwise. Any other element of Qa ']Pub is represented by a matrix w = (a d) with c > 0. The relation
li) C1
(a+cm
(1 1n
(a d)
)
=
c
d+')
2. Automorphic Forms in General
38
shows that the double coset Qd/a = B d) B determines c uniquely and d modulo integral multiples of c. Moreover, given c, d with w = (c d) E Qa 1FUb, the double coset 1d/c does not depend on the upper row of w. Indeed, if w' = I c d
E Ua
y = Ua
1
Ua 1,
'ya=a, yEra,w'w1EB,w'= (11''w,a'=a+ cm for sgmemEZ. We conclude the above analysis with the following:
Proposition 2.7. For any pair of cusps a, b for r the set Qa lrub is partitioned into disjoint double cosets
Ua1rUb=babBUU U B(* d)B
(2.32)
c>O d (mod c)
where babB = B if a = b or else it is the empty set, and the union is taken over c > 0 and d (mod c) such that (* d) E o'a 'rub . In applications we need to control the number of elements in the cosets
B(*")B. Put (2.33)
C(a,b)={c>0:
(*
I E oa'rub
c
J
Let c(a, b) denote the smallest element of C(a, b), and put c(a) = c(a, a). That c(a) exists is seen from the construction of the standard polygon for the group oa 1rua; c(a)1 is the radius of the largest isometric circle. Hence c(a, b) also exists; in fact one can show that (2.34)
c(a, b)2 >, max{c(a), c(b)} = cab, say.
Since the standard polygon contains the semistrip {z : 0 < x < 1, y > c(a)1} whose area is c(a), it follows that c(a) is bounded by the area of the fundamental domain, c(a) < ID1.
(2.35)
Proposition 2.8. For any X > 0 we have c1 {ci(mod c) : (c d) E oa1FUb }
(2.36)
<, Cab X.
o
Hence for any c c C(a, b) we have
{ d (mod c) : (c d) E ora 'rub
.
cab c22
2.5. Double coset decomposition
39
Proof. By symmetry we can assume without loss of generality that c(a) > c(b). If w = (* d) and w' = d) with 0 < c, c' < X are both in 0a1F0.b,
then w" = w'w1 = (C * ) with c" = c'd  cd' is in a 1raa. If c' = 0 then the cusps a, b are equivalent, so equal, w" c' = c and d' = d. If
c0then Ic"I >c(a) and
 dc d c'
(2.37)
c(a)
>
c(a) .
cX Summing this inequality over 0 < c < X and 0 < d < c, where d'/c' is chosen to be the successive point to d/c, we get (2.36). cc'
Applying (2.36) for X = c(a, b), we infer the inequality (2.34) from the trivial bound
{d(mod c): (c d) EOra1rubJ > 1,
if c E C(a, b).
Proposition 2.9. Let a be a cusp for r, z E IHI and Y > 0. We have (2.38)
I{yEFa\F:IM aQlyz>Y}I <1+
10
c(a)Y
Proof. Conjugating the group, we can assume that a = oo, o'a = 1 and Fa = B. Then the strip P = {z : 0 < x < 1, y > 0} is a fundamental domain of Fa. Let D be the standard polygon of IF, so D consists of points in P of deformation less than 1. For the proof we may assume that z E D, so icz+dl > 1 for any y = (* d) E r with c > 0. Since Im ryz = ylcz+d12 > Y, this implies y > Y, c < (yY) and lcx+dl < (y/ Y) 12. By the last inequality and the spacing property (2.37) are estimate the number of pairs {c, d} with 21
C
1+
)
(Yy'
2
)
<
10C y1 c(a)
(Y) 2
Adding these bounds for C = 2'(yY)21 with n > 1, we get 10/c(a)Y. This is an estimate for the number of relevant y's not in Fa. Finally adding 1 to )account for Fa, we obtain (2.38). As an example consider the Hecke congruence group Fo(q) (note that
1 E Fo(q)) and the cusps at oo, 0. The scaling matrices are a = (1 1) We have a001Fo(q)aoo = Qo'ro(q)ao = ro(q), and the set a.1Fo(q)ao = v0 1Fo(q)o. consists of matrices of type
and ao = (2.39)
(').
(c / ale ) y Vq
6 /
with a, ,Q, y, b F_ Z, abq  0y = 1.
Hence c(oo) = c(0) = q and c(oo, 0) = c(0, oc) = ,Fq.
2. Automorphic Forms in General
40
Remark. Propositions 2.7, 2.8 and 2.9 are formulated for a Fuchsian group of motions rather than for a discrete group of matrices. Remember to take into account the factor 1 = (1 _1) when translating the correspondence between the linear fractional transformations and their matrix representations (if 1 belongs to the group of matrices).
2.6. Multiplier systems For a complex number z 0 we choose its argument in (7r, 7r]. We denote the principal branch of the logarithm by log z, so it is real for positive z and
logz=logIzI +iargz
ifzEC*.
Then we define the power zs for any s E C by z8 = exp(s log z).
Let A, B E SL2(R). By the chain rule jAB (Z) =jA(Bz)jB(z) it follows that the expression (2.40)
27rw(A, B) = arg jAB(z) + arg jA(Bz) + arg jB(z)
does not depend on z c H. More precisely, w(A, B) takes only three values 1, 0, 1, because Iw(A, B) I is an integer 5 2. By examining numerous cases one can establish the following properties (cf. [Pet] or [Ran]):
(2.41)
(2.42)
w(AB, C) + w(A, B) = w(A, BC) + w(B, C) w(A, B) = w(B, A)
if A, B commute
(2.43)
w(DA, B) = w(A, BD) = w(A, B)
(2.44)
w(AD, B) = w(A, DB)
(2.45)
w(A1DA, B) + w(A, A'DAB) = w(A, B)
2.6. Multiplier systems
41
(2.46)
w(A, D) = w(D, B) = 0
(2.47)
w(ADA1, A) = w(A, A'DA) = 0
where A, B, C are arbitrary and D = (1 i) Here (2.45) is just a special case of (2.41) with A, B, C replaced by A, A1DA, B respectively. Moreover .
we have
w(AD, A') = w(A, DA1) = w(A, A') = 0 except for A = (o d ) with d < 0, in which case w (A, A`1) = 1. For any real number k we define the factor system of weight k by setting w(A, B) = e(kw(A, B)).
(2.48)
Note that w(A, B) depends on k (mod 1) and w(A, B) = 1 if k is an integer. We have
w(A,B)7AB(z)k =7A(Bz)k7B(z)k.
(2.49)
For any
tions f : H , (2.50)
SL2(]R) we define the "slash" operator JA acting on funcby
fIA(z) = iA(z)k f
(Az).
This satisfies the rule of composition (2.51)
fIAB = w(A, B)(fIA)18
Let IF c SL2 (]R) be a discrete subgroup. A multiplier system of weight k for r is a function 19 : F > C such that (2.52)
I,9(7)I = 1
(2.53)
'9(7172) = w('Yl,72)'d(71)?)(72).
Since w(1, 1) = e(k), the above properties imply that if 1 belongs to IF, then i9(1) = ±e(k/2). Throughout we shall require that (2.54)
i9(1) = e(k/2)
if  1 E IF,
which is called the consistency condition (the other choice yields the zero automorphic form only).
2. Automorphic Forms in General
42
Observe that if 19 is a multiplier system of weight k, then it is also a multiplier system of any weight k' = k (mod 2). The complex conjugate 19 is a multiplier system of weight k. If 191,192 are multipliers of weight k1, k2 respectively for a group I', then 191192 is a multiplier system of weight k1 + k2 for F. If 19 is a multiplier system for IF and a E SL2(R), then',(2.55)
19a(Y) = i9(a'Ya1)w(Q'YQ1, a)w(a,'Y)
is a multiplier system for the conjugate group a1FQ. Note that _9 UT = (19 U)T
(2.56)
19a=19
(2.57)
ifaEF.
A multiplier system of weight k E 7G is just a unitary character of IF which satisfies the consistency condition 19(1) = (1)k. For example, given a Dirichlet character X to modulus q, we set (2.58)
19('y)
= X(d)
if ry =
(a
d) E I'0 (q).
This is a multiplier system for I'o (q) of any integral weight k such that (2.59)
k =_ 1(1 
X(1)) (mod 2).
2.7. Automorphic forms Throughout I' is a discrete subgroup of SL2(Is). It is not at all obvious that a multiplier system for r of given weight k exists. This problem is equivalent to the existence of a nonzero meromorphic function f : Ill1 > C satisfying the transformation rule (2.60)
fv = 19('Y)f
for any yEF.
Notice that (2.60) is obvious for y = 1, due to the consistency condition (2.54).
If I' has no parabolic elements, then a multiplier system of weight k exists for P if and only if k = 47rn/ JD J[ml, ... , me]
where n E Z, IDI is the volume of the fundamental domain, m1,. are orders of elliptic generators of I, (see Proposition 2.4) and [ml, .
.. , mp . . , me]
2.7. Automorphic forms
43
denotes the least common multiple (equal to 1 if £ = 0). In particular if r is a hyperbolic group of genus g > 2, then this condition asserts that 4(g 1)k is an integer. The holomorphic functions satisfying (2.60) are called automorphic forms for r of weight k with respect to the multiplier system V. We shall construct automorphic forms for groups having parabolic elements in due course (Poincare series). In this book we shall require an automorphic form to be
holomorphic not only in H butlso at every cusp. The latter condition needs some explanation.
Suppose a is a cusp for F. Let Qa E SL2 (Il) be a scaling matrix, so aaoO = a and oa 11Faaa is the group of integral translations generated by Q = (1 11, together with /3 if 1 E F. Then l iJ (2.61)
rya=Qa
(1
1) 1 1)O'
generates the stability group ra, together with rya if 1 CI`. By (2.60) we obtain flaa
('3z)
=
jaa
(/3z) kf (cal3z) = jaa (/3z)kf (ryaaaz)
= .7va (Qz)kiYa
=
(O'az)
jaa (Qz)k7_ya (O'az)k.7aa (z)k19(7a)fj. (z)
Here the product of jfunctions is equal to
Qa) by (2.49), and each of the wfactors is 1 by (2.46) and (2.47) respectively (insert rya = Qa/3o'a 1) . Therefore we have (2.62)
fiaa(/3z) = 9(rya)f1aa(Z)
We set (2.63)
19(rya) = e(rca)
with 0 < rca < 1.
By (2.62) it follows that e(rcaz)fjaa(z) is periodic of period 1, so we can write e(rcaz)flaa(z) = g(e(z))
where g(q) is holomorphic in the punctured plane C*. We say that f is holomorphic at the cusp a if g(q) is holomorphic at q = 0. In this case the power series expansion of g(q) at q = 0 yields W
(2.64)
fiaa (z) = e(rcaz) E fa(n)e(nz) n=o
2. Automorphic Forms in General
44
where fa(n) are complex numbers called the Fourier coefficients of f at the cusp a. The above series, called the Fourier expansion of f at a, converges absolutely and uniformly in halfplanes Im z > E. Denote by Mk (F,19) the linear space of automorphic forms for r with multiplier 19 of weight k. If 19 is trivial on the subgroup Fa, i.e. (2.65)
19(ya) = 1,
so Ica = 0, then a is said to be a singular cusp for the multiplier system 19. An automorphic form f E Mk(F,19) is said to be a cusp form if
f.(0) = 0
(2.66)
for any singular cusp.
Thus, by Fourier expansion, a cusp form decays exponentially at every cusp (whether singular or not). Hence a cusp form f (z) has exponential decay as z tends to any point on the boundary i9H = R U {oo}. We denote the linear subspace of cusp forms by Sk (I',19) The space Mk(I",19) has finite dimension, but in general one does not know its exact value. If 19 is the trivial multiplier and k is even integer > 2, then we have .
where (g; m1, , mj; h) is the signature of F. Hence by the GaussBonnet formula (2.17) we get the bound . . .
dimMk(I') < (47r)1IDlk+1. In full generality one can prove that (2.68)
dimMk(F,19) < c(k + 1)(IDI + 1)
where c is an absolute constant.
2.8. The etafunction and the thetafunction We shall conclude our general considerations by giving two examples of automorphic forms of weight 1/2 to present two interesting multiplier systems. The first one is constructed from the discriminant function (2.69)
0(z) = (27r)12e(z)fl(1  e(nz))24, 1
2.8. The etafunction and the thetafunction
45
which is a cusp form of weight 12 with respect to the trivial multiplier on the modular group. Since 0(z) does not vanish in IH[, one can define for every k > 0 a holomorphic function f (z) = (27r)k0(z)k/,2,
which is a cusp form of weight k, but, of course, its multiplier system is no longer trivial. For k = we obtain the etafunction 2
i(z) = e (_z 2) fl(1  e(nz)).
(2.70)
oo
Dedekind determined the multiplier system for q(z). Precisely, we have 7)((z) = 19(y)jy(z)27)(z)
(2.71)
if y c SL2(7G)
where 19(y) = e(1/4)19(y) for any y, 19(y) = e(b/24) if y = (1 b) and a + d  3c
t9(y) = e
24c

1
2
s(d c)
)
a
if
c
bl ,c>0.
d
Here s(d, c) denotes the Dedekind sum
s(d' c) _ O<_n
c. (dc )
where,O(x) = x  [x]  2. The Dedekind sums can be computed by a simple algorithm based on the following reciprocity law: s(d, c) + s(c, d) = 12
(++_3)
for c, d > 0, (c, d) = 1. Moreover, the etamultiplier can be expressed in terms of the LegendreJacobi symbol. For example, we have 19(y)
= (Idl )
if y =
(a d)
E F(24), c > 0.
The Fourier expansion of 77(z) is a lacunary series e(n2z/24)
77 (z) =
n=±1 (mod 12)

E nf5 (mod 12)
e(n2z/24).
2. Automorphic Forms in General
46
The inverse
rj(z)1 has the Fourier expansion given by 00
00
e(z/24)77(z)1 = fl(1 
e(nz))1
= > p(n)e(nz) 0
1
where p(0) = 1 and p(n) is the number of unrestricted partitions of n > 0 into sums of positive integers. The Fourier coefficients of are quite i7(z)1 has weight z but it is not holomorphic at oo). large (note that G.H. Hardy and S. Ramanujan invented the circle method, by means of which they established the asymptotic formula 77(z)1
p(n)  (4,43n)  1 exp (r 2n/3)
as
n  oo.
A close relative of the Dedekind etafunction is the classical thetafunction defined by its Fourier series:
8(z) = E e(n2z).
(2.72)
00
It is an automorphic form (not a cusp form) for the group ro(4) of weight k = 1/2 with respect to a multiplier system slightly different from that of the etafunction. Precisely, we have (see Theorem 10.10) (2.73)
8('Yz)
=
Ed
(C) d,7.y(z)ao(z)
if y = ( c db) E ro(4)
where (2.74)
Ed =
51
ifd = 1(mod 4)
i ifd  1(mod 4) and (a) denotes the extended quadratic residue symbol; namely, it is the Jacobi symbol if 0 < d  1 (mod 2) extended to all d  1 (mod 2) by (2.75)
d (0
(2.76)
d)
if c # 0
d
Icl 1
{0
ifd=fl otherwise.
The thetafunction has the Jacobi product representation 00
(2.77)
0(z) = fl(1  e(nz))(1 + e((n + 1/2)z))2 1
which shows that 0(z) does not vanish in H.
Remark. The theta multiplier system coincides with the eta multiplier system on the subgroup r(24).
Chapter 3
The Eisenstein and the Poincare Series 3.1. General Poincare series A very important class of automorphic forms is constructed by the method of averaging. Suppose a is a cusp for r which is singular with respect to a multiplier system e9 of weight k. Let p : H ' C be a holomorphic function which is periodic of period 1. Define 7r: r x H  C by (3.1)
ir('y, z) = 9(y)'W (aa 1,'Y) joa 1ry(z)kp (Qa 17z)
Actually ir(y, z) depends only on the coset ray. To prove this, consider 7r (y', z), where 'y' = iy with rl E ra, so 17 = va/3va 1, where Q is an integral translation. We have p (ca ,7,z) = p (Qka "yz) = p (ca 1`yz)
ja.
(z)k
11,(z)k
j".
l.r(Z)k
t9('y') = i9(t17) = w(17,'Y)'9(7) by (2.53) and
w(71,1')w (cn 1,1'x) = w (aa , 7)
by (2.45). Collecting these results, we arrive at n(y, z) = ir(ry, z). This property allows us to write without ambiguity the infinite series
pn(z) = E irl'y, z) 7Ero\r 47
3. The Eisenstein and the Poincare Series
48
provided it converges absolutely. For example, if p(z) is bounded the series (3.2) is majorized by 17(z),_k
Ijao
k
(Im oalryz)'
= yk
,
yEro.\r
'YEr,\r
and this converges absolutely if k > 2 by virtue of Proposition 2.9. For any singular cusp b we deduce the following:
Palab(z) = jab(z)kPa(abz)
(Z) k E
=
7r ('Y,abz)
7Era\r iab
(Z) k
1
,
?f (aaryab 1 , abz)
7EB\aatrab =lab
(z)k
(aaryab 1) w (aa 1, aa'Yab 1) i7a61(abz)kp(7z)
7EB\ap lrab
Since ja(z)'kj7a1(az)k = w(rya1,a)j7(z)k, this gives
Pa1,b (z) = E
t9ab(7)j7(z)kp('Yz)
7EB\aa 1rab
where (3.4)
l9ab('y) = t9 (aa1'97b 1) w (aa
aaryab
w (ryab 1,ab)
Using the relation (2.41), one can derive a handful of expressions for i9ab(ry). For example, we have (3.5)
i9ab('y)W(aa,'Y) =19 (aaryab') to (aaryab n, 0'b)
Hence, in particular, for a = b the series (3.3) becomes (3.6)
Paiaa(z) _ E
'j'('Y)j7'(z)kp('Y'z)
' Er, 00 \r, where t9' is the multiplier system for the conjugate group I" = aal'aa given by (3.7)
19'(ry')w(aa, 7) = '9('y)w(ry, aa)
if ry' = as lryaa.
The function Pa(z) defined by (3.2) is called the Poincare series associated with the cusp a and the generating function p (a is required to be singular with respect to the multiplier system, and p is periodic such that the series (3.2) converges absolutely).
Proposition 3.1. The Poincar6 series Pa(z) is an autotnorphic form, P0Ir(z) = t9(T)Pa(z) if T E F. (3.8)
3.2. Fourier expansion of Poincare series
49
Proof. By conjugating the group we may assume that a = oo and aoo = 1, in which case the series (3.2) looks simpler, namely Poo(z) =
(3.9)
9(Y)jry(z)kP('Yz)
ryEra,\r
Hence for r E IF
Poo(rz) _
F, '9('Y)jy('rz)kP(YTZ)
.yEr.\r
= C`
j7r_i (rz)kp(yz).
ryEroo\r
Here we have
(yri) = w('y, T1)d(7)d(T1) = j.yr1(Tz)k = w(7,r1) jry(z)k jri
w('y,r1)w(,r, ri) 9(r),y(7),
(rz)k = w('y,T'),w(,r,r1) jr(Z)k j l(z)k,
and by these expressions we arrive at (3.8). Clearly Pa(z) is holomorphic in H. To prove the holomorphy at cusps we need to expand Pa(z) in Fourier series.
3.2. Fourier expansion of Poincare series Let a, 6 be singular cusps for a multiplier system t9 on r. We seek the Fourier expansion of Pa(z) at the cusp b, i.e. for the series (3.3). Applying the double coset decomposition (2.32), we split the series into Po1g (Z) = aabP(Z) +
l9ab(7)Ir(Z) 1; 7ES\oo 'rab/B
where the first term comes from the contribution of 7 = 1 (which exists only if a = b), and for any 'y = ( :)Ean'rcbwithc>Owehave
I.y(z) = > jyr(Z)'P(7rz) rEB nEZ nEZ
c z+n +d )
P
)
a
1
c
c(c(z + n) + d)
By Poisson's summation we get
I.y(z) = E J uEZ
(c(z + v) + d) kp I 00
a
 c(e(z + v) + d)) e(nv)dv.
3. The Eisenstein and the Poincare Series
50
In the sequel we specialize the generating function to p(z) = e(mz)
where m is a nonnegative integer. For this function we can compute the Fourier integral quite explicitly. First by a linear change of variable we obtain / ma C nd)\
1,(z) E e I nz + nEZ
where
Jc(m., n) =
roo+iy
J
/ m
(ev)ke I
\\
oo+iy
(TV
 nv I dv.
Notice that this integral does not depend on y by Cauchy's theorem. If n < 0, then, moving the horizontal line of integration upwards, we see that the integral vanishes,
3c(m, n) = 0
if n <, 0.
If n > 0 but m = 0, then we have (see [GR], 8.315.1) (3.10)
nk1
27r
J.(0, n) = (ic)
T(k)
For n > 0 and m > 0 we have (see [GR], 8.412.2) (3.11)
Jc(m, n) =
21r
i
(n) a
Jkl
(4)
where J.y(x) is the Bessel function of order v, defined by 00
(3.12)
(1)t :C V I.2f J (x) = F Q!r(e+ 1 + v) (2) e=o
Exercise. Derive (3.11) from (3.10) using power series expansion for e(z).
Collecting the above computations, we obtain the desired Fourier expansion for the Poincare series generated by the function p(z) = e(mz), namely
P0!, (z) = Sahe(mz) + > e(nz) n=1
Spb(m, n; c)J,(m, n) n>n
3.2. Fourier expansion of Poincare series
51
where J (m, n) are given by (3.10)(3.11), and Sab(m, n; c) is the Kloosterman sum defined by (3.13)
''ab(l)e
Sab(m, n; c) _
/ ma + nd (
\
c
'Y=( d)EB\°O'rab/B
Recall that i9ab(y) is given in terms of the multiplier system by (3.5). Since there are no negative terms in the Fourier expansion at any cusp, it proves that the Poincare series is an automorphic form in our strict sense. For m = 0 we denote Pa(z) by (3.14)
Ea(z)
(7)w
_
a
1,'1) 700
y(z)k
'yEr,\r
which is called the Eisenstein series of weight k. This has the Fourier expansion 00
(3.15)
jab (z)kEa(O bz) = bab + > rlab(n)e(nz) n=1
with k
(3.16)
%b (n)
(2z)
k
1
T(k) 0 c ksab(0, ni c).
For m > 0 we denote Pa(z) by
(3.17)
Pam(z) _ E ('Y)'w (Qa 1, ry) joo ,Y(z)ke (mca 1ryz)
7Er,\r and call Pa,,,(z) the mth Poincare series of weight k. This has the Fourier expansion 00
(3.18)
jab(z)kPam(Qbz) =
ab(m,9b)e(nz) n=1
with (3.19)
pab (m n) r l
()
kI
/41r
k b abbmn + 27riE c1Sab(m, ni c)Jk1 l\
c>0
Since there is no constant term in the Fourier expansion (3.18), we obtain
Proposition 3.2. For m > 1 the Poincare series
is a cusp form.
3. The Eisenstein and the Poincare Series
52
3.3. The Hilbert space of cusp forms Let f, g be automorphic functions with respect to a multiplier system '0 of weight k for a group r. Thus fh7 = &(y)f,
g17 = 'd(7)g
for all y E r. Hence the expression (a measure on IIll) (3.20)
(f, g) (Z) = ykf (z)9(z)dµz
is rinvariant. Therefore we can define the inner product (due to H. Petersson) (3.21)
(f, g) =
J\a(f, g) (Z)
provided the integral converges absolutely. For f = g we set
r\
IIf112 = (f, f) = f Ykif(z)I2dµz. A cusp form f has exponential decay at cusps, so IIf II < oo. The linear space of cusp forms Sk(r, i9) equipped with the Petersson inner product is a finite dimensional Hilbert space.
The Poincare series Pa,,,(z) associated with a singular cusp a and a positive integer m belongs to Sk(r, z9). We shall compute the projection of
any f E Mk(r, i9) onto Pa,,, by the unfolding technique. Without loss of generality we can assume that a = oo and as = 1 (take the conjugate group a; lraa if necessary). In this case we obtain
(f,
_k
f \MYkf (z)
f
e(myz)dpz
(IM yz)k f (yz)e(myz)dµz \8i 7Er.\r 0o
I
= f f yk f (z)e(mz)dµz. 0
o
Inserting the Fourier expansion of f at cusp a = oo, say
f (z) _
=o
f (n)e(nz),
3.3. The Hilbert space of cusp forms
53
we get a contribution from the mth term only: 00
(f, Pr) = f (m) J
yk2 exp(41rmy)dy
= r(k  1)
0
(41rm)k1
f
In general, for a singular cusp this argument yields
Theorem 3.3. Let Pp,,, be the mth Poincare series attached to a singular cusp a for a multiplier system 0 of weight k > 2. Let f be an automorphic form with respect to the same multiplier system. Then (3.22)
(f, 1'asn) =
r(k 1)1 (4Trrrn) '
fa(m)
where fp(m) denotes the mth Fourier coefficient off at the cusp a, i.e. one has the expansion
fp(n)e(nz).
(z) _ n=0
Combining (3.22) with (3.19), we obtain Corollary 3.4. Let a, b be singular cusps for a multiplier system i9 of weight k > 2. For m, n positive integers we have (3.23)
(1 am, Pbn)
(4uj)k_1
c
Sab(m, n, c)Jk1
c>O
In particular, (3.24)
=
((4irm){1+2iri>c'Saa(mm;c)Jk.l k  1)
rli1 c
}
c>0
Corollary 3.5. Let a be a singular cusp for a multiplier system 17 of weight k > 2 on r. The space of cusp forms S1. (r, i9) is spanned by the Poincare series Pp,,, for m = 1, 2, 3, ... .
Proof. The linear space spanned by Pa,,, is closed in Sk(F, i9) because the whole space has finite dimension, and a function orthogonal to this subspace must be zero because all its Fourier coefficients vanish by virtue of (3.22). This proves Corollary 3.5. Since Sk(I', z9) is spanned by with in = 1,2,3,... and on the other hand s, (r, d) has finite dimension, it. follows that the Poiucar6 series
3. The Eisenstein and the Poincar6 Series
54
are linearly dependent. There are many open problems about Poincarb series such as:
Problem 1. Find all the linear relations between Pam(z). Problem 2. Find a basis of Sk(r, t9) consisting of the Poincar6 series. Problem S. Which Poincar6 series do not vanish identically? For the modular group it is known that the first Poincar6 series Pm(z) with m < dim Sk (r) span the space Sk(r). Recall that dim Sk(r) = k/12 +
0(1) by Theorem 1.4. R. Rankin has proved that Pm(z) does not vanish identically if m < c(e)k2f. C.J. Mozzochi extended this result for the group ro(q), showing that P,,,(z) # 0 if m < c(e)(kq)2. These results depend on Weil's bound for Kloosterman sums.
Exercise. Prove in general that Pam(z) does not vanish identically for any m < ckIDI, where IDS is the volume of the fundamental domain. Hint: estimate (3.24) using the trivial bound for Kloosterman sums (4.2), and apply (2.35). Choose an orthonormal basis of Sk(r,19), say .P, and expand Pam into this basis. By (3.22) we get (3.25)
P..(z) =

r(k  1) (41rm)k  1
f a(m)f (z)
fEF
where fa(m) denotes the mth Fourier coefficient of f at the cusp a. Comparing the nth Fourier coefficients at the cusp b on both sides of (3.25), we obtain by (3.19) Theorem 3.6. Let a, b be singular cusps for a multiplier system 19 of weight k > 2 on r. Let Jr be an orthonormal basis of Sk(r,19). Then for any positive integers m, n we have (3.26)
I'(k 1) (47r mn)k_1
fE
fa(m)fb(n)
=bmnaab+21rikEc 1Sah(m,n;c)Jk_1 c>O
(47 r
mn' Vr
.
e
Remark. For k = 2 the Poincar6 series (3.17) does not converge absolutely; nevertheless, the formula (3.26) remains valid (the series of Kloosterman sums and Bessel functions converges absolutlely).
3.3. The Hilbert space of cusp forms
55
Example 1. Let IF = SL2 (Z) and k be an even integer, k > 2. The Kloosterman sum for the trivial multiplier system and for the cusps at oo becomes the one which was originally introduced by Kloosterman [Klol, namely
ma + nd
S(m, n; c) _
(3.27)
eC
c
ad=1 (mo d c)
For in = 0 this degenerates to the Ramanujan sum 's(0' n; c)
d(mod c) e
\dc
/
E . ()a
6I(c,n)
where the star restricts the summation to the classes prime to the modulus. Hence for k > 2 slk = aki(n)/n k '((k) E C kS(0, n; c) _ ((k) c>0
61n
so by (3.16) the nth Fourier coefficient of the Eisenstein series of weight k is
77(n) =
(27ri)k
((k)r(k) Qk1(n)
This agrees with the previous result (1.49). Now suppose in > 0. Then the Poincar6 series
P,n(z) _
7y(z)ke(m7z)
ryer.\r
can be written more explicitly as
PMW = 2
(cz + d) ke I and (c,d)=1
 c(cz + d) )
The Fourier expansion (3.18) becomes 00
Pm(z) = E p(m, n)e(nz) n=1
where
(!)
p(m, n) = in
{6rnn +
27rik
E
c1S(in,
47r cmnl
n; c)JkI
,
c>o
and S(m, n; c) is the original Kloosterman sum (3.27). By the Selberg relation (4.10) for S(m, n; c) and by a similar relation (3.33) for b117 we deduce
that (3.28)
p(m, n) =
dk1 p (1, mnd2)
in1k
.
dl(m,n)
This formula belongs to the theory of Hecke operators (see Chapter 6).
3. The Eisenstein and the Poincarc Series
56
Example 2. Let r = SL2(Z) and k = 12. The space of cusp forms S12(r) is onedimensional, spanned by
O(z) = (27r) 12 E r(n)e(nz). 1
Therefore any Poincar6 series P7 (z) is a multiple of 0(z), say ,,,,(z) c(m)A(z). To compute the constant c(m) we appeal to (3.22), getting c(m)IIOII2 = (0, Pm) = 27rF(11)(2m)"r(m) Hence (3.29)
P,,,.(z) = 27rP (11 ) r(m) A (Z)
(2m)" II0II2
.
Applying (3.26), for f (z) = 0(z)/IIAII we obtain (3.30)
r(m)r(n) = v(mn) z 6m,n + 27r> c1S(m, n; c)J11
(4)
r>0
where v is an absolute constant. One can express this constant by the value at s = 1 of the symmetric square Lfunction associated with f ; namely, we get v = IIIII2/47r13r(11) =
by (13.62). In particular for n > 1 we have (3.31)
r(n) = 27rvn 21 > c'S(1, n; c)Jll
(47r c
c>o
Ramanujan Conjecture (proved by P. Deligne in 1974). Ir(n)I <, n 2 d(n)
(3.32)
Applying Selberg's formula (4.10) to the Kloosterman suns in (3.30) and the simple identity (3.33)
S1,nind2
dj(m,n)
we obtain the relation (3.34)
r(m)r(n) _
d11r (mnd2) dI(n,,n)
.
Chapter 4
Kloosterman Sums
Kloosterman sums are intimately related to Fourier coefficients of automor
phic forms. They also play a very important role in the modern analytic number theory. Most recent results and applications of Kloosterman sums lie beyond the scope of this book. We shall only prove a few elementary estimates which, nevertheless, yield quite good bounds for the Fourier coefficients of automorphic forms if r' is a congruence group.
4.1. General Kloosterman sums For a general group and multiplier system the Kloosterman sum (3.13) is quite intricate; we cannot even describe explicitly the set of moduli (2.33). We can only estimate trivially (4.1)
SO(m,n,,c)
cab c2
for any c E C (a, b) (see Proposition 2.8). This is a rather poor bound; however we can do slightly better on average (apply (2.36)): c 1ISab(m,n;c)I
(4.2)
cab X.
O
Hence the series (4.3)
Z(s) _
c23Sab(71l, tn; c) c>O
converges absolutely in Re s > 1. In the spectral theory of automorphic forms one shows that Z(s) has meromorphic continuation to the whole complex splane. This is a very deep result (which is due to A. Selberg, [Se1]) and technically difficult to establish in full generality (cf. (1w2)). 57
4. Ifloosterman Sums
58
4.2. Kloosterman sums for congruence groups Kloosterman sums associated with a congruence group and the trivial multiplier system can be expressed in terms of the classical Kloosterman sum
ma + nd ad=_ 1(mod c)
c
Consider the Hecke congruence group r = ro(q) and the cusps 00, 0 (recall the last paragraph of Section 2.5). The sets of moduli are C(oo, oo) = C(0, 0) = {c = eq : e E N}
C(oo, 0) = C(0, oo) = {c = £ ,/ : e E N, (e, q) = 1} ,
and the Kloosterman sums are given by (see (2.39)) (4.5)
S.. (m, n; c) = Soo(m, n; c) = S(m, n; eq)
(4.6)
So. (m, n; c) = Sao(m, n; c) = S(mq, n; e)
where q denotes the multiplicative inverse of q modulo e, i.e. (4.7)
qq = 1 (mod e).
This notation for the multiplicative inverse will be used throughout the book. Be aware that q depends on the modulus e. Exercise. Let r = ro(q) with q = rs, (r, s) = 1. Consider two cusps, oo and 1 /r with the respective scaling matrices a. _ (1 t) and at /,. Show that the set of moduli are
C(1/r,1/r) = {c = 1q: e E N} C(oo, 1/r) = {c=ervrs :QE N, (e, s) = 1}
and the Kloosterman sums are given by S11,.,11,.(m, n; c) = e ((m  n) s)
S(m, n; eq)
Soo,1/,.(m, n; c) = e (mr) S(ms, n; er). s
4.3. The classical Kloosterman sums
59
4.3. The classical Kloosterman sums In this section we shall evaluate the Kloosterman sum (4.4) for special moduli. First we record the following elementary properties:
S(m, n; c) = S(n, m; c),
(4.8)
(4.9)
S(am, n; c) = S(m, an; c)
(4.10)
S(m, n; c) =
if (a, c) = 1,
dS(mnd2,1; cd1) dj(c,m,n)
The third property is due to A. Selberg [Se2]. In particular this gives (4.11)
S(m, n; c) = S(mn,1; c)
if (c, m, n) = 1.
Moreover we have (4.12)
S(m, n; c) = S(qm, 4n; r)S(rm, fn; q)
if c = qr with (q, r) = 1, where q, r are multiplicative inverses of q, r to moduli r, q respectively. The last property (twisted multiplicativity) allows one to reduce the problem of evaluating S(m, n; c) for any c to that for prime power moduli.
Lemma 4.1. If c = pea with a >, 1 and (c, 2n) = 1, then (4.13)
(2c )
S(n, n; c) = c2 (e
+e
2n)
)
Proof. Put d = a(l + bpa), where a ranges mod pea, a is prime to p, and b ranges freely mod pa. Then every primitive class mod plc is covered pa times. We have d  a(l  bpa) (mod p2c), so
S(n n, p2a)=p a
E e (na \ ++ d+ n (a  d)b)/I
I:.
a (mod p 2a )
E*
a (mod pa)
L
(mod
p)
elna+d
`. P2a l
a=_a (mod pa )
The solutions to the last congruence are a = ±1 + tpa with t ranging freely modulo pa, and a = ±1  tpa. Hence we deduce (4.13).
4. Kloostermazi Sums
60
Lemma 4.2. If c = p2a+1 with a >, 1 and (c, 2n)) = 1, then (4.14)
S(n, n; c) = 2
( c } c2 Re e,e (?)
where (n) is the LegendreJacobi symbol and Ec = 1, i according to whether
c  1 or 1 (mod 4). Proof. Putting d = a (1 + bpa+i), we obtain as before a+a
e n p2a+1)
S(n, n; pea) _
.
a (mod pea+I )
a=a (mod pa)
The solutions are a = ±1 + tpa with t ranging freely mod
pa+i, and a =
±1  tpa ± t2p2a. Hence
e(n
S(n, n; pea+') = Me
2+ t2 2a ) pea
t (mod p°+i )
= 2Re pae
2n
9(n, p)
p2at t
where 9(n, p) is the Gauss sum 2
(4.15)
g(n,p) _ t (mod p)
e
(_) = EP (p) p2.
This formula will be established later in Lemma 4.8. Hence (4.14) is proved. We put both formulas (4.13) and (4.14) (which are due to H. Salie, 1936) in a unified form
Proposition 4.3. If c = pa with J3 > 2 and (c, 2n) = 1, then (4.16)
S(n, n; c) = 2 (c) cz Re Eye
2cn
One can generalize this result slightly for sums S(m, n; c) with c = plj, Q
2, such that (c, 2mn) = 1. We have S(m, n; c) = 0 unless m
(mod c), in which case (4.9) and (4.16) yield S(m, n; c) = S(Cn, Cn; c) = 2 ( In any case we conclude
 c)
c^ Re e,e
(261). c
C271
4.4. Powermoments of Kloosterman sums
61
Corollary 4.4. If c = p3 with Q > 2 and (c, 2mn) = 1, then IS(m, n; c) I <, 2c!2.
(4.17)
The same arguments work for c = 213, giving similar results.
The Kloosterman sum S(m, n; p) to a prime modulus resists any explicit evaluation in the above fashion. In 1948 A. Weil proved the Riemann hypothesis for curves over finite fields, from which he deduced that S(m, n; p) = ap + Op
(4.18)
where ap, Op are algebraic numbers such that ap = Op and apfp = p. Hence
Iapl=I1pl=pa and (4.19)
I S(m, n;p)I 5 2p2
Combining all cases, one can derive by the multiplicativity property (4.12)
that
Theorem 4.5. For any c > 1, m, n we have (4.20)
I S(m, n; c)I
(m, n, c) Z c2,r (c)
where r(c) denotes the divisor function.
4.4. Powermoments of Kloosterman sums Today the Riemann hypothesis for curves, hence the Weil bound for Kloosterman sums, can be established by elementary means (due to S.A. Stepanov, W. Schmidt and E. Bouibieri), yet the arguments are quite involved. It. is much easier to estimate S(m, n; p) on average with respect to na, n. In this section we estimate a few of the powermoments defined by E,.
(4.21)
VA (p) _
Sk(a,1;p)
a (mod p)
By (4.9) we obtain
(p  1)Vk(p) = EY: Sk(a, b;p) abAO (niod p)
J:E
SA(a,b;p)
 (p 1)A
2(1)'`(p 1)
a,b (mod p)
= r'k(p)2r'  (p 
1)
4. Kloosterman Sums
62
where vk.(p) is the number of solutions to the system
(modp) 0
(modp).
For k = 1 we have no solutions, so vl (p) = 0 and V1(p) = 1.
(4.22)
For k = 2 the solutions are x1 = x2  0 (mod p), so v2(p) = p  1 and V2(p) =p2  p  1.
(4.23)
For k = 3 the solutions are x2 = xly, x3 = X1(1 +V) where xl $ 0 (mod p) and y2  y + 1 0 (mod p). The number of solutions in y is 1 + (=3), so
V3(p) = (p  1) (1 + (=3)). Hence p (4.24)
3) p2 + 2p + 1.
V3 (P) _ (
\ p
For k = 4 we write
V4 (P) _ EE 7I(u>v)2 u,v (mod p)
where rj(u, v) is the number of solutions to the system
x+yu (modp), xJ1+y1v (modp). If u=0then v0and r7(0,0)=p1. Ifu 0then v$0,x+y=u, xy  uv1, so x, y are solutions to the quadratic congruence z2
 tiz + uv1 = 0 (mod p).
ty1, so the number of solutions is The cliscriminant. is A = v2  42(A) y(u,v)=1++(14/uv)
p
p Hence
EY: uvOO (mod p)
?(U, V)2
?(U, 1)1
= (p  1)
u O (mod p)
(i+()) =(p1) E ml (mod P) = (p  1) 4L ::l + 1
4
(p  1) (2p  5),
4.4. Powermoments of Kloosterman sums
and v4(p) = (p (4.25)

63
1)2 + (p  1) (2p  5) = 3(p  1) (p  2), giving
V4(p) = 2p3  3p2  p  1.
By (4.25) we infer by dropping all but one term in (4.21) that (4.26)
IS(m,n;p)I <2pi,
if p f 2mn. This result and the above methods of proof are due to H.D. Kloosterman (from his thesis of 1926; see [K1o]). By (4.23) and (4.25) we get V4(p)V2(p)1 > 2p 2, whence (4.27)
IS(a,1;p)I > (2p  2)21
for some a (mod p), showing that the constant 2 in Weil's bound (4.19) cannot be replaced by any number < V2. Therefore some of the roots ap are closer to the real axis than to the imaginary one. Incidentally, one can show that S(m, n; p) does not vanish, for S(m, n; p) _ 1
(mod 7r)
where 7r = 1  (p is the prime factor of p in the cyclotomic field Q((p). In 1931 H. Salie and H. Davenport (independently) estimated the sixth powermoment (still elementarily) (4.28)
V6 (p) << p4
from which they deduced that (4.29)
S(a, l; p) < p:
.
Higher moments of Kloosterman sums resisted any attack until 1979 when N. Katz skillfully employed P. Deligne's profound theory of exponential sums over varieties over finite fields. It turns out that more natural than the powermoments are the meanvalues of Tchebyshev polynomials Uk.(x) at normalized Kloosterman sums. In view of Weil's bound we have (4.30)
p ^i S(a, l; p) = 2 cos Op(a)
where 0 < Op(a) <, 7r is called the Kloosterman sum angle. The Tchebyshev polynomials are defined recursively by Uo(x) = 1,
U1(x) = x
4. Kloosterman Sums
64
xUk(x) = Uk1(x1 + UkF1(x)
One shows that
(1)t(k  E)ixk2t
Uk(x) _
8I(k2811
Another way of defining the polynomials Uk(x) is by sin(k + 1)0 Uk(2 cos 0) =
sin 0
The generating series of Uk(x) is 00 y2)1
Uk(x)yk = (1  xy + k=0
Theorem 4.6 (N. Katz). For k
1 and p > 2 we have
I(k+1)pa.
1 E* Uk(2cosOp(a))I
(4.31)
a (rnod p)
The key feature of the Katz estimate lies in the fact that the Tchebyshev polynomials Uk(x) form an orthonormal basis in G2([O,ir],µ) with respect to the SatoTate measure µ = 21r1 (sin O)2d0.
(4.32)
Thus Theorem 4.6 implies that the Kloosterman angles Op(a) are asymptotically equidistributed for the SatoTate measure as a ranges over the primitive classes (mod p) and p  oo. In other words, for any continuous function f on [0, 7r] we have linl
op
f (O (a)) _
1
a
? ff(O)(sinO)2dO.
(mod p)
A much harder question is, how is the Kloosterman sum angle distributed with respect to the modulus? Conjecture. For a fixed a 0 0 the Kluusterrnan angles Op(a) are asymptotically equidistributed for the Sato Tate measure as p ranges over primes,
i.e.
limn
00
E f (Op(a)) _
()p<x x
_
f (0)(sin0)2d0. fo
This conjecture seems to be out of reach by present methods. We do not even know if (the case of f (0) = 2 cos O) (4.33)
line
1 E S(a,1; p) = 0 7r (X) p
VP
4.5. Sums of Kloosterman sums
65
4.5. Sums of Kloosterman sums In this section we establish estimates for sums of IS(n, n; c) I over special moduli by completely elementary means. These estimates are strong enough to serve as a substitute for Weil's bound (4.20) in various important applications (see our elementary proof of (5.19)). We begin by showing that (4.34)
r a (mod r)
IS(an, an; r)12 <, 4(n, r)rr(r).
Indeed, opening the Kloosterman sum, squaring it out and then summing over a (mod r), we see that the left side is equal to the number of solutions to the congruence n(x +.t) _ n(y + y) (mod r). This is equivalent to n(x  y)(xy  1) _ 0 (mod r) in (xy,r) = 1, whence by a local consideration one can easily deduce that the number of solutions is S 4(n, r)rr(r), as claimed. Next we infer by (4.34) that A(Q, R) _ EY: I S(4n, 4n; r) 12 q
r
(Qr + 1 I
IS(an,an;r)12 a (mod r)
S E 4(Q + r)(n, r)rr(r) 5 4r3(n)R2(Q + R) log 4R. r
Hence by (4.12) and Cauchy's inequality
13(Q,R) EE IS(n,n;gr)I q
4r3(n)QR(Q + R) log 4QR. Finally we remove the condition (q, r) = 1, getting A(Q, R) 2 A(R, Q)
C(Q, R) = 1: 1: I S(n, n; qr)I q
> > 13(Ql a, R/b) 5 3r.3(n)QR(Q + R) (log 4QR)2. a
Hence by partial summation we deduce
Lemma 4.7. For Q, R > 1 we have (4.35)
EE q
(qr)1 {S(n, n; qr)I < r3(n)(Q + R)(log4QR)3.
4. Kloosterman Sums
66
Remark. By Weil's estimate (4.20) one could derive (4.35) with Q + R replaced by (QR) 2 , which has the same order of magnitude if Q = R.
4.6. The Salie sums They are obtained by twisting the classical Kloosterman sums with the JacobiLegendre symbol:
(d) e (m2 +nd)
T(rn,n;c) _
(4.36)
d (mod c)
c
c
J
Salie sums occur in the context of automorphic forms with respect to the theta multiplier system (see (2.73)) by means of the quadratic reciprocity law. These sums can be evaluated explicitly in terms of the Gauss sum (4.37)
g(n,c) _
e t (mod c)
We begin by evaluating the Gauss sum.
Lemma 4.8. If (c, 2n) = 1 then g(n, c) =
(4.38)
where
ec _
(4.39)
11 i
ifc ifc
1 (mod 4)
1 (mod 4).
Proof. Put c = qr2, where q is squarefree, and change the variable t (mod c) in (4.37) to t = x + qry with x (mod qr) and y (mod r). We get t2 = x2 + 2qrxy (mod c), whence
g(n,c) _
j e (nx) L t\) C2n.xy
x (mod qr)
y (mod r)
r
The innermost sum is equal to r if x  0 (mod r), and it vanishes otherwise. Therefore (4.40)
g(n, c) = rg(n, q)
For q a squarefree number we have
#{x (mod q)::r.'=y (mod
q)}(1 +\p// rlr
/
(d).
Ir
4.6. The Salie sums
67
Hence
9(n,q)=E E (d)e(q ) dlq y (mod q) The inner sum vanishes save for d = q, giving (4.41)
(nq)
9(n,q)=
e(q) =
q
y (mod q)
g(1, q).
ve Combining (4.40) and (4.41), we der(nC) (4.42)
9(n, c) =
9(1, c).
Finally we get by (1.10) (4.43)
9(1,c) = e,c2
and complete the proof.
Now we evaluate the Salie sums. Suppose (c, 2n) = 1. Let us consider the function
F(x)_
E
d (mod c)
(d)e(md+cndx2 ) J
defined in x (mod c). The Fourier transform of F(x) is P(y)
=xE F(x)e (f!) (mod c) _
d d
(
r
(c)
e
and
a
(ndx2_ C
x (mod c )
)
xE /
Complete the square; the innermost sum becomes e (Qndy2) g(nd, c). If we substitute 9(nd, c) = (c) g(71, c)
the LegendreJacobi symbol disappears in the Fourier transform F(y), so that it becomes the Ramanujan sum F(y) = g(n,c)
e
(md 
I
C
d (mod c)
= g(n, c)
E_
e
((4mn
d (mud c)
= g(n., c)
E
dV2 )
dI(4,nni 2,c)
 y2)., dc
dtt(c/d).
4. Kloosterman Sums
68
Hence by the Fourier inversion
F(x) _ 1 y (mod c)
F(y)c ` c
9(c+c)>dµ(c) r l
l
r
e\xc/
y (mod c)
d1c
y24mn (mod d)
If (x, c) = 1 this simplifies to F(x) = g(n, c)
e
F%Zmn (mod c)
Taking x = 1, we obtain the following.
Lemma 4.9. If (c, 2n) = 1 we have (4.44)
E
T (m, n; c) = (c) g(c)
y2ctnn (mod c)
e (v).
Corollary 4.10. If (c, 2n) = 1 we have (4.45)
IT(m,n;c)l <, car(c).
In the case of m = n we can write explicitly all the solutions to y2 = 712 (mod c), namely y = (ad  bb)n, where a, b range over integers with ab = c and (a, b) = 1. Thus (4.44) becomes
Corollary 4.11. If (c, 2n) = 1 we have (4.46)
T (n, n; c) = (c) g(c)
c (2n 1 ttb=c,(tyb)=l
G

J
I
111 /
Remark. This result was first established by H. Salie (Sal), but only for c a prime power. Originally we [Iwl] extended the Salie formula to the above form by an appeal to the quadratic reciprocity law; however, the present derivation (due to P. Sarnak [Sar]) is more direct. It is important that our formula does not depend on prime factorization of the modulus. The very specific shape of (4.46) (a kind of Dirichlet convolution) will be crucial in its application to estimate the Fourier coefficients of cusp forms of half integral weight (see Section 5.3).
Chapter 5
Bounds for the Fourier Coefficients of Cusp Forms For various constructions to be carried out in later chapters, such as the Lfunctions associated with a cusp form, we need to control the growth of the Fourier coefficients. In this chapter we shall establish the required estimates. Throughout we assume that r is a discrete group for which oo is a cusp of width 1, i.e. the scaling matrix is v00 = 1 (the more general case of an arbitrary cusp can be obtained by conjugating the group). We also assume that i9 is a multiplier system of weight k > 0 for r which is singular at the cusp oo.
5.1. General estimates Let f be a cusp form for r with respect to the multiplier system 19. Thus f has the Fourier expansion
f (z) =
L a(n)e(nz).
Our problem is to estimate the coefficients a(n) = a f(n). Quite strong results are deduced rather easily from the observation that (5.2)
F(z) = y4lf(z)I
is a bounded function on the whole upper halfplane H. This is clear because F(z) is Fperiodic and has exponential decay at every cusp. Therefore, we 69
5. Bounds for the Fourier Coefficients of Cusp Forms
70
have f (z) G< yk/2
(5.3)
if z E H
where the implied constant depends on f. Conversely, if F(z) is bounded in H then for any a E SL2(R) we have I fro (z) I = y12F(az) << y 3 . Hence fla,(z) vanishes as y  oo, proving that f is a cusp form. Therefore we have established the following criterion.
Criterion for Cusp Forms. I. Suppose f is an automorphic form of weight k > 0 for a group r. Then f is a cusp form for r if and only if (IM z)1/2I If (z) I is bounded in the upperhalf plane.
By the Parseval formula and (5.3) we obtain 1
la(n)
I2e4any = f If 0
n
(z) I2dx <<
yk,
whence
E la(n)I2 << yke4nNy n
for any y > 0. Choosing y = N1, we obtain
Theorem 5.1. For any N >, 1 we have (5.4)
E la(n)I2 K Nk n
where the implied constant depends on f.
Remark. The upper bound (5.4) is best possible, for one can prove (by the RankinSelberg method, see Section 13) the more precise asymptotic formula (5.5)
E Ia(n)I2  cNk n_
as N > oo, where c is a positive constant which depends on f. The upper bound (5.4) shows that the Fourier coefficients of cusp forms satisfy (5.6)
a(n) << n2
on average. For any individual coefficient it yields a(n) << n 2.
Using Cauchy's inequality, we deduce by (5.4) the following estimate:
71
5. 1. General estimates
Corollary 5.2. For any N >, 1 we have (5.8) n<'N
This bound can be improved substantially if we drop the absolute value to allow cancellation between the terms. Indeed we established the following:
Theorem 5.3. For any real a and N 3 1 we have (5.9)
a(n)e(an) << N L21 log 2N n<,N
where the implied constant depends only on f (not on a). Proof. The Fourier coefficients are given by
a(n) =
'I
J0
f (z)e(nz)dx.
Hence the sum of coefficients twisted by the additive character e(an) is equal to
a(n)e(an) = J f (z + a)SN(z)dx 1
n<,N
0
where
SN(z) = E e(nz) = e(Nz)
1
K e27rNyl1  e(z)l1.
O
Since f (z + a) << yk/2 and 1
11  e(z)lldx << log(2 + y1),
Jo
we obtain
E a(n)e(an) << yk/2e2iNy log(2 + y1) n_
where y is arbitrary positive number. For V = N1 this yields (5.9).
The improvement in the estimate (5.9) is due to the variation in the sign of a(n), since (5.8) is essentially best possible. The logarithmic factor in (5.9) cannot be reduced to a small constant, since it would contradict (5.5) by Parseval's identity. Theorem 5.3 shows that the Fourier coefficients of a cusp form are asymptotically orthogonal to additive characters. The fact that the bound (5.9) does not depend on a makes it possible to derive (by means of additive characters) the same result for a sums of the Fourier coefficients restricted to any arithmetic progression.
5. Bounds for the Fourier Coefficients of Cusp Forms
72
Corollary 5.4. For any q 3 1 and a (mod q) we have a(n) << N 2 log 2N
(5.10) n_< N
na (mod q)
where the implied constant depends only on f.
The argument used in the proof of Theorem 5.3 can be modified to estimate a bilinear form in the Fourier coefficients of the following type:
Theorem. For any complex numbers a,,,,13n we have
EE
(5.11)
m) << NZ IIaII IIf3II
m
where hall, 111311 denote the £2norms of a = (a,n) and /3 = (13n) respectively.
Proof. The restriction m < n can be ignored because a(e) = 0 if t 5 0. We estimate the resulting bilinear form by applying the CauchySchwartz inequality and the Parseval identity as follows: 1
in
fo
n
E,Qne(nz) dx
f (z) E arne(rnz)
a,n13na(n  m) =
m
<< y 2 a >
n
(Ia,2)
t
2
22
Iaml2e4amg
in
N4
t IQn12eaany
n
2
(l/3l2 l 2
=
N4IIall11,11
n
by choosing y = N.
5.2. Estimates by Kloosterman sums In the general case the bound (5.7) is essentially the best possible; definitely (5.6) cannot hold for all n. However, if one has a nontrivial estimate for Kloosterman sums, then one can improve the individual bound (5.7) by an appeal to the Fourier expansion of Poincare series ro
(Y)?y(z)tie(nvyz) _
Pm(z) _ 'yEr,\r
p(m, n)e(nz).
=1
5.2. Estimates by KIoosterrnan sums
73
We assume that k > 2, so any cusp form is a linear combination of P,,,(z) with m 3 1; therefore, it suffices to estimate p(m, n). By (3.25) and the CauchySchwartz inequality we obtain (5.12)
Ip(m, n) I2 < (m
p(m, m)p(n, n)
Hence it suffices to estimate the diagonal coefficients (see (3.19)) (5. 13)
p(n, n) = 1 + 2 rik E cS(n, n; c)Jk_1
(!) C
c>O
Recall that (see (3.25)) p(n,n) =
(5.14)
r(k  1)
(47rn)k1
11(n) 12 ! E.F
where 2 = (f) is an orthonormal basis of Sk(r, z9). Suppose the Kloosterman sums S(n, n; c) are bounded by c2°1 on average; more precisely, suppose we have E c2a IS(n, n; c) I << ne
(5.15)
c>O
< or < 1 and any e > 0 with the implied constant depending on or, e and the group r. We have for some (5.16)
J" (X) < lain{x",x z }
x6,
if  2 < 6 < v.
By (5.15) and (5.16) with v = Ic  1 and 6 = 2a  1 we derive (5.17)
p(n,n) < n
Hence by (5.14) we get
Proposition 5.5. Suppose the Kloostersaun sums associated with a multiplier 79 for a group IF satisfy (5.15). Then the Fourier coefficients of any cusp form f E Sk (r, 79) of weight k > 2 are bounded by (5.1$)
a(n) <
If r is the Henke congruence group and 19 is the trivial multiplier system, we have the Weil hound (4.20), which gives (5.15) with any a > In this case (5.18) becomes .
(5.19)
iE a(71) «71.2i.
5. Bounds for the Fourier Coefficients of Cusp Forms
74
We shall prove (5.19) by elementary means without appealing to the Weil bound for individual Kloosterman sums. Our argument makes use of (4.35) instead of (4.20), but not directly. Suppose f is a cusp form for ro(N) with respect to the trivial multiplier system of weight k > 2. Then f is also a cusp form for any ro(q) with q = 0 (mod N). Let IIfIIq denote the norm of f with respect to the group ro(q), so we have (5.20)
IIfIIq = [ro(N) : ro(q)]IIf1IN
where the index is determined by (2.25). If a(n) is the nth Fourier coefficient of f (see (5.1)), then (5.21)
f (n) = a(n)/II f II q
is the nth Fourier coefficient of the normalized form fq = f /IIf IIq. Take an orthonormal basis F of the space SkjI'o(q)) which contains fq. By (5.13), (5.14) and (5.21) we get
c1S(n,n;c)Jx_u
1+2iri_A
(5.22)
(47rn)k)IIIfII2'
(.±)
c0 (mod q)
by dropping all but fq from F. Next we sum this inequality over q = 0 (mod N) with Q < q <, 2Q. Since for fixed N the sum 7n f(Q) _
(5.23)
IIf IIq 2
Q
q=_0 (mod N)
is bounded by positive constants (see (5.20) and (2.25)), we obtain 00
n1Aa(n)2 << Q +
>(qr)'IS(n,n;gr)Jk_1 Q
4I. ()
By (5.16) for v = k  1 > 1 and (4.35) we get (5.24)
nl _k
a(n)2 < Q + (Q + R) min QR, (Q)
(nQR)c
for some R > 0. The worst bound occurs in (5.24) for R = nQl, namely (5.25)
nlka(n)2 < (Q + nQ1)n
Here the best choice is Q = n°, giving (5.19).
5.3. Coefficients of cusp forms with theta multiplier
75
5.3. Coefficients of cusp forms with theta multiplier In the study of representations by quadratic forms in an odd number of variables (see Chapter 11) we shall encounter cusp forms with respect to a multiplier system derived from the standard theta function (2.72). By the theta multiplier of weight k = 1/2 + 2 where t E Z we mean (5.26)
(cll2k (Ed
if r =
kill
a c
b
d
E re(4).
The Fourier coefficients of cusp forms in Sk(ro(N), i9) satisfy the bound
a(n) <
(5.27)
by virtue of the Salie estimate (4.45) and Proposition 5.5. This result is essentially best possible; namely the exponent 2  a cannot be reduced if n = m2. However if n is squarefree, one expects that hit a(n) « n Z +E
(5.28)
but this (an analog of Ramanujan's conjecture) has not yet been proved. We shall do slightly better than (5.27) by exploiting a cancellation in sums of Salie sums.
Theorem 5.6. Let k
z, 41N and f E Sk(ro(N),V). Then for n square
free the nth Fourier coefficient of f satisfies the bound (5.29)
a(n)<
az3r(n) logn
where the implied constant depends on f. This improvement on the standard bound (5.27) is very slight; nevertheless it is vital for applications, in particular to establish the equidistribution of integral points on the sphere (see Section 11.6). One can assume that 8IN by considering the subgroup Fo(2N) if necessary. We begin by modifying the arguments used in the proof of Proposition 5.5. For any p we have the inclusion Sk(ro(N),19) C Sk(F1)(pN),19), from which we deduce the following basic inequality (by positivity as in the proof of (5.22)):
F(k 
1)(41rn)1kIa(n)l2lIf112 [Fo(N)
I+
:
Fo(pN))`1
c1
27ri.k
c_0 (moil pN)
S(n, n; 641
(!!)
5. Bounds for the Fourier Coefficients of Cusp Forms
76
where S(n, n; c) is the Kloosterman sum,
S(n, n; c) = E* v9(r)e I nd + d
(5.30)
I
d (mod c)
Our goal is to create some kind of bilinear forms in the Kloosterman sums. To this end we average the above inequality over primes p t n in the interval
P < p < 2P, each such p being weighted by logp. We shall choose P later subject to N(logn)2 < P < ne . Since [to(N) : Po(pN)] = p + 1 and
E (p + 1) ' log p "log 2 P
the resulting left side is asymptotically pn' Ia(n)12, where p is a positive constant which depends only on f. On the right side of our basic inequality we obtain a sum of Kloosterman sums to moduli c  0 (mod N) weighted by
loge.
w(c) = P
Note that w(c) < log c if c > 0 and that the constant term 1 (the diagonal term) on the right side of our basic inequality yields w(0) P. Therefore we have
nlloa(n) I2 << P + I SI
(5.31)
where S is the weighted sum of Kloosterman sums,
w(c)c_.1S(n,n;c)Jkt
S
(n)
cu (mod N)
c
Recall that the individual Kloosterman sum satisfies
S(n,n;c) << (c,n)2c2r(c), and the Bessel function is bounded by
Hence the terms with c < C contribute to S at most =2I
(5.32)
Sb << n
J>, n)21,r(c) loge << Cn. .i (7(n) log n)2, c<,C
5.3. Coefficients of cusp forms with theta multiplier
77
and the terms with c 3 D contribute to S at most (5.33)
SO «n2 >(c, n)2T(c)c`2 loge << Dln2(T(n) log
n)2.
c>_D
We shall choose C and D quite close to n, with C < n < D, so the above bounds will be admissible. Now we are left with the central terms
S* =
w(c)c1S(n, n; c)Jk
C4C 1rn 1
C
JJ
We set c = qr, where q denotes the largest factor of c coprime with nN, so r has its all prime factors in nN and is divisible by N. Therefore w(c) = w(q), and S* splits into
S* =
(5.34)
E
r' T,
rl(nN)°O r=_0 (mod N)
where 47rn
Tr= E w(q)9lS(nn;qr)J ,
q1'
C
First we estimate Tr trivially as follows:
qrnqr {()
3
1
T, <<(n,r)Zr2T(r)T(q)q z' min 9
logq
« T (')') (11, 1 )'l 'it 1 (lug 7)2.
This will he used only for r sufficiently large, say r > R. Hence the contribution to S= of terms with r > R is bounded by (5.35)
711T, « R =1 n2 (log n)2
r(1') (n, I) ^ r'
r>R
78
It remains to estimate T, for each r < R. The Kloosterman sum in Tr factors into S(n, n; c) = S(nq, nq; r)T (nr, nr; q)
(5.36)
where qq  1 (mod r), rr = 1 (mod q) and T(nr, nr; q)
(5.37)
x (mod q)
(T)
e q(nr(x
+ x))
is the Salie sum. Observe that the Kloosterman sum S(nq, nq; r) depends only on q (mod r). Splitting into residue classes, we obtain
Tr S E_ 1S(n3, ns; r)T,.sj
(5.38)
s (mod r)
where
w(q)91T(nr, nr; q)Jk1
Trs
C
4(irn
qr
) /J
q=s (mod r),(q,n)=1
The Salie sum was evaluated in Corollary 4.11; namely, we have
T(nr, nr; q) = eq I
q
I
\ /
, aEq a (2nr
bQ
(a,b)=1
In the sequel we omit writing the condition (a, b) = 1 whenever it is redundant visavis the meaning of a (mod b) and b (mod a). By the reciprocity formula
x+y y
x
xy
(mod 1)
we write (
`E e (2n? q
\
//
= 2Re ae (In a + 2n a
a
b)
Inserting this expression into !rs we get L
2r
Trs=ES(s )(n)
:;
Re
w(ab)lable(2n C
b
+2n
)j \ _
5.3. Coefficients of cusp forms with theta multiplier
79
where
j(x) = s e(x)JA._1(27rx). Next we remove j(2n/abr) by partial summation in b. Using the estimates j(x) «x2, j(x) << 1 and j(x) << 1 we get
EI
Trs << C1(nr)1
`"'(ab) (b) a (2n(1 + rr) a)
I
a< A a< b < B (a,nr)=1 abas (mod r)
for A = (D/r) 2 and some B which depends on a such that C < arB < D. Since w(ab) = w(a) +w(b), the innermost sum splits into Va +w(a)VQ, where
E w(b)(n)el2n(1+rf)a ),
Va=
a
and Va' is given by the same sum without the weight w(b). Thus
E (jVaj+w(a)jVQj)
T,. << Cl(nr)2
a
We shall estimate each Va separately, so the dependence of B in a is irrelevant (the case of Va is similar, actually slightly simpler). We have Va = P
(loge) (p.)
a/p
()
e1
2n(a+r))
.
\\\
apl_s (mod r)
If a < P we split the summation over a into residue classes (mod ar) and apply the PolyaVinogradov estimate n 1
E t
(
«n= logn
f=a (mod ar)
(note that the character (e) is nontrivial on any arithmetic progression to modulus ar because n is squarefree and larger than ar). We obtain
Va «aPn2 logn.
(5.39)
If P < a < A we regard Va as a bilinear form in 2 and p. By Cauchy's inequality
/
f
Pi ape=$ (mod r)
p)(n)a(2n(1+rr)L!)l2 P \
5. Bounds for the Fourier Coefficients of Cusp Forms
80
where L = BP1, P1 = max(P, a/e) and P2 = min(2P, B/e). Squaring out and changing the order of summation, we get
E
Vat << L(log P)2
e
1
P
(2n(1 + rr)(p  p')we
I.
a/p'
p=p' (mod r) apt=s (mod r)
The terms with p = p' contribute O (L 2p log p). If p p' the innermost sum over a is an incomplete Kloosterman sum to modulus a. After completing the sum by Fourier technique we can apply Weil's bound (4.20) to deduce
that
E <<(pp',a)a1LlogL+(pp',a)la21 T(a)loga. t Summing over p # p', we obtain a2 << (L2P + a1 L2P2
LP2) (,r (a) log n)2.
+ a2 Since a > P the middle term can be ignored; hence
Va << (BPI +a4B2P2)7(a)logn.
(5.40)
The same estimate holds true for Va' by similar arguments. Summing over a (use (5.39) if a < P and (5.40) if P < a < A), we infer
that
T,,, «C1(nr)2 (n2p3+DP 21 +DSP2) (logn)2. Inserting this into (5.38) and then summing over s (mod r), by the trivial estimate IS(ns,ns;r)I < r we get
Tr<
<
RLC1 (DP z
+ DRP2) n2 (7(n) logn)2.
r
From (5.41), (5.35), (5.33) and (5.32) we deduce that
S<[Cn21 +D1nz +R z1n2 +R2C1 (DP We choose C = (5.42)
1151
112
14
nil], D = n lit , P = n.1 T 11
S < 711
+DBPz)nz](T(n)logn)2. 2
and R = nil, , getting
(T (71) log n)2.
Finally, inserting (5.42) into (5.31), we deduce (5.29).
5.4. Linear forms in Fourier coefficients of cusp forms
81
Remark. We could easily get a slightly sharper estimate if we did not waste various factors in r. However, any further improvement seems to be
insignificant for applications. Our choice of the parameters C, D, P, R seems to be very delicate, leaving the impression that we were lucky that at the end of the proof we have room to accommodate these parameters. However, this could be predicted quite easily before computations if one assumes there is a better estimate for sums of Salie sums (over the moduli) than that which follows from the estimates for individual Salies sums. Our improvement is obtained from cancellation due to the variation in the sign of Salies sums, which we were able to exploit by creating bilinear forms.
5.4. Linear forms in Fourier coefficients of cusp forms Choose an orthonormal basis of cusp forms in Sk(F, i9), say F. Any f E F has the Fourier expansion (we assume that oo is a singular cusp of width 1) 00
f(z)
=
f(n)e(nz).
n=1
The orthogonality of distinct forms in F has profound impact on the Fourier coefficients. One of these can be seen in estimates for general linear forms (5.43)
,Cf(a) = > an'/'J(n) n
where (5.44)
iJ, f(n) =
(I'(k  1)/(47rn)k
1) z
f (n)
are normalized coefficients. Given a sequence of complex numbers a = (a,,), one can show that there is a considerable cancellation between the terms of £f(a) for almost all f E.P. We shall give precise estimates of L f(a) for the Hecke group F = Fo(q) in terms of the level.
Theorem 5.7. Let F be an orthonormal basis in S(Fo(q)) with k > 2. Then for any complex numbers a,, we have (5.45)
E 1G f(a)12 = {1 + O(N/q)} 1IaIh2 JE.F
where the implied constant is absolute.
5. Bounds for the Fourier Coefficients of Cusp Forms
82
Proof. By (3.26) the lefthand side of (5.45), say L(q), is equal to IIaII2 +
47r mn
c1 1:1: a .,,,a,,S(m, n; c)Jt;_1
27rik
c= o (mo d
m,n, N
q)
Opening the Kloosterman sum, we derive by Cauchy's inequality
I EE
C dm
n; c)I <, E I E ame
11 E ane
d (mod c) m<,M
m,n<_N
dn(l JI
n<,N
ane
C
(dn)I2 C
d (mod c) n<N
l
= C EL aman <,(c + N)IIa1I2. m=n (mod c)
In order to attach the factor Jk_ 1 (4 7r VCmn) we expand the Bessel function into power series: 00 Jk1(x) _ E (1 e!r(k + e)
)e(x/2)k1+ze
and apply the above inequality for each term separately, getting
I E: amanS(m,n;c)Jk_1 \C47r mnl) c
m,n,
where
00
Ik1x) = e=o
Ik1
I
(47rNl (c+N)IIa1I2
\
c
J
(x/2)k1+2e
e!r(k + e)
Next we estimate (1 + c1N)Ii,1 (mod q)
C47rN)
\
c
C1 +
N1
qJ
IR_1
(47rN) q
JJ
C(k
Hence (5.46)
L(q) = (1+0(N/q)}IIaII2
with IB(x)I S 21r((k  1)Ik_1(47rx)(1 + x). This proves (5.45) if q >, N. For q < N the estimate (5.46) is rather poor because Ik_ 1(x) has exponential growth. To improve the result we apply the inequality (5.47)
L(q) < (p+ 1) L(pq),
5.5. Spectral analysis of the diagonal symbol
83
which follows from the embedding Sk(ro(q)) C Sk(ro(pq)) and the estimate for the index [ro(q) : I'o(pq)] 5 p + 1. By (5.47) and (5.46) we obtain L(q) S (p + 1){1 +9(N/pq)}11x112.
Here we choose p with N < pq < 2N, getting (5.45). To get some perspective on what Theorem 5.7 offers, first one needs to know the size of the normalized Fourier coefficients (5.44) and how many forms are in the basis T. Applying (5.45) for the sequence consisting of one term an, we get (5.48)
E I'+Gf(n)I2 = 1 + O(n/q) fE.F
Hence I f(n)I has the size (5.49)
ITI112 on average provided n << q, where
IFl = dim Sk(ro(q)) x kq H (1 + p Pi q
\\
.
/
In view of these results Theorem 5.7 asserts that the linear forms L f (a) are bounded by (5.50)
Ilall = i lanl2 n<,N
no matter what the coefficients an are, provided only that N is not too large; namely N « q. For example, for the von Mangoldt function an = A(n) this assertion is as strong as the Riemann hypothesis for the Hecke Lfunction (5.51)
Lf (s)
c*
Of (n)n'
but, of course, only for almost all cusp forms f.
5.5. Spectral analysis of the diagonal symbol In this section we consider the space Sk(Fo(q)) of cusp forms with trivial character. The effect of orthogonality on estimates for general linear forms in the Fourier coefficients of cusp forms, which we have already seen in Theorem 5.7, can be strengthened if one employs a larger spectrum. There are several possibilities. One can perform extra averaging over the level q, over the weight k, or both. Since the qaspect. was the issue in the previous
5. Bounds for the Fourier Coefficients of Cusp Forms
84
section, we focus here on the kaspect. Note that either k is even or else the space S.(ro(q)) is zero. For notational convenience we put e = k  1.
(5.52)
We begin by recalling Theorem 3.6 for cusps a = b = oo. The left side of (3.26) is (5.53)
De(m, n)
(m)'bf (n) f C Fl.
where .Fk, is an orthonormal basis in Sk(ro(q)) and ib f(n) are the normalized Fourier coefficients given by (5.44) (we do not display the dependence on q because the level is fixed throughout this section). As we know, De(m, n) is
basis independent. Indeed, (3.26) asserts that (5.54)
De(m, n) = 6(m, n) + ie+i Jt(m, n)
where 6(m, n) is the diagonal symbol (5.55)
6(m, n)
1,
j` 0,
if m=n otherwise
(4).
and Je(m, n) is given by the series of Kloosterman sums and Bessel functions (5.56)
c 'S(m, n; c)Je
Jt(m, n) = 27r c0 (mod q)
One may think of (5.54) as a kind of spectral decomposition of S(m, n), since Jt(m,, n) can be shown to be quite small uniformly in m, n « e. This range of uniformity is satisfactory for a few applications; however, it is not sufficient for some advanced problems. Therefore we shall refine (5.54) by averaging over e. To control the range of e we choose a test function g(y) which is smooth and compactly supported on R4, and we count (5.54) with weight g(e). It is of interest to average separately over (5.57)
k=0
(5.58)
k = 2 (mod 4),
(mod 4),
f  1 (mod 4) and e  1 (mod 4) respectively. This distinction is particularly important in the context of the Hecke Lfunction (5.51) because it has the functional equation with sign ik = 1, 1 respectively (see Theorem 7.2). For a given a = ±1 we consider i.e.
(5.59)
D(na, n) =
n), fair (mod 4)
5.5. Spectral analysis of the diagonal symbol
(5.60)
1:
G(m, n) =
85
g(2)Je(m, n).
P=_a (mod 4)
F om (5.54) we obtain (5.61)
>2
D(m, n) = S(m, n)
g(e) + iai"1G(m, n).
t =a (mod 4)
Our goal is to estimate G(m, n). Inserting (5.56) into (5.60) and changing the order of summation, we arrange G(m, n) as (5.62)
G(m, n) = 27r
>2
c1S(m, n; c)G
147r
mryi I
c=_0 (mod q)
where (5.63)
G(x) = > g(f)Je(x) e=_a (mod 4)
Hence we are led to the problem of computing the series (5.63) of Bessel functions of integral order, which is called the Neumann series. Various expansions into series of Bessel functions are thoroughly developed in the context of the spectral theory of Kloosterman sums (see the comments at the end of Chapter 9 of [1w2J). A complete characterization of the functions G(x) which can be represented by absolutely and uniformly converging Neumann series was given by G. H. Hardy and E. C. Titchmarsh (HTJ. Our task is simpler, since we start from the coefficient function g(y). We shall not compute G(x) exactly, but rather give a strong approximation, which is friendly in applications. We begin by applying the integral representation (see [GR]) 1/2
Mx) = ill /2 e(et)e"'ill This can be regarded as the Fourier transform of the periodic: function exp(ix sin 21rt); therefore, by Poisson's formula, g(C)JJ(x) =
00
J
9(t)e
where g(t) is the Fourier transform
g(t) _
[()etY)1i.
5. Bounds for the Fourier Coefficients of Cusp Forms
86
The above formula can be automatically generalized for series of Bessel functions twisted by an additive character
g(Q)e(cxt)Jt(x) = J
g(t)eixsin2n(at)dt
by change of variables. Now we use the additive characters to detect the congruence e = a (mod 4), getting 4G(x) =
J
g(t)c(t)dt 00
where
c(t) = eiasin2nt + ia eixcos21rt + i2a eixsin2nt + 2i1a 2i sin(x sin 27rt) + sin(x cos 21rt)
because a is odd. Since we think of the original function g(y) as being supported in a large segment, the Fourier transform g(t) concentrates on a short segment. Therefore, there will be no loss if we simplify c(t) by applying the approximations sin 27rt = 27rt + O(ItI3) cos 27rt = 1  27r2t2 + 0(t4). Hence
c(t) = 2i sin(27rtx) +
2i1a
sin(x
27r2t2x) + O(xjtl3).
The error term O(xItl3) contributes to G(x) at most O(xc3(g)), where (5.64)
cl,(.9) =
By the involution formula 9(y) contributes to G(x) exactly
f
oo ao
g(y) we see that the term 2isin(27rtx)
9(x)  9(x) = 9(x)
(5.65)
because g is supported on R+. To compute the contribution of the term 2i1a sin(x  21r2t'x) we need the formula XI
(5.66)
J
a g(t)e(t2x)dt = (1 + i.)
Jn
g(2 :ry)e(y)dy.
5.5. Spectral analysis of the diagonal symbol
87
This can be established by using the rule f §f = f g f with f (t) = e(t2x), for which we have (see (10.11) )
f(y) =
(p2) 2Ve
Inserting this into the integral and changing the variable y to 2 xy, one obtains (5.66). Put (5.67)
h(x) = 2
J
g(t) sin(x  27r2t2x)dt. 00
By (5.66) we derive that 0C
(5.68)
h(x) = 1
g(
2xy) sin(x + y  4 )(iry)1'2dy.
0
Finally adding (5.64), (5.65) and (5.68), we obtain
Lemma 5.8. Suppose g is a smooth, real function compactly supported on R+ and a = ±1. Then (5.69)
4 E g(e)Jt(x) = g(x) + iiah(x) + O(xc3(g)) tea (mod 4)
where It is given by (5.68), and the implied constant is absolute. One can refine (5.69) by using more terms in the power series for sin 27rt and cos 27rt. For example, the error term O(xc3(g)) can be replaced by Tgn. (x) + O(xc.4(g) + x2cc, (g))
From now on we assume that the test function g satisfies the following conditions: (5.70)
(5.71)
supp g C j K, 2K] g(i) << K
for all j > 0, with the implied constant depending only on j. Then we have (5.72)
g(t) < Ii (1 + jtj ).
This yields (5.73)
Ig(t)t"Idt < K".
5. Bounds for the Fourier Coefficients of Cusp Forms
88
) « (xK2)', whence we deduce by
Moreover, we have (O1/8yi)g( partial integration that h(x) <<
(5.74)
(xK_2)j
for any j > 0. By virtue of these estimates Lemma 5.8 yields (5.75)
4G(x) = g(x) + O(xK3 + (xK2)j)
for any j >, 0, where the implied constant depends only on j. Now we return to D(m, n) and G(m, n). Inserting (5.75) into (5.62) and applying the trivial bound for Kloosterman sums, we derive the approximation (5.76)
4G(m, n) = 9(m, n) + O
)lo2mn) qh + (=mn)j) K 2K2
where (5.77)
g(m,n) = 27r
E C'S(m,n;c)g (47r c
c=0 (mod q)
mnl /J
Next, by Poisson's formula we have (5.78)
9(e) = 9(0) +
4
O(K?).
e ma (,nod 4)
Combining (5.61), (5.76) and (5.78), we obtain
Theorem 5.9. Assuming the conditions (5.70) and (5.71), we have (5.79) If
4D(m, n) = g(0)S(mn., n) +ia'''lg(rn, n) +O
mn
+
' `m.n ` ' (d
log 2mr,.
where D(m, n) is the spectral sum defined by (.558), (5.59) and 9(m, n) is the sum of Kloosterman sums defined by (5.77). This approximation holds for any j >, 0, with the implied constant depending only on j.
Note that, g(m, n) is void unless V7nn > qK, in which case we have
the trivial estimate g(m, n) < i/qK. A slightly better estimate follows from Weil's bound for Kloosterman sums. However a more interesting result can be established on average by applying the large sieve inequality; namely (5.80)
EE
N<m. ,,<2N
(1t + NK ')IloII111311.
5.5. Spectral analysis of the diagonal symbol
89
On the other hand the diagonal terms contribute
g(0) E an Nn << KIIaIIIIQII N
Note that the bound (5.80) does not beat the contribution of the diagonal terms, although it comes tantalizingly close. There are problems (such as evaluation of power moments of central values of Hecke Lfunctions) which require the coefficients a,n, 3n to be special. In such cases one can do considerably better to get bounds for g(m, n) which beat the contribution of the diagonal. Unfortunately these subtle and important cases lie beyond the scope of this book. So far we have employed separately each half of the spectrum corresponding to (5.57) and (5.58). However, in many problems one can afford to employ both halves together, which amounts to summing up (5.79) for a = +1. Then the sum of Kloosterman sums g(m, n) cancels out, and we are left essentially with the diagonal term (5.81)
2 E g(l) /i(m, n) = g(0)5(m, n) + D 9 +
(=mn)j) log 2mn q
.
1 odd
At this point it is interesting to observe how the synthesis of two spectra of distinct parity dramatically improves the approximation of the diagonal symbol in terms of Fourier coefficients of cusp forms. In Section 11.4 we shall express the symbol b(m, n) in terms of additive characters by using the circle method of Kloosterman. We apply (5.81) to extend Theorem 5.7 (a similar result was established first in [1w4)).
Theorem 5.10. Let y be a function satisfying (5.70) and (5.71). Let £1(a) denote the linear form (5.4y) with complex coefficients a for 1 n < N. Thcn. (5.82)
g(k  1)
2
k even
l
f(a)12 =
fE.F,
where g(0) is the integral mean value of g(y) and 1t(K, N) satisfies
rl(I ,N) < {N/qK3+(N/gIi2)j}Nlog2N for any j >, U, with the implied constant depending only on j.
Chapter 6
Hecke Operators 6.1. Introduction It was observed long ago that the Fourier coefficients of basic classical modular forms have remarkable arithmetical properties. A fascinating example is the multiplicativity of the Ramanujan Tfunction,
T(m)T(n) =
d11T(mnd2). dj(m,n)
We have derived this formula by an appeal to the Fourier expansion of Poincare series using a similar property of Kloosterman sums (4.10). However, in quite a different way, this formula was first established by E. Hecke (1936) by means of certain selfadjoint operators. The theory of Hecke operators explains numerous other identities. More important, Hecke's original ideas proved to be vital for developments of modern fields such as Galois representations. In a general setting the Hecke operators are averaging operators over a suitable finite collection of double cosets with respect to a group: therefore a great deal of the Hecke theory belongs to linear algebra. But when one considers the spectral analysis of these operators the problems become more delicate, and complete results are known only for arithmetic groups. In this chapter we shall present the theory of Hecke operators in the context of the congruence group I'o(q) and the Dirichlet character (mod q). For economy of exposition we replace the double coset constructions with specific representatives. Throughout we assume that k is a fixed integer. For any A E GL.; (R) the slash operator is defined on functions f : H + C by (6.1)
ff.t(:) = (detA)R/JA(z)k f(Az). 91
6. Hecce Operators
92
This is the same operator as in (2.50) if A E SL2(R); it is invariant under multiplication by positive scalars. For any real number a # 0 we have lA (1a
flaA
a/
fill
Since k is a integer, the slash operation is associative:
flits = (flA) IB Obviously the slash operator maps a holomorphic function into a holomorphic function.
6.2. Hecke operators T Throughout r = SL2(Z) is the modular group. For a positive integer n we consider the set (6.2)
d):
a, b,c,dEZ, adbe=?t}.
In particular, G1 = r. The modular group acts on G from both sides, so that (6.3)
G = ran =
Lemma 6.1. The collection l
d): ad=n, 0<,b
(6.4)
fwms a complete set of right coset representatives of G modulo F. i.e. we have the disjoint partition
an = U Pp. PEA,,
Proof. Let p = ( ) E G,,.
First one can find integers y, b such that.
ya 4 6c = 0 and (y,6) = I (take y = c/(a,c) and h = a/(a, c)). Hence there exists r = (7 ) E r which gives rp Changing the sign of r if necessary, we get rp = (n I,) with ad = n, d > 0. Finally, multiplying
6.2. Hecke operators T
93
on the left by ( 1 1) with a suitable u E Z, we bring b to 0 < b < d. As p ranges over 0 the cosets rp are disjoint, for (a
(y 6)
d) =\a' d'/
implies y = 0, a = S = 1 and 3 = 0. This completes the proof of Lemma 6.1.
Note that A,, is a finite set with the number of elements o, (71) = > d.
,ti There exists a onetoone correspondence between 0,, x r and r x 0,,, i.e. for any p E 0,,, T E r there exist unique r' E r, p' E 0 such that PT =T'p.
(6.7)
For a fixed T E F, as p ranges over the whole set 0,,, p' does also. This intertwining relation is the key for resolving problems of automorphy of many functions and operators. We shall need (6.7) in a somewhat more explicit form. Writing (6.7) as
(a'
a
(6.8) a
d)
'y
6)
y'
d'/
b')
we get by multiplication (6.9)
Since
as + yb
,d
1°
bd
V+
= I we deduce that
a' = (aa+yb,yd) a' = (aa + 'yb)/(aa + yb, yd)
y' = yd/(aa + yb, yd) d _ n./(aa + yb, yd)
because a'd' = n. Furthermore, notice that a'6S' = I (mod Iy'I) by the determinant equation, so 6S' is given modulo Iy'I. Then, equating the right lower entries in (6.9), we find b' modulo d'. If we require 0 <, b' < c!' then such b` is determined uniquely, and so is 6'. Note that our argument. for
6. Hecke Operators
94
the uniqueness proves also the existence of the intertwining relation (6.7)
(except for the case y = 0, which can be verified directly; see (6.16)).
Let X : Z + C be any function. This yields a function on GL2(Z) by setting (6.10)
X(p) = X(a)
if p
For a given x and a positive integer k we define the operator Tn on functions
f : H + C by the formula (6.11)
Tf = nz1
X(P)fip
Using (6.4) we shall write T formally as (6.12)
T. = n E X(a)ak > { a nd=n
b
d
0_
This expression means that for any function f : H , C (6.13)
f
(Tnf)(z) = n Y X(a)ak ad=n
(t)
0<,b
Remember that T depends on X (well, also on k); we shall display this fact by writing T,i whenever it is necessary to avoid confusion.
6.3. The Hecke operators on periodic functions From now on we assume that (6.14)
ti(1) = I and k(
IN
_ (
1)r.
We begin to examine the Hecke operators T on functions f which are automorphic with respect to the group (6.15)
r, =
1
:uEz},
i.e. satisfying fir = X(7) f for any r E rte,. This is a fancy way of saying that f is periodic of period 1, i.e. f (z + 1) = f (z). Theorem 6.2. The operator T,, maps periodic functions to periodic functions, i.e. T :.M,:(r;>:,, X) # M1 (r.., .N:).
6.3. The Hecke operators on periodic functions
95
Proof. The correspondence (6.7) holds between 0 x ra and r. x 0,,; more explicitly, we have the identity (6.16)
Ca d)(1 1)=(1 1)\a d/
where b' = b + au  dv. Hence, if pr = r'p' with p, p' E On and T, T' E ro, then flp. = fir'p' = X(r')flp, and X(A)X(r') = X(p')X(r); therefore (Tnf),T = J1 E X(A)fp,pE A nk
=
2
X(r)X(A )fop' = X(T)Tnf
1
p'EA This completes the proof. Of course, one can see Theorem 6.2 more directly; nevertheless the above argument prepares us for more subtle considerations. Next we examine how the operator T. acts on the coefficients of a Fourier series
f (z) =
(6.17)
a(m)e(mz). m=o
Proposition 6.3. Suppose f is given by the series (6.17) which converges absolutely in H. Then Tn f is given by the series (T,L f)(z) _ E a (mn)e(mz)
(6.18)
rn=0
whose coefficients are (6.19)
a,,(m) = > X(d)dkla(mnd2). dj(n,,n)
Proof. Applying (6.12) we get 1
n n,=0
a(m)
:i(a)ci all=n
.
c b (nod (1)
00
_
E
z)
t=0 ad=,r OG
E E x(a)a'''a(de) n,=0
e(mz)
ud
which yields (6.18) with coefficients given by (6.19).
nz+b (711 c1
6. Hecke Operators
96
Corollary 6.4. For n >, I we have an(0) = ak_I(n,X)a(0)
where Q3(n,X) =
(6.20)
X(d)d8. din
Corollary 6.5. For m, n > 1 we have an(m) = a,n(n).
(6.21)
From now on we assume that X is completely multiplicative, i.e.
X(ala2) = X(a,)X(a2)
if al, a2 E Z,
and we show that the Tn are also multiplicative in the following style.
Theorem 6.6. For any m, n > 1 we have
T.T. =
(6.22)
X(d)dk1T,nnd2. dI(m,n)
Proof. By (6.12) we get mnT,7n _
E
b2
dbi
X(ala2)(ala2)A
ald1=,n
b1 (rood (11)
a2d2=n
Cal
1) \aL d2)
,
b2 (m,od
and after multiplying we reduce the resulting matrix by c5 =
(al,d2) to
obtain mnT,.,,T,r = (al,d2)=1
(aia2
(a, a2)'
Y(b)bk
ajd1=m/h
b, (mod d1)
a2r12=,r./h
G2 (mod d2)
b
dld2
where b = al b2 + bldg. Given a,, a2, d, , d2 as above, the upperright entry b covers every class modulo dl d2 exactly b times when bl,b range over all classes nlodulo d1, bd2 respectively. Indeed b determines b2  bal (mod d2) and bl = (b  (.,b2)d2 l (mod di). Therefore
.
91b927;mTr, _
aI a.2
(<5)bk+l
(11111 =14's
(rr1 .d2)=1
n.d,=m/,'I
b (mod(1td_1
b
d, d,
6.3. The Ilecke operators on periodic functions
97
Put a = aja2 and d = d1d2i so ad = mn62. Conversely, given a factorization ad = mnb"2, there exist unique factorizations a = aja2,d = d1d2 with (al, d2) = 1, a1d1 = m/b and a2d2 = ?t/b; indeed al = m/(m, bd) and d2 = 6d/(m, bd). Hence we can write
a ad=mnd2
dl(m,n)
d)
G (,t,od d)
which yields the formula (6.22).
Corollary 6.7. The Hecke operators commute: T,nTn = TnT,n .
(6.23)
One can invert the multiplicative relation (6.22) into (6.24)
,L(d)x(d)dkIT..IdTnid
T,,,,, =
dl(m,n)
where p(d) is the Mobius function. Hence in particular Tmn = IM T,,
if (m, n) = 1.
Taking in = p" and n = p, we get by (6.24) the following second order recurrence formula:
71,Tp,  x(p)p"T,,,i.
(6.25)
This property can be captured at once in the formula for the formal power series (6.26)
X(p)pk1X,)
T1,"X _ (1  T1,X +
JJ
For X = p' this becomes (6.27)
(1  I1,p
1
s
+
X(p)pl:i2sl
If X = 1 then the recurrence formula (6.25) is similar to that satisfied by the Tchebyshev polynomials U,,(X); in this case we obtain (6.28)
'T,,, =
U,,(p1;..1T,).
Let T denote the algebra over C generated by all T,, (called the Hock o algebra). By the above properties we conclude that T is a counuutaa(ivr' algebra generated by the operators 7 ,tn primes p.
6. Hecke Operators
98
6.4. The Hecke operators for the modular group Next we consider the Hecke operators T on the linear space Mk(r) of modular forms of weight k. Since r = SL2(Z) contains 1, we assume that k is an even integer; otherwise the space Mk(r) is zero. Under the restriction of Tie to Mk (r) one can exchange every g in (6.11) with anything in the coset rp, so one can write without ambiguity
T,tf =
(6.29)
n1
flp
pEr\G
because flip = flp if 'r E r and f E .Mi. (r). Theorem 6.8. The Hecke operator Tn maps a modular form to a modular form and a cusp form to a cusp form: Tn .Mk(1') M k(r), T,t : s, (r) , Sk,(r).
Proof. By the correspondence pT = r'p' we get (Tnf)l,r =
n21
flpT
=
flp,
n21
PEA
f
p'E0,.
since fipr = flr'p' = flp,. A cusp form f (z) has the property that it decays exponentially as z tends to any point on the boundary of H. However this property is preserved by the slash operators, so it is carried over to Tn f . Since we already proved that T, ,f is a modular form, it must be a cusp form.
Let us recall that the space of cusp forms SR(r) with k > 2 is spanned by the Poincare series jT(z)
(6.30)
k
e(nir z)
TEr.;;.\r
for m  1, 2, 3, .... Therefore T,, Pnl is a linear combination of various Poincard series (not unique). We shall find a short combination in the following.
Theorem 6.9. For k > 2, m >, 0 and n 3 1 we have. (6.31)
1 Pm = E dl(t,n)
uI=
6.4. The Heclce operators for the modular group
99
Proof. First notice that (6.32)
nR'1
(TTP,)(z) =
E j9(z)ke(mgz). gErx\G,,
Let f and 9 be any sets of right coset representatives of r,,,\F and r\G respectively. Then naturally fG is a set of right coset representatives of r \Gn, but also !9H is a set of right coset representatives of I',,,,\G,,. Hence
nk1 >: E jpr(z)ke(mfrz)
TTPni(z) =
pEc rE?{
ad=n
nk1
> jr (z)ke \(m arz + b
dk
nk1 >
=
E ad=n dim
d
b (mod d)
dlk E jr(z)`ke ( am Tz
\d
rE?t
which yields (6.31).
For m = 0 we get the Eisenstein series jr(z)k =
Ek. (z) =
rEr,,\r
(cz +
d)k
(c.d)=1
In this case Theorem 6.9 implies that Ek is an eigenfunction of all the Hecke operators Tn with eigenvalue ak1(n), i.e.
T,Ek = akl(n)Ek
(6.33)
Corollary 6.10. For m, n >,I we have
mk1 T Pvn = n 1;1 T,,, p".
(6.34)
A similar symmetry occurs for the inner product of T,,f against P,,,. Indeed, if f has the Fourier expansion (6.17), then T, ,f has the expansion (6.18), and by Theorem 2.12 and Corollary 6.5
h(k Hence
1)'(41rm)kI (T,,f,
Pm) = (6 (m) _ (t,,,(11).
6. Hecke Operators
100
Proposition 6.11. For in, n > 1 and f E Mk(r) we have p"). rn11(Tnf, p,,.) = (6.35) Let S be a finite dimensional Hilbert space over C. Any linear operator T : S , S has the adjoint operator T* : S S defined by
(Tf,g) = (f,T*g) T is said to be normal if it commutes with T'. If T = T* then T is said to be selfadjoint.
Theorem 6.12. The Hecke operators T acting on the space of cusp forms for the modular group are selfadjoint, i.e. (f,Tng) for all f,g E Sk.(I').
(6.36)
Proof. Since Si.(I') is spanned by the Poincare series (6.30), it suffices to verify (6.36) for the Poincare series. By Corollary 6.10 and Proposition 6.11 we get Pt) = (T.PP, P,n) = (P.,,T.P,) because the Fourier coefficients of Poincare series, and so the inner products, are real. Now we can appeal to the following result from linear algebra.
Proposition 6.13. Let A C
be a commuting family of normal ma
trices. There exists a unitary matrix U E M,,(C) such that U'AU is diagonal for every A E A. This result can be stated in terms of linear operators.
Proposition 6.14. Let S be a finite dimensional Hilbert space over C and let T be a commuting family of normal operators T : S , S. Then there e:c:isis an u,
basis .P of S which consists of common eigenfunctions
of all the operators in T.
Proof. Let
be an orthonornial basis of S. Then
Tft = E A7.i(T)fj with some \jj(T) E C. In this way T is represented by a matrix A(T) _ (Aij(T)) which commutes with its adjoint A(T)" = A(T)', i.e. A(T) is a normal matrix. These matrices commute; therefore by a suitable linear transformation with a unitary matrix the system T can be diagonalized. Applying Proposition 6.1I to the flecke algebra, we get.
6.5. The Hecke operators with a character
101
Theorem 6.15 (Hecke). In the space Sk(F) of cusp forms for the modular group there exists an orthonormal basis .F which consists of eigenfunctions of all the Hecke operators T,,.
Let f E Sk(r) be a Hecke eigenform,
T, ,f = .(n) f for it = 1, 2, 3, .. .
(6.37)
Suppose f has the Fourier expansion 00
f (z) _
a(m)e(mz).
Combining (6.37) with (6.19), we obtain dk_..ra(mndl).
A(n)a(m) = E
(6.38)
dI (nt,n)
For m = 1 this gives a(n) = A(n)a(1).
(6.39)
Hence a(1) # 0, as otherwise f would vanish identically. Therefore the Fourier coefficients of a Hecke cusp form are proportional to the eigenvalues of the Hecke operators. As an example, consider the space S12(f), which is onedimensionally spanned by 0(z); therefore 0(z) is automatically a simultaneous eigenfunction of all the Hecke operators, namely
r(n)0(z)
(6.40)
where r(n) is the Ranianujan function. Hence one obtains another proof vi the formula (3.34).
6.5. The Hecke operators with a character The significance of Hecke theory is most evident in the context of the group
f ( q) =
n
EP:y0(moclq)
and a Dirichlet character N, (niod q) (here h is the full modular group). NotsV that X(p) vanishes for p = (`; ;) E GL.,(Z) with (a, q) > 1, so the set of right
6. Hecke Operators
102
coset representatives An in the definition of Tn (see (6.11)) can be reduced to (6.41)
On=11 a
d
I EOn:(a,q)=1}.
Observe that the correspondence between On x r and r x 0,a given by (6.7)
projects to that between on x ro(q) and r'o(q) x On. Indeed, if r'p' = pr with (p,r) E On x r'o(q), we deduce by (6.9) that
a'a' = as (mod q)
(6.42)
so (a', q) 1, i.e. p' E An, and y' = 0 (mod q), i.c. r' C r'o(q). Moreover (6.42) shows that (6.43)
X(p)X(r) = X(r')X(P) if pi = 'r A'
for (p,r)EAn' xro(q). As before we require that the character x (mod q) satisfies the consistency condition x(1) = (1)k; otherwise the space Mk(r'o(q), x) of automorphic forms of weight k for the group ro(q) with respect to the multiplier x is zero.
Theorem 6.16. The Hecke operator Tn = Tn maps automorphic forms to automorphic forms and cusp forms to cusp forms: Tn : Mk(ro(q), X)  Mk (IFO (q), X)
T Sk(ro(q), X)  Sk(ro(q), X) Proof. We proceed as in the proof of Theorem 6.8 by using (6.43), namely
x(P)fipr = x(p)fir,p, = x(p)x(r')fiP' = X(r)x(p )fid, whence (Tn f )1r = x(r)Tn f for any r E ro(q). Moreover if f is a cusp form then so is Tn f , because any slash operation preserves exponential decay of a function at any boundary point.
Recall that we have already established the multiplicativity (6.22) of the operators Tn, actually in a bigger space of periodic functions, hence in Mk(ro(q), x). This shows that the Hecke algebra T is commutative, generated by Tp for all primes p. We have p
1
(6.44)
Tp = p
+ X(p)pkl
b (mod p)
1
r
/
6.5. The Hecice operators with a character
103
If p divides the level, this expression reduces to (6.45)
Tp = 1
1
E
if plq,
p
P 6 (mod p)
P,
and (6.22) yields Tp = (Tp)"
(6.46)
if p1q.
The situation becomes somewhat complicated with the action of T,z = Tn on the Poincar6 series (6.47)
E
Pm(z) =
rEroo\ro(q)
One needs these for all m to span the space of cusp forms, but only the TnPm with (mn, q) = 1 possess the exact symmetry which lies behind the selfadjointness. More precisely, TnPm is a combination of the Poincare series of the following type.
Theorem 6.17. Suppose k > 2 and X(1) = (1)k. For all m >, 0 and n 3 1 we have (6.48)
TnPm =
X
rnl
l\nd
1k
1 Pmnd2
ld
The proof of Theorem 6.17 is similar to that of Theorem 6.9 (one needs obvious modifications of the argument to accommodate the character X). For m = 0 we see that the Eisenstein series X(T)jr(z)k
Ek(z) = E rEroo\ro(q)
is an eigenfunction of all Tn with eigenvalue
ak1(n,X) =
X(d)dk1 dIn
If nlq°O then vanishes unless nom, in which case TnPm = (n, q) = 1, we can write (6.49)
TnPm = X(n)nk1 E di(rn,n)
Hence
X(d)d1'Pmnd
.
/n. If
6. Hecke Operators
104
Corollary 6.18. If m, n > 1 and (mn, q) = 1, then (6.50)
X(m)mk1TnPm = X(n)nk1TmPn
Our next aim is to find the adjoint of T. One could argue as in the proof of Theorem 6.12 using the symmetric relations (6.21), (6.50) and Theorem 2.12 to establish that (6.51)
(Tnf, g) = X(n)(f,Tng)
if (n, q) = 1,
but this method works only for special f, g E Sk(ro(q), X) which are in the linear subspace spanned by the Poincare series P,,, with (m, q) = 1. Since
not every cusp form can be represented as a linear combination of such Poincare series, we cannot conclude that Tn = X(n)Tn. To establish (6.51) completely we make use of the original idea of Hecke. To this end one needs the concept of the absolute inner product. We begin by recalling the expression (3.20), (f,9)(z) = ykf(z)9(z)dµz.
Let t9 be a multiplier system of weight k for a group r (here I' is arbitrary). If f , g E Sk (r, t9), then (f , g) (z) is a rinvariant measure, and the inner product is just the volume of a fundamental domain for this measure,
(f, 9) = Notice that for any a E GLZ (R) we have (6.52)
(fig, 9i) (z) = (f, g) (az).
One can also use any subgroup r' C r of finite index and scale down by the index to get the same value: (6.53)
(f,9) = [r
1
w(f,9)(z)
For computational convenience we wish to have r' sufficiently small so that the multiplier t9: r + C becomes trivial on r, i.e. f, g E Sk(r'). This is certainly possible if r is a congruence group and t9 is a character (take the kernel of t9). Furthermore, for any given a E GL2 (Q) one can choose
r' small enough, but of finite index, such that the aslash operator maps Sk(r, 9) into Sk(r'), i.e. (6.54)
fI, r. Sk(r') if f E Sk(r,19).
For example, if a E G,, and r = ro(q), then r' = r(nq) is the group to take.
6.5. The Henke operators with a character
105
Proposition 6.19. Let r be a congruence group and a E GL2 (Q). Then the aslash operator is an isometry of Sk(r, o9) with respect to the absolute inner product, i.e. (6.55)
(fl0, 910) _ (f, 9)
for all f, 9 E Sk(r,'o)
Proof. Let r' be a sufficiently small subgroup of r of finite index such that (6.54) holds true, and let D' denote the fundamental domain of r. Then
(60,910) = [r : I'll 1 fDhhla)(x)
_ [r: r')' ID'
g)(az)
= [r : r')1 f D,(f,9)(z) where OD' is a fundamental domain for the conjugate group r" = ar'Q1.
Choose r' small enough that r" C r. Then [r: r"J = [r: ar'alj = [r: r] by computing volumes. Hence we get (6.55). One can write (6.55) in the following way: (fro, 9) = (f, 9101)
saying that the a1slash operator is adjoint to the aslash operator with respect to the absolute inner product. Notice that gl01 = gI0,
where a'a = det a.
Now we are ready to prove (6.51) in all of Sk(ro(q), X).
Theorem 6.20. If (n, q) = 1 and f, g E Sk(ro(q), x), then (6.56)
(Tnf,9) = X(n)(f,T,19)
In other words, the adjoint of Tn is Tn = k(n)T,; thus T is normal while (n)1/2Tn is selfadjoint.
Proof. Computing via the absolute inner product, we get (T,,.f,9) =
rte' (X(P)fIp,9) pEOn
6. Hecke Operators
106
Here (fip,g) = (f,glp,), where pp' = n. If p = (a d), then p' _ (d X(p) = X(a) = X(n)X(d) = X(n)X(p') Therefore
Q)
and
(X(p)fjp) = X(n) (f, X(P )91p,)
For any r, r' E Fo(q) we deduce by (6.55) and the automorphy of f, g that (f, X(P )9ip') = (f, X(P ")9Ip")
where p" = r'p'rr. Assuming (a, d)1b, we can make p" = p. To this end we choose integers a, b such that
as+6d+bq0 (mod n) a6
1 (mod q).
Then we put
a' _ (aa + bq)d1,
6' _ (bd + bq)a1.
Note that a', b' are integers with a'6' = ab + bq(aa + bd + bq)n; 1 (mod q). Therefore
r \q
r,
ab
q
are in ro(q). We check that r'p' = rp, so p" = p. This proves that (x(p)fip, 9) = x(n)(f, X(p')9ip')
for p = (a d) with ad = n and (a, d) fib. Suppose that n is squarefree. Then the condition (a, d),b is automatically satisfied for any p E 0,,. Thus, summing over p, we get (6.56). If n is not squarefree, then the above construction needs some modifications. However for the proof of Theorem 6.20 we do not need extra work because the algebra of Hecke operators Tn is commutative and generated by the operators Tp, for which (6.56) is already established.
Applying (6.56) to an eigenfunction f = g # 0, we infer that (6.57)
x(n).1(n)
if Tnf = \(n)f, (n, q) = 1.
By Theorem 6.20, Corollary 6.7 and Proposition 6.14 we obtain the following fundamental result, due to Hecke.
Theorem 6.21. In the space of cusp forms SA(I'o(q), x) there exists an orthonormal basis .F which consists of eigenfunctions of all the Hecke operators Tn with (n, q) = 1.
6.6. An overview of newforms
107
6.6. An overview of newforms Let f E Sk(ro(q), X) be a Hecke eigencuspform, i.e. a cuspform which is a common eigenfunction of all Tn with (n, q) = 1, say
Tn f = \(n) f if (n, q) = 1.
(6.58)
The condition (n, q) = 1 is a hassle in practice. Suppose f has the Fourier expansion
f (z) = E00 a(m)e(mz).
(6.59)
M=1
Combining (6.58) with (6.19), we obtain
A(n)a(m) _ E X(d)dk1a(mnd2) dl(m,n)
but only if (n, q) = 1. For m = 1 this gives (6.60)
a(n) = A(n)a(1)
if (n, q) = 1.
Unfortunately we cannot deduce from (6.60) that a(1) 0 0, because the condition (n, q) = 1 does not allow us to control all the coefficients in (6.59). In fact there exist Hecke eigencuspforms f 0 with a(1) = 0. They come from lower levels if x is not a primitive character. Let q* denote the conductor of X. Take any positive integers q', d such that q*lq' and q'dlq. Let X' (mod q') be the character induced by X. Then
f (z) E Sk(ro(4 ), X) = f (dz) E Sk(ro(q), X). This follows immediately from the identity (6.61)
d 1)('Y 6)(7/d
1) 6)(d
Now if f (z) has the Fourier series (6.59), then
f (dz) = E a(m)e(mdz) = E +n
a(m/d)e(mz);
n &=_O (mod d)
therefore f (dz) has first coefficient zero if d # 1. We denote by SR(ro(q), X), the linear subspace of Sk(ro(q), x) spanned by all forms of type f (dz) where djq and (6.62)
f E Sk(ro(4'), X')
6. Hecke Operators
108
for some q' < q such that q'dlq. Here X' (mod q') is the character which is induced by X (mod q). Then we define Sk(ro(q), x) the linear subspace of Sk(ro(q), X) orthogonal to SS(ro(q), X) with respect to the inner product. Thus we have the orthogonal decomposition (6.63)
Sk(ro(q), x) = Ss(ro(q), x) ® Sk(ro(q), x).
Proposition 6.22. The subspaces Sk, So are stable under the Hecke operators T with (n, q) = 1.
Proof (sketch). That Tn : Sk  Sk can be verified directly on individual forms f (dz). Moreover, T preserves the orthogonality to Sk, because its adjoint is Tn = X(n)T,,. Hence T : Sk ' S. By Proposition 6.22 it follows that each of the two subspaces Sk, Sk has orthonormal bases consisting of eigenfunctions of the Hecke operators Tn with (n, q) = 1. The Hecke eigencuspforms in the space Sk(ro(q), X) are called newforms.
A cusp form f E sk (ro(q), X) which is an eigenfunction of Tn only for (n, q) = 1 is not quite adequate for the study of various Lfunctions attached
to f. For this reason we wish f to be a common eigenfunction of all the Hecke operators. An easy way of getting this property would be to show that the joint eigenspaces for all of the Tn with (n, q) = 1 are onedimensional, i.e. for each sequence {A(n)}(n,q)=1 there is at most one cuspform f 96 0 (up to a scalar) which is an eigenfunction of Tn with prescribed numbers A(n)
for (n, q) = 1 as eigenvalues. Then any other operator T which commutes with all Tn for all (n, q) = 1 preserves each a(n)eigenspace, so T f must be a multiple of f, i.e. f is automatically an eigenfunction of T. One can show that the multiplicityone principle holds true in the space SS(ro(q), X) Consequently, if f is a newform then (6.58) extends to all n. By (6.60) we 
deduce
Proposition 6.23. Suppose f is a newform. Then (6.58) holds for all n. The first coefficient in the Fourier expansion (6.59) does not vanish, so one can normalize f by setting a(1) = 1. In this case
a(n) = \(n) for all n.
6.7. Hecke eigencuspforms for a primitive character We shall give details for the assertions made in the last section in the case of a primitive character X (mod q). Thus in this case SR is zero and SO is the whole space Sk(ro(q), X).
6.7. Hecke eigeucuspforms for a primitive character
109
For any d the operator Ad defined by Ad
1
d
( b (niod d)
b/d 1
acts on the Fourier series f (z) = >2 a(m)e(mz)
(6.64)
in
by selecting the terms with m  0 (mod d), i.e. we have
E
(Adf)(z) =
(6.65)
a(m)e(mz).
,n_0 (mod d)
Hence the sifting operator Sq defined by (6.66)
µ(d)Ad
Sq = dIq
acts on the Fourier series (6.64) by selecting the terms with (m, q) = 1, i.e. we have
(Sq f) (z) = E a(m)e(mz).
(6.67)
(m,q)=1
Remark. One can show that S. : Sk(ro(q), X) , Sk(ro(gg1), x), where q1 is the product of prime factors of q. Lemma 6.24. If f E Sk(ro(q), X) has the Fourier coefficients a(n) = 0 for all n with (n, q) = 1, then f = 0. In other words, the sifting operator Sq has zero kernel.
Proof. We begin with the equation (CZ)'f
(ca
cz) _ X(d)f (
d
+ c)
for any T = (a * ) E I'0 (q), which is just the automorphy relation fi. = X(T) f written at the point (z  d)/c. Expanding both sides into Fourier series, we obtain a kind of Poisson's summation formula,
(CZ)`k E m=1
a(m)e
cam C

 rnl = X(d)>a(n)e (dn + nz 00
cz J
n=1
c
C
6. Hence Operators
110
for any 0 < c  0 (mod q) and ad  1 (mod c). Hence, summing over a (mod c with (a, c) = 1 for c = q, we get 00
(qz)""E a(m)S(m, 0; q)e
\
=
m=1
n=1
a(n)Sx(n, 0; q)e l
z
\q J
where S(m, 0; q) is the Ramanujan sum and Sx(n, 0; q) = X(n)T(X) is the Gauss sum. Since every term on the righthand side vanishes by our hypothesis, we conclude that every term on the lefthand side also vanishes (the uniqueness of Fourier series expansion), i.e. a(m)S(m, 0; q) = 0 for every m.. If q is squarefree, then the Ramanujan sum never vanishes, namely we have S(m, 0; q) = µ(q/(m, q))W((m, q));
therefore a(m) = 0 for every m, proving Lemma 6.24 in this case. If q is not squarefree then the assertion of Lemma 6.24 remains true, but our proof needs substantial modifications. To this end we generalize the above relation between the Ramanujan and Gauss sums. First we multiply the Poisson formula by a function V;(a), and only then we sum over the primitive classes. We obtain (qz)_A
E a(m)S(m)e
(ml
m=1
where
S(m) =
*
a (mod q) e
Sx(n)
=
qz J
E SS(n)e (nz"\ q
n=1
(am q) V (a)
d (mod q) X(d)e
(
q')0(a)
and ad  1 (mod q). For a given m we want S(m) # 0, whereas SS(n) = 0 for all n with (n, q) # 1. We can permit i 1(a) to depend on m. To construct such 0 we factor the modulus q = q, q2 so that q1 has primes in m and qq is coprime with m. We put
qo= II p. pl(m,q)
We use only the residue classes a (mod q) of type a=bg141+ I 1+cq1 I g242 qo/
6.7. Hecke eigencuspforms for a primitive character
111
where b runs over the classes modulo q2 with (b, q2) = 1 and c runs over the classes modulo qo with (1 + cql/qo, ql) = 1. For O(a) we choose a character of conductor q2 restricted to the above classes. Then we have
S(m) = E i'(b)e (bmL)
e
b (mod q2)
(m
c (mod qo)
(mi!) qo [I 1 1 p
11+
41 \
qo
0
as desired. In order to compute S((n) we factor X (mod q) into the primitive characters Xl (mod qj) and X2 (mod q2). Since
d = bglgl + (1 + c41)
g2g22,
go
where the ' is the mapping to the multiplicative inverse modulo ql, we obtain
X1(1+c9o)e(nql (i+C)').
TX2(b)ebn I
Sx(n) b(mod g2)
c(mod go)
Assuming that 1/.'X2 is primitive, the first sum is equal to z/.X2(41n)T(zbx2), so it vanishes if (n, 92) 1. Since Xl is primitive, the second sum vanishes if (n, qj) # 1. Combining both facts, we conclude that SX(n) = 0 if (n, q) 96 1. We still have a problem if X2 is the only primitive character to modulus
q2. This is the case for 92 = 3 and q2 = 4. If q2 = 3 we take b the principal character, and the above argument still works because (b)e (bm 3
)
= 2cos
(3 mql)
0.
b (mod 3)
If q2 = 4 then X2(b) = X4(b) We apply the above arguments for the modulus 2q in place of q. We take z/i(b) = X8(b) = (2). Since X8 and X8X4
are both primitive, the previous result is valid. This completes the proof of Lemma 6.24 for any primitive character X (mod q).
Remark. The above arguments are sensitive to the primitivity of the character X, as they should be. Now suppose that f E S,.(ro(q), X) is an eigenfunction of all the Hecke
operators T with (n, q) = 1, say (6.58) holds, and that f has the Fourier expansion (6.59). Then we have (6.60). Hence by Lemma 6.24 it follows that the first coefficient of f is a(1) 0. Therefore we can normalize f by setting a(1) = 1. Moreover, Lemma 6.24 implies the multiplicityone principle. These observations yield the following conclusion.
6. Hecke Operators
112
Lemma 6.25. A Hecke eigencuspform f E Sk(ro(q), X) can be normalized to have its first Fourier coefficient a(1) = 1, and the normalized form is determined uniquely by the eigenvalues of T with (n, q) = 1. Since Tn commutes with T,,, for any m, we obtain
Tn(Tmf) = Tm(Tnf) = A(n)Tmf if (n, q) = 1. In other words, Tmf is a Hecke cusp form with the same eigenvalues as f for all (n, q) = 1. By the multiplicityone principle (Lemma 6.25) it follows that f is a multiple of f . Therefore we get
Theorem 6.26. If f E 3k(ro(q), X) is an eigenfv.nction of every T with (n, q) = 1, then it is an eigenfunction of the whole algebra T = {T,,,; m E N}, i.e. Tn, j = ).(rn)f for all r E N.
Besides Hecke operators Tn there is yet another important operator acting on cusp forms for ro(q), namely the l ricke involution, which is defined by (6.68)
W f= few
where w = (q
1
\ I
(cf. [AL)), i.e.
(Wf)(z) = qk/2z`kf(1/qz) Note that w2 = q, so = (1)k. Moreover w normalizes the group ro(q), i.e. wro(q) = ro(q)w or, explicitly, W2
WT = T W
where
/ l T=I bq aI4I
if
\ T=(c d).
Hence it follows that for f E Sk(ro(q), X) and T E ro(q) (fiw)lr = f1wr = fj71w = X(T')ffw =
Therefore (6.69)
W : Sk(ro(q), X) , Sk(ro(q), X)
Since w E GL2 (Z), the rricke involution is an isoinetry, i.e. we have the equality (W f, W g) = (f, g).
Here (,) on the left and right are the inner products in Sk(ro(q), k) and Sk(ro(q), X) respectively (one can see that this equality holds, because either side coincides with the corresponding absolute inner product).
6.7. Hecke eigeiicuspforms for a primitive character
113
Theorem 6.27. If (n, q) = 1 we have WT,x = X(n)TTW.
Proof. We have WTn =
n2'
X(P)fl, PEOn
Tn W = nf
X(P)flwp. PEAn
We also have wpw;
if P=
_
d)
Hence for r = (a 6) we compute that P! = Tiwpw1 _
6d +,3bq
yd
,f3a
as
Let (n, q) = 1 and suppose p E On. Choose a = d/(b, d) and 'y = bq/(b, d). For this choice there are unique 0 < Q < a and 6 such that T

P
d/(b, d) bql (b, d)
*
n/(b,
E rfl(q)
d)
E On.
This establishes a onetoone correspondence on the set On given by p 14 P, = T1wpw1. We have flwp = flrp'w = X(T)flp'w and
X(T)X(P) = X(dl(b,d))X(a) = X(n)X((b,d)) = X(n)x(P') Hence Theorem 6.27 follows.
In order to bring the character X back to X we shall compose W with the complex conjugation operator K defined by
(Kf)(z) = 1(2).
(6.70)
Note that K acts on a Fourier series by simply conjugating its coefficients,
f (z) =
a(n)e(nz) ' (K f) (z) =
a.(n)e(nz).
6. Hecke Operators
114
Be aware that K is not linear over C, since
K) f = AK f if A E C. We have (6.71)
(6.72)
(6.73)
(6.74)
K2 = 1, WK = (1)kKW K : Sk(ro(q), X) ' SL(ro(q),
(K f, Kg) _ (f, 9) KT,x =
for all n.
Set (6.75)
W = KW.
Note that W = KW. From the above properties of K and W we deduce that (6.76)
W2=1, Wrrf =j)Wf if
(6.77)
W : Sk(ro(q), X) + Sk(ro(q), X)
(6.78)
Tn.W = X(n)WTT if (n, q) = 1.
77 EC
Now suppose f E Sk(ro(q), X) is a nonzero eigenfunction of all the Hecke operators T,, with eigenvalues A(n) for (n, q) = 1 (we have called f a Hecke eigencuspform). Then by (6.58) and (6.57) we get
f = X(n)WTn f = X(n)WA(n)f = X(n) \(n)Wf = \(n)W f showing that W f is also an eigenfunction of all T,, with the same eigenvalues A(n) for (n, q) = 1. Therefore by the multiplicityone principle (Lemma 6.25) we obtain
6.7. Hecke eigencuspforms for a primitive character
115
Theorem 6.28. A Hecke eigencuspform f E SR;(ro(q), x) is also an eigenfunction of the involution operator W, i.e. (6.79)
ITV f = ref.
By (6.76) we deduce that the eigenvalues of W (for nonzero functions) are complex numbers on the unit circle; indeed f = W 2 f = Wrl f = t7Wf = rin f , i.e. (6.80)
I771 = 1.
The. eigenvalue of the involution W from Theorem 6.28 will appear as a factor in the functional equation for the Lfunction attached to f. We shall compute this eigenvalue in terms of the Gauss sum (6.81)
r(X) _ a (mod q)
X(a)e
(q)
and the qth Fourier coefficient of f. Suppose that f is normalized, i.e. its first Fourier coefficient is one. Then all the Fourier coefficients of f coincide with the eigenvalues of the Hecke operators T, i.e. 00
f (z) _
(6.82)
A(n)e(nz). n=1
Recall that for all m, n > 1 A(m)A(n) _
(6.83)
X(d)dk1A(mnd2).
In particular, for any m and djq we obtain the exact factorization A(dm) _ A(d)A(m). Hence
(Ad f) (z) = E
A(n)e(nz) = A(d) f (dz);
n_0 (mod d) in other words
(6.84)
Adf = A(d)d k/2 f, (d 1)
if dlq.
Theorem 6.29. The eigenvalue of the involution W of a normalized Hecke eigencuspform f E Sk(ro(q), X) is given by (6.85)
47 = T(X)A(q)q_'
l'
6. Hence Operators
116
Proof. We begin by writing the identity
uiq) (q 1/q) (1 v/q) _ (u (uv 1)/q\
(1
.
We choose uv = 1 (mod q) to bring this matrix to ro(q), then we divide by (1 v/q) and we slash f with the resulting matrix to obtain 1
fl(1 u19)(9 1/q) = X(v)fl(1 1/9)
by the automorphy of f. Next we sum over the reduced residue classes modulo q, getting
g = E* fl(1 u/q) ( 1/q) _ i X(v)fl(1 v/9\. u (mod q)
l
it (trod q)
q
1
1
/
If we apply the Fourier expansion (6.82), the righthand side yields
g(z) = E X(v)f (z  9) _ EX(n)
X(v)e
n=1
it (mod q)
it (mod q)
(_!!)) q
e(nz)
00
= T(X) E X(n)A(n)e(nz). rt=1
Hence K9(z) = T(X)(S9f)(z)
because X(1);r(X) = r(X) and a(n) for (n, q) = 1 (see (6.57)). On the lefthand side we relax the condition (u, q) = 1 by the Mobius formula and apply (6.84) as follows:
E{ fl(1 u/q) =ad=q L /_t(a)
u (mod q)
1
L
u (mod d)
fl(1 u/d) 1
1: µ(a)dAdf =
E
ad=q
ad=q
1
Slashing with ((I 1/q), since (d 1) (9 1/q) = (q 1) (1 1/a)I we obtain
9=
E* fl(1 u/q)(
u (mod q)
1
µ(a)X(d)d1A/2
1/q) = q
ad=q
1a) /
6.7. Ifecke eigencuspforins for a primitive character
117
Now suppose f is an eigenfunction of W, say (6.79) holds. Then K fj,, _ KWf = W f = f and K fl (1 ) = ri f (1 _ rjak/2f(az); hence 1/a
R
1/a )
Kg(z) = rl E
(az).
ad=q
Comparing this with the former expression, we obtain the identity (6.86)
T(X)(Sgf)(z) =,q E µ(a)A(d)(a/d)k/2df(az).
ad=q
From the first Fourier coefficients of (6.86) one gets (6.87)
T(X)
=
Hence by virtue of (6.80) and (6.88)
IT(X)I = q2
we first deduce that (6.89)
JA(q)l = q42',
and applying this to (6.87) we complete the proof of (6.85).
From the identity (6.86) one can also infer Theorem 6.30. For plq the eigenvatues A(p) of TT in the space Sk(Fo(q), X) satisfy (6.90)
IA(p)I =
pT.
Proof. Comparing the pth Fourier coefficients in (6.86), one gets (p)p(q)g1k/2
= A(q/p)(p2/q)k/2q/p
which yields (6.90) because \(p)A(q/p) = A(q).
6. Hecke Operators
118
6.8. Final remarks If X is not primitive, then the space of newforms So(I'o(q), X) is not necessarily the whole of Sk(ro(q),X); nevertheless the basic results remain true in this subspace. In particular, a newform is automatically an eigenfunction of all the Hecke operators Tn, n = 1, 2, 3, ... , as well as of the involution W (but the formula (6.85) still requires X to be primitive). As a matter of fact the converse is true, i.e. if f E SS(ro(q), x), f 0 0, is an eigenfunction of all
the Hecke operators T, n = 1, 2, 3, ... , and of W, then f is a newform. A newform has the first Fourier coefficient different from zero; hence, letting this coefficient be one (normalization), we have the Fourier expansion f(z) = j a(n)e(nz)
(6.91)
1
with coefficients equal to the cigenvalues of Hecke operators. By virtue of (6.83) the associated Dirichlet series 00
L f(s) = E,\(n)n'
(6.92)
I
(which converges absolutely for Re s > k21 by Corollary 4.2) has the Euler product (6.93)
Lf(s) = 11(1  A(p)p s P
Winnie Li [Li] established a kind of converse assertion by showing that if f E Sk(ro(q), X) is an eigenfunction of W and the associated series L1(s) has the Euler product of type (6.93), then f is a normalized newform. Suppose X is a real character. Then both operators K, W map the space Sk(ro(q), X) to itself. Moreover K commutes with all T,a for (n, q) = 1. Therefore we can make a Hecke basis r of Sk(ro(q), X) which consists of eigenfunctions of K, say K f = ±f since K2 = 1. Replacing f by if, we can change the eigenvalue 1 to 1, so we can always require that (6.94)
Kf = f.
This means f has real Fourier coefficients. For such f we have W f = WI, and if f is a newform then (6.95)
W f = 77f
with q = fik because W2 = (1)k. Recall that n is also given by (6.85). However, even in the case of a real charecter X (mod q) for which the Gauss sum T(X) is computed in Lemma 4.8, the exact determination of the eigen
value q = fik remains a problem since the Fourier coefficient A(q) of f is intractable (except for some forms f such as these which are associated with elliptic curves).
Chapter 7
Automorphic Lfunctions
7.1. Introduction A connection between automorphic forms and Lfunctions can be traced in the celebrated memoir of B. Riemann (1860) on the zetafunction 00
(7.1)
C(S)=En8,
Re s > 1.
i Starting from the gamma function oo
r(s) = je_1'Y3_1d by termwise integration Riemann established the formula (7.3)
(27r)sr(s)c(2s)
=
2
j(O(iy) 
where 0(z) is the standard theta function (2.72). Then by the Jacobi inversion (10.55) he arranged this formula in a symmetric fashion: (7.4)
izr (2) ((s) +
s(1
s) =
where
jt(v) (y2 +y'$) y'dy 00
2I
u(y) =
(0(,Y)
ejrn2y
2 i
119
7. Automoiphic Lfunctions
120
Since u(y) << e'y the above integral converges absolutely and uniformly on
vertical strips, giving analytic continuation of ((s) over the whole splane with only a simple pole at s = 1. Moreover (7.4) implies the functional equation (7.5)
1r
2r
`2l ((s) = .7r Y r (1 2 s) S(1  s)
(actually Riemann's original transformations were slightly different). Conversely, in 1921 H. Hamburger showed that any Dirichlet series 00 a(n)n_9
L(s) _
satisfying (7.5) and some regularity conditions is necessarily a multiple of
These ideas were greatly generalized by E. Hecke (1936). We shall present this profound theory in the context of automorphic forms for row) of integral weight k > 0 and a character X (mod q), not necessarily primitive.
7.2. The Hecke Lfunctions We begin with a cusp form f E Sk(ro(q), X). Consider simultaneously the Fourier expansions at cusps oo and 0, say f(z) _ 00 a(n)e(nz)
(7.6)
00
g(z) = Eb(n)e(nz)
(7.7)
1
where (7.8)
g
with w =
(q1}
Recall that g E Sk(ro(q), X) by (6.69). To these Fourier series we associate the Hecke Lfunctions (7.9)
L1(s) _
00 1
a(n)n'
7.2. The Hecke Lfunctions
121
00
b(n)n'.
L9(s)
(7.10)
By virtue of Corollary 5.2 the above Dirichlet series converge absolutely in
the halfplane Re s >. In this halfplane we can show by termwise integration that (7.11)
(27r)8r(s)Lf(s) = (21r),r(s)L.9(s)
(7.12)
=
f f
oo f(iy)y8ldy
00 g(iy).ysldy.
0
Since f and g have exponential decay at 0 and oo, we have
Theorem 7.1. The Lfunctions L f(s) and L9(s) associated to the cusp forms f E Sk(ro(q), x) and g = flw E Sk(ro(q), X) have analytic continuation over the whole splane, and they are entire functions. Actually the socalled complete L junctions
A f(s) =
(7.13)
12
s 1
r(s)L1(s)
I
r(s)L9(s)
\\
A9(s) =
(7.14)
are entire functions, bounded on vertical strips.
Both Af(s) and A9(s) are connected. Indeed, by (7.8), for z on the imaginary line we get
f(iy) = qk/2(i/y)kg(z/qy) Hence
f00 0
f
(iy)y9ldy
= ikgk12
f
9(iy)yksldy
0
which together with (7.11), (7.12) yields
Theorem 7.2. The complete Lfunctions satisfy the functional equation (7.15)
A f(s) = ikA9(k  s).
If f is an eigenfunction of the operator W = KW (see (6.75)), say (7.16)
Wf = nf,
7. Automorphic Lfunctions
122
then b(n) = ra(n), so Lg(s) = 77L f(s), and the functional equation (7.15) becomes VY
(7.17)
(2
k
s
VY
ks
r(k  s)L f(k s).
r(s)L1(s) = i
In particular, if X is a real character and f has real Fourier coefficients and is an eigenfunction of W, say Wf = fikf,
(7.18)
then (7.17) becomes
(7.19)
= f ()'r(k_8)Lf(k_s).
Of
Thus for k = 2 (this includes the IlasseWeil Lfunction; see the next chapter) the eigenvalue of W coincides with the sign of the functional equation.
The assumption that f, g are cusp forms is not absolutely necessary; however without this requirement A f(s), Ag(s) may have simple poles at s = 0 and s = k. Most of the arguments used above have little to do with automorphic forms and can be reversed by Mellin's transform to establish the following result.
Theorem 7.3 (Hecke). Suppose f and g are given by the Fourier series 00
(7.20)
f(z) = E ane(nz) 0
(7.21)
g(z) = 00 E bne(nz) 0
with coefficients an, bn bounded by O(nk) for n > 1, where a is a positive constant. Put 00
(7.22)
L f (s) _
(7.23)
L9(s) _
anns
00
bnn s
and (7.24)
Af(s) =
(7.25)
A9(s) =
()3r(s)Lf(s) ()Sr(s)Lg(s)
7.2. The Hecke Lfunctions
123
where q is a given positive number (not necessarily an integer). Put w = = (fz)k f (1/qz), where k is a given positive inte( q 1) and (fl..,) (z) ger. Then the following assertions are equivalent: (Al) The functions f, g are connected by
9 = Al.
(7.26)
(A2) Both Af(s) and A9 (s) have meromorphic continuation over the whole splane, (7.27)
A f(s) + aos1 + boik(k 
(7.28)
A9(s) +
bos1 + aoik(k

s)1
s)1
are entire and bounded on vertical strips, and they satisfy (7.29)
A f(s) = ikA9(k  s).
Proof. Assuming (A1), we establish the following integral representation: 00
A1(s) + aos1 + boik(k  s)1 = J
(.f (iv/vT) 
ao)y'ldy
00
+ ik 1
bo)yksldy
1
Since f (z)  ao and g(z)  bo have exponential decay as y = Imz > oo, the above integrals converge absolutely, and they are entire functions, bounded on vertical strips. A similar integral representation holds true for (7.28). By symmetry we infer the functional equation (7.29), but with the polar terms included. However the simple fractions of the polar terms alone satisfy (7.29), so we can subtract these. This completes the proof of the implication (A1) = (A2). The converse implication goes by routine application of the Mellin inversion, the PhragmenLindelof convexity principle and Stirling's estimate for the gamma function. From the functional equation (7.29) one
first deduces (A1) on the imaginary halfline z = iy with y > 0, and one extends this relation to the upperhalf plane by analytic continuation. Note that the holomorphy of (7.27) and (7.28) implies that ao = L f(0),
bo = L9(0) and (s  k)L f(s) N bo(2iri/ f)k, (s  k)L9(s) ti ao(21ri/ f)k ass >k.
7. Autolnorp.hic Lfunctions
124
7.3. Twisting automorphic forms and Lfunctions Automorphic forms behave nicely with respect to twists by primitive characters.
Theorem 7.4. Let f E Mk(ro(q), X), where X is a character of conductor q' with q* l q. Let ip be a primitive character of conductor r. If f has the Fourier series 00 f (z) a(n)e(nz), then the twisted series fp(z) _ E00bb(n)a(n)e(nz) 0
belongs to Mk(ro(N), X'iG2), where N is the least common multiple of q, q'r,
r2. Moreover if f is a cusp form, so is f,p.
Proof. Let r(V)) be the Gauss sum T(O)
/,
_
`d(u)e
(±).
u (mod r)
Since i/i is primitive we have IT(V))l = r1/2, and for any n T(`Y)Y'(n) _ u (mod r)
'd(u)e (ur )
Hence we can express fp in terms of f as (7.30)
T( )ff _ E G(u)fi(1 u (mod r)
u/r 1
Also, one has the identity due
T =
C1
Hence if ry = therefore
r
1) \c d) (a b
1_(a Cr
b
uc
bcdu
r
E ro(N), then r E ro(N) and T
T(')).f
I7 = E
(a d) (mod q*);
(u)X(d)fl(1 d' /r ) \
u (mod r)
= X(d)I2(d) E ' u (mod r)
= X(d)ik2(d)T(' )fo.
r u'::du'
I
(u)fI(1 u/r) 1
7.3. Twisting automorphic forms and Lfunctions
125
This completes the proof of the automorphy of f V,. That f o is a cusp form
whenever f is can be seen by writing the Fourier expansion of fo explicitly at every cusp. However, we only need to check that f p does not grow too fast at the boundary of H, or for that matter f V, satisfies the criterion of Section 5.1. This is seen immediately from (7.30).
Theorem 7.5. Let f E Sk(ro(q), X), where X is a character to modulus q. Let rk be a primitive character of conductor r with (q, r) = 1. Then (7.31)
APIWN = w(Ogi*L
where wN = (N 1) for N = qr2, g = fl,,, for w = w4 = (9 1) and (7.32)
w(1P)
=
X(r)'tP(q)T('O)2r1
Proof. /Recall that g E Sk(ro(q), x) by virtue of (6.69). Check the identity (7.33)
1 l
(9 1\ (uq *) (1
u1 r) (N 11 _r
uir
We choose uvq = 1 (mod r) to bring (_rUq *) into ro(q), so this identity yields (7.34)
f, (1 u/r)WN = X(r)gj(1 u/r). 1
1
Moreover we have
T('P)*(n) u (mod r)
'd(u)e ( ru)
for any n, because 1/i is primitive. Multiplying (7.34) by (u) = Eli(vq) and summing over the reduced residue classes to modulus r, we obtain
?P(1)r.
This yields (7.31), since
Now let L f(s, ,O) denote the Hecke Lfunction associated with the twisted
form fj,, i.e. 00
(7.35)
L f(s, tG) _i(n)a(n)n9,
and put (7.36)
A f(s,,O) =
(,,IN
By Theorems 7.5 and 7.3 we deduce
s
r(s)L f(s, ).
7. Automorphic Lfunctions
126
Theorem 7.6. Suppose f E Sk(ro(q),x),
is a primitive character of conductor r with (q, r) = 1, N = qr2 and w(O) is given by (7.32). Then
A f(s,1/)) is an entire function, bounded in vertical strips, and it satisfies the functional equation Af(s,i)) = ikw(V))A.9 (k  s, i)
(7.37)
where 9= fl,,, forty=wq= I q1 1. In the particular case of k = 2, X principal, 7li real, W f = 77f (this includes the HasseWeil Lfunctions), the functional equation (7.37) becomes (7.38)
2
(2
s
2s
r(2  s)Lf(2 
where N = qr2 and (7.39)
since
n1L(q) I7(iP)I2
= r and T('+J,) = Zb(1)T(ir) =C1)T(O).
7.4. Converse theorems For q = 1 (the case of the modular group) Theorem 7.3 shows that an automorphic form is fully characterized by a single, suitable functional equation; precisely, we have
Theorem 7.7 (Hecke). Let k be a positive, even number. Suppose f is given by Fourier series 00
f (z) _ E ane(nz) 0
with the coefficients an « n° for all n >, 1. Then f E Mk(r0(1)) if and only if the function
Af(s) =
(27r)sr(a)00E
anns
1
can be analytically continued over the whole splane, Af(s) + a,,s1 + ikao(k  s)1 is entire and bounded in vertical strips, and satisfies the functional equation (7.40)
A f(s) = ik A f(k  s).
7.4. Converse theorems
127
Proof. The modular group is generated by the translation Tz = z + 1 and the involution Sz = 1/z. If q > 1 one needs more functional equations to show the converse theorem; these are obtained by twisting with characters. There are many possibilities for choosing a sufficient collection of characters. We shall follow closely the original version established by A. Weil [Wei].
Theorem 7.8 (Weil). Let k be a positive, even number, and X a character to modulus q >, 1. Suppose f, g are given by the Fourier series (7.20), (7.21) with coefficients bounded by O(na) for all n 3 1, where a is a positive constant. Suppose A f (s), A9 (s) defined by (7.24 ), (7.25) satisfy (A2). Furthermore let 1Z be a set of prime numbers coprime with q which meets every
primitive residue class, i.e. for any c > 0 and any a with (a, c) = 1 there exists r C 1Z such that r  a (mod c). Suppose for any primitive character zJi of conductor r E 1Z the functions A f(s,1J)), A9(s,,O) defined by (7.36)
with N = qr2 are entire, bounded in vertical strips and satisfy the functional equation (7.37) with w(7b) given by (7.32). Then f E Mk(ro(q), X), g E Mk (ro(q), X) and g = fl,,,, where w = wq = (q 1 ). Moreover f, 9 are cusp forms if L f(s) or L9(s) converges absolutely on some line Re s = a
with 0
fop=Eafla if /3=>act a
or
where a ranges over GLZ (1R). We begin by constructing special O's. For any v with (v, r) = 1 there exists u such that (7.41)
ry(v/r) =
r
uq
* ) E ro(q). v
Put (7.42)
a(v/r) _
and (7.43)
O(v/r) = (X(r) 'Y(v/r))a(v/r)
Actually a(v/r) E GLl (Q) and /3(v/r) is in the group algebra C[GL; (Q)].
7. Atitomorphic Lfunctions
128
Lemma 7.9. Suppose r is a prime number with (r, q) = I. Suppose f and g = fl,,, satisfy (7.37) for any primitive character ip of conductor r with the constant w(O) given by (7.32). Then glp(v/r) does not depend on v for (v, r)
= 1.
Proof. First recall the formula (from the proof of Theorem 7.4)
r( )f+, = E VP(u)fIa(u/r).
(7.44)
u (,nod r)
Slashing with WN, we get by (7.31)
w('+G)TW9o = E '
(u)fla(u/r)wr4.
u (niod r)
Then applying (7.44) to g,y, we arrive at (7.45)
X(r)V)(q) E is(u)91(u/r) = E (u)fWu/r)wN u (mod r)
u (mod r)
Next for each (u, r) = 1 we take v with uvq = 1 (mod r); this yields ry(v/r) E I'o(q) (see (7.41)). Recall the identity (see (7.33))
a(u/r)WN = rwgry(v/r)a(v/r) Hence fla(u/r)WN =
and we can write (7.45) as follows: {v}gla(vlr)
= 0.
v (mod r)
Since this holds for every 0 other than the trivial character, it implies by orthogonality that 9jp(,,,/r) = 9100v2/r) if (VI V21 r) = 1 (to see this, multiply by O(v1)  ' ,(v2) and sum over 0 (mod r)). Lemma 7.10. Let m, n be primes with (mn, q) = 1. Suppose f and g = few satisfy (7.31) for any primitive character 0 of conductor r = m or n with the constant w(,O) given by (7.32). Then (7.46)
9iy = X('Y)9
it  n) . for all ry E I'p(q) of type ry = ( uq
7.4. Converse theorems
129
Proof. Applying Lemma 7.9 for y and 11 = ("` v } we get uqn 91(x(m)y)a(vlm) = 9I(X(+n)y,)a(v/vn)5
whence (7.47)
9I(X(m)y)  9I(X(m)7j)a(2v/in)
Similarly, applying Lemma 7.9 for y,1 = (_' ,n) and
ry1
=
(n t) , we
get
(7.48)
91(x(n)y, ') =
9I(X(,n)y1)a(2v/n).
Since X(mn) = 1, we have the relations
X(n) 'yl 1 = X(n)(X(m)  Yl)7i 1 X(n) y1 = X(n) (X(m)  7)'Y1 and inserting these into (7.48) we get (7.49)
91(X(m)7i)y1 T 9I(X(m)y)j1a('2v/n).
Combining (7.49) with (7.47), we arrive at 9I(X(m)y) = 9I(X(m)7)7la(2v/n)7ia(2v/m).
Putting h = gi(x(,n)y) and tion asserts that
yla(2v/n)yja(2v/m), the above rela
(7.50)
hIQ = h.
We have
Q_
1
2v/m
2uq/n 3 + 4/mn
hence Itr/3I = 124/mnj < 2 and itr/31
'
0, 1, and so Q is an elliptic element
of SL2(Q). The eigenvalues of /3 are not roots of unity. Indeed, since they are in a quadratic field, they would have to be among the numbers ±1, ±i, ±(6, ±(3, giving Itr/31 = 0, 1, 2, which is not possible. Let zo be the fixed point of /3 in the tipper halfplane H. Put p = (z  Nv)1 (1
zo) E GL2(C),
7. Autoulorphic Lfunctions
130
so p brings f3 to the Jordan form,
\
pop1 = (
(7.51)
with n E C.
n_1 J
Moreover p maps IHl onto the \\unit ball B = {w E C : IwI < 1}. Therefore the function P(w) _ (hip1 )(w) = jp1 is holomorphic in B. By (7.50) and (7.51) we get (w)kh(p1w)
P(n2w) = n"P(w) Looking at the Taylor expansion of P(w) at w = 0, we find that all the coefficients vanish because n is not a root of unity. Therefore P(w)  0. Hence h(z)  0, which means gay = X(m)g = X(n)g = X(ry)g, and this completes the proof of Lemma 7.10. Finally we are ready to prove Theorem 7.8. We need to show that (91f
=X(7)9
if
'Y

dbl ac
E ro(q)
This is obvious if c = 0, so let c > 0, c = 0 (mod q). Applying suitable integral translations on both sides of 'y, we can bring y to a matrix of the type considered in Lemma 7.10. Indeed, by the prime number theorem for arithmetic progressions (due to Dirichlet) there exist m = a + ct E 7 and
n=d+csE7.,andwehave
d)
( rn
1t) uq
n
\1
1sl with u = c/q and v = b + sm + to + cst. By Lemma 7.10 (note that the Cc
hypothesis g = fl,,, is satisfied since (A2) implies (A1) by virtue of the Hecke converse Theorem 7.3) we get 91, =X(n)9 = X(d)9 = X('y)9
since the integral translations do not alter the character nor the multiplier factor. This relation shows the automorphy of g with respect to ro(q) and the character X. Hence f is automorphic with respect to ro(q) and the character X by virtue of (6.69). Suppose L f(s) converges absolutely on the line Re s = a, the case of Lg(s) being the same by applying the involution w. Then we have 00
jamIe2nmy << 1 + max
If (x)I
(n°e2xny)
n=0
Hence f is a cusp form by virtue of the following
<< 1 + yO.
7.4. Converse theorems
131
Criterion for Cusp Forms. H. Suppose f is an automorphic form of weight k > 0 for a group r such that (7.52)
ykf(z)>0
as
y>0
uniformly on H. Then f is a cusp form. Proof. The zeroth coefficient of f at a cusp a satisfies lao! S fo I fto(z)I dx =
f
o
1(y1Im az) z l f (az)ldx
where a E SL2(R) is a scaling matrix. We can assume that a is not at oo (by choosing an equivalent cusp on R), so a is not an upper triangular matrix.
Thus as y  oo we have y1 : Im az  0 so the above integral vanishes. Hence ao = 0, proving that f is a cusp form. Remark. If L f(s), L9(s) converge absolutely in Re s > k  6, then A f(s), A9(s) are holomorphic in Re s > k  6, so ao = bo = 0 by (7.27), (7.28); but the vanishing of the Fourier coefficients at cusps other than 0 or oo does not follow by the holomorphy of Af(s), A9(s). For q prime every cusp is equivalent to 0 or oo, so in this case the holomorphy condition alone permits one to deduce that f, g are cusp forms.
Chapter 8
Cusp Forms Associated with Elliptic Curves 8.1. The HasseWeil Lfunction An elliptic curve E is an algebraic curve (a projective algebraic variety of dimension 1) of genus 1 over a field K. If char(K) yc 2, 3, then E is given by the Weierstrass equation (8.1)
y2=x3+Ax+B
with A, B E K. The discriminant of E is the discriminant of the cubic polynomial (8.2)
g(x) = x3 + Ax + B,
and it is equal to (8.3)
A = 16(4A3 + 27B2).
It turns out that (see Figure 9) E is nonsingular
A
0,
Ehasnode taA=0,A0, Ehascusp s.A=0,A=0. 133
8. Cusp Forms Associated with Elliptic Curves
134
y2 = x3 + x
y2 = x3  x
smooth
smooth
y2 = x3
y2 = x3 + x2
Cusp
node
A=64
A=64
00
=0
Figure 9. Elliptic curves
If char(K) = 2 or 3, the Weierstrass equation and the description of singularities are slightly different. Suppose E is given by the Weierstrass equation (8.1) with A, B E Z and
A 36 0. For each prime p consider the reduced curve E/Fp over the field Fp of p elements. Let v(p) denote the number of points on E/Fp, i.e. the
8.1. The Hasse Weil Lfunction
135
number of solutions to the congruence y2
= 9(s) (mod p).
We do not count the point at infinity. It turns out that v(p) is well approximated by p; more precisely, the difference (8.4)
\(P) = p  v(P)
satisfies
IA(P)I < 2 j. This estimate is due to H. Hasse and is essentially best possible. In order to understand how the X(p) vary with p, Hasse began and Weil continued to investigate the Lfunction for E defined by the following Euler product: \(P)P9)1 [j (1 (8.6) LE(S) = 11(1  \(P)P' + pl2s)1 (8.5)

PIo
PIo
Remarks. Although we assumed that E/Q is smooth (since A ,E 0), the reduced curve E/]FP is singular if pl0, and E is said to have bad reduction at such primes. One can show that for primes of bad reduction a(p) = 0, 1, 1
according to a type of singularity which occurs, namely a cusp, a node with rational slopes for the tangents, or a node with quadratic irrational slopes for the tangents. If p f i then the reduced curve E/1FP remains smooth, so E is said to have good reduction at p. In this case the local factor 1 
\(P)P8 + pl2s
appears naturally in the socalled congruence zetafunction of E defined by 00
CE/Fn(s) = exp(E, m1V(pm)P m4) M=1
where v(q) denotes the number of points on E over the finite field E. (it is not the same as the number of points modulo q). We have 2s). (E/Fp(s) = (1  pls)l(1  )1(P)P' +P1 The estimate of Hasse (8.5) can be interpreted as the Riemann hypothesis for CE/FP (s), which asserts that the roots are on the line Re s = 1/2. Write LE(s) as the Dirichlet series 00
LE(s)
Note that the Euler product (8.6) and the Dirichlet series (8.7) converge absolutely for Re s > 3/2.
8. Cusp Forms Associated with Elliptic Curves
136
Conjecture (Hasse). LE(s) has analytic continuation to an entire function, and it satisfies the functional equation (8.8)
(2)2sr(2
(A)r(s)LE(s) 2= 77
 s)LE(2  s)
where q is a positive integer composed of prime factors of A, the socalled conductor of E, and 71 = ±1 is called the root number.
Conjecture (ShimuraTaniyama). The Fourier series 00 f(z)
(8.9)
A(n)e(nz)
is a cusp form of weight 2 for ro(q) and the principal character; it is a newform with
T f = A(n) f Wf =77f.
(8.10) (8.11)
Recently these conjectures were proved by A. Wiles (at least if q is squarefree). We shall give a simple proof for special curves, the socalled congruent number curves.
8.2. Elliptic curves Er In this chapter we shall examine a family of elliptic curves Er given by the equation y2 = x3
(8.12)
 r2x
where r is a positive, squarefree integer. These curves were studied by J. `I unnell in connection with the ancient problem of the socalled "congruent
numbers." A positive rational number r is called a congruent number if it is the area of some right triangle with rational sides. Equivalently, there exists x such that all three number x  r, x, x + r are squares of rationals. This also means there are infinitely many rational points on E,.. Multiplying by suitable squares, we may require r to be a positive, squarefree integer. Tunnell used the curve Er to establish an effective method of checking if r is a congruent number. The smallest congruent numbers are r = 5,6,7. Note that the discriminant of E,. is A,. = 64r6, and if plA,. then vr(p) _ p, so Ar(p) = 0. In this case the HasseWeil Lfunction reduces to 11 (1  a,.(p)P s +p12s)1 = (8.13) L1=r(s) = ,(n)ns
142,'
(, 2r)_I
8.2. Elliptic curves E,.
137
In general if a curve E is given by
y2 = 9(x) with g E Z[x], then the number of points of E/]Fp is equal to
V(p) _ E (mod p)
px
(i+(i\
where (p) is the quadratic residue symbol (the Legendre symbol). Hence gy(p)=
(8.14)
(g(x)\.
p
x (mod p)
In particular, if p fi 2r, then
r(p) x_(mod p) (x3 pr2x/  (Pr) x (mod p) \x3p x/ = (Pr)
(P)
by changing x > rx. Hence for all n (8.15)
Ar(n) = Xr(n)A1(n)
where Xr(n) is the Jacobi symbol, (8.16)
X, (n) = (n)
This shows that the .function for Er is obtained from that for El by twisting with the character Xr, (8.17)
Xr(P))'1(p)Ps
LE,.(s) = [J(1 
(p)p'2S)1
+ X?
p#2
= 1: Xr(n)AI (n)ns 2in
By virtue of the above connection it will be sufficient to prove the Masse
and the ShimuraTaniyama conjectures for El. In this case we simplify notation by omitting the subscript r = 1, so we write E = El, A = X1 and (8.18)
LE(s) =
rj(1  A(p)p +pl2s)1 p712
= .
2{,i
A(n)n'.
8. Cusp Forms Associated with Elliptic Curves
138
After establishing the conjectures for the curve E, one can extend the results for E, by an appeal to a general principle about twisting automorphic forms with characters (see Section 7.3, Theorem 7.4 and the formulas (7.32), (7.33)).
8.3. Computing A(p) The curve E given by the equation (8.19)
has many automorphisms; for example, if (y, x) is on E then so are the points
(y, x) and (iy, x). We do not dwell on explaining what really happens here, but only say that this observation is tacitly used in the course of computing \(p). The discriminant of E is A = 64. For p = 2 we have v(2) = 2, so A(2) = 0.
(8.20)
For p = 1 (mod 4), since (p1) = 1 and g(x) = g(x), we derive by (8.14) that \(p) = .\(p), and so (8.21)
A(p) = 0
if p = 1 (mod 4).
In the remaining case p = 1 (mod 4) we shall carry out computations by passing to another curve E' given by the equation
Y2 =X4+4.
(8.22)
There is a map from E  (0, 0) to E' given by
(y,x) > (2x 
y2x2, yx1)
This has the inverse from E' to E  (0, 0) given by (Y,X) +
(x(Y + X2), 2(Y+X2))
Therefore the number of points on E/Fp and E'/Fp are related by v(p) 1 = v'(p). The key advantage of dealing with E' is that E' has a diagonal equation. Let p  1 (mod 4). The multiplicative group FP* is cyclic of order p1 0 (mod 4), and so is the character group F. For any z E FP* we have
#{x E Fp : x4 = z} = E X(z). X4=1
8.3. Computing A(p)
139
Hence
v'(p) = 2 + > J(X) X4=1
where
E
J(X) =
X(Y2
 4).
Y (mod p)
There are four characters of exponent 4, all given b y X = 1 , n, 72, rj is a fixed character of order 4. For X = 1 we get
7
7, where
J(1) = p  2.
For X = 7?2 (it is the Legendre symbol) we get j(172)
_
E X((Y  2)(Y + 2)) Y
=EX((Y4)Y)=> x(YY4) Y Y:A0 _ 1: X(1  4Y) _ 1 + > X(Y), Y
YOO
whence
j(r12) = 1. For X = n3 we get j('73) =
,7(n) From the above evaluations
we infer that &J(P) = p  1 + ,7(m) + .707),
whence
v(P) = p + J(r!) + Y(rl), and (8.23)
)(P) _ ,T(rl)  .707)
Now we proceed to compute J(r?) (the Jacobi sum). First we establish
that (8.24)
P'.
S. Cusp Forms Associated with Elliptic Curves
140
Indeed, by squaring, factoring y2  4 = (Y  2)(Y + 2) and changing the variables several times we derive the following expressions: 2
_
77((x  4)x)
IJ('7)12 =
77
X
x,y9E0,4
_ E 77(z) z
E
77
l
(
\ (y  4)y/
(z_4) _ z
=p2+>77(z) >
4(z  1) Cz + y4 /
y90,4
77(z+(z1)v)
V00,1
z5E1
= p  2  > 77(z) (17(z) + 1) 201
=p> z
Z
Next we determine the argument of J(77). There are not many possibilities to choose from. Since 774 = 1 the terms of J(77) take values 0, ±1, ±i; therefore ,7(77) is a Gaussian integer, J(77) E 7G[i]. On the other hand, p  1 (mod 4) factors in Z[i] into
p=7r7; where 7r is determined up to complex conjugation (7r is not distinguished from ir) and a unit c = ±1, ±i (by the unique factorization in the ring Z[i]). Combining the above facts, we deduce that J(77) = 7r
(8.25)
for some prime factor of p in Z[i]. To determine which factor (out of eight possibilities) is correct, we test the equation (8.25) modulo the ideal
a = ((1 + i)3) = 2(1 + i),
Na=8.
Since the character 77 takes values 0, ± 1, ±i, each of which except for 0 is congruent to 1 (mod (1 + i)), we infer that 7(17) =
'J(Y2
 4) =
Y (nod p)
77(Y2
0<1'<'P2 '
\
1+21
p2111
Hence for p  1 (mod 4)
/
(8.26)
1+2E Y#2
p2 (mod a).
J(77) = 1 (mod a).
 4)
8.4. A Hecke Grossencllaracter
141
The above congruence together with (8.25) determines J(77) up to complex
conjugation (surely one cannot be more exact as long as rl is not distinguished from n).
We say that a Gaussian integer a is primary if a  1 (mod a). Every odd a (i.e. coprime with a) is conjugate to exactly one primary integer. The only primary unit of 7L[i] is 1. The product of primary numbers is primary, and every primary number factors uniquely (up to permutation) as a product of primary numbers which are Gaussian primes. By (8.25) and (8.26) it follows that ,7(7) is a primary prime. Finally by (8.23), (8.25) and (8.26) we conclude that (8.27)
A(p) = it + Tr
if p  1 (mod 4)
where 7r: = p and ;r is determined up to conjugation by the congruence (8.28)
it  1 (mod a),
i.e. IT is a primary factor of p.
8.4. A Hecke Grossencharacter Consider the multiplicative group (7L[i]/a)* of residue classes in 7L[i] to mod
ulus a and prime to a; it is a cyclic group of 4 elements represented by the units. For a odd we define p(a) to be the unit which makes p(a)a primary, i.e. p(a) = 1, i, i2, i3 is such that (8.29)
p(a)a  1 (mod a) if (a, a) = 1.
If (a, a) 1 we set p(a) = 0. Thus p is a character on Z[i] to modulus a. Then we put (8.30)
X(a) = p(a)a.
The function X is one of many kinds of Grossencharacters which have been invented by E. Hecke. This can be regarded as a character on ideals r C 7L[i]. Every ideal is a principal ideal, say r = (a), with generator determined up to a unit. If (r, a) = 1 we can fix a by requiring a 1 (mod a), and we set
X(t) = a.
(8.31)
If (t, a) # 1 we put X(r) = 0. With the character X Hecke associated the Lfunction defined by the Euler product (see Chapter 12) (8.32)
L(s, x) = [J(1  X(p)(NP)`)V
8. Cusp Forms Associated with Elliptic Curves
142
Verifying the local factors in (8.18) with these in (8.32) by means of (8.20), (8.21), (8.27), we see that the HasseWeil and the Hecke Lfunctions coincide,
LE(s) = L(s, X).
(8.33)
Then comparing the coefficients in the Dirichlet series we get
a(n) = E X(r) =
(8.34)
Nr=n
4
E p(a)a IaI2=n
where t ranges over the ideals of 7L[i) and a ranges over the elements of 7L[i) (there are four units in 7L[i), whence the factor 14).
8.5. A theta series By (8.34) the Fourier series (8.9) associated with the curve (8.19) is expressed as a sum over Gaussian integers
AZ) = 4 E p(a)a e(zIaI).
(8.35)
aEZIi)
This expression constitutes a vital step in showing that f (z) is an automorphic form; it marks a transition from the multiplicative to the additive structure of 7L[il. At this point it is worth recalling that the sign of X(p) is captured by the congruence (8.28), which allows us to determine a(n) without inspecting the factorization of n into primes. Now, having f (z) expressed in additive terms, we can apply tools of harmonic analysis. Considering Z[i] as a lattice, we shall treat the series (8.35) by using Poisson summation; the arguments are standard but details are cumbersome because of powers of 2 in the conductor. Much more general theta functions will be considered in Chapter 10. We begin by recalling the formula (1.9), E e7r(n+r)2v = v1/2 n
En
e7r?t2t?1
e(nx).
Differentiating in x, we derive another formula:
E(n + x)e'r(71+x)2v =
iv3/2
71
En
ne7rn2v1
e(nx).
Combining both formulas, we find that J :(m + x + i(n + y))e7r[(m+y)2+(n+y)2}v 7)t
7T
_ iv2 E E(m + 771
71
in)e'r(?n2+n2)t,1
e(mx + ny).
8.6. The automnorphy of f
143
This formula is established for x, y, v real with v > 0; however it remains true for any complex numbers with Re v > 0 by analytic continuation. In the complex numbers notation we rewrite the above as follows (put ce =
m+in,)3 = x+iy,v = 2iz): aEZ[i]
2
E
(a + /3)e(zIa + $12) = 4z2
aEZ[i]
ae
4z+ Re a,(3)
.
This holds for any complex numbers ,Q, z with Im z > 0. Hence we infer our basic formula for the following thetaseries: (8.36)
F(z; 0, y) = E
ae(ZjaI2);
aEZ[ij
ap (mod y)
namely, for any 6, y E Z[i] with y 36 0 we have (8.37)
F(z; N y) = aEZ(ii]
ae (4 II7II2 + Re aR)
.
We shall exploit the transformation rule (8.37) to establish automorphic properties of the thetaseries f (z) twisted by the character p(a) by splitting into residue classes to suitable moduli. Our target is to show that f E S2(r(,(32)).
(8.38)
8.6. The automorphy of f First we show that f is an eigenfunction of the involution of weight 2 and level 32 with eigenvalue 1, i.e.
fl,,, = f for w = More explicitly, we need to show that (8.39)
32z f
(32z)
z f (z).
Let y = 2(1 + i), so Iry12 = 8. By (8.37) we obtain
E p(Q)F(z; 0,,y)
f (z) =
/3 (mod y) iY
(322)2
aEZ(i]
ae ( I al ,i G(a,y) 32z
8. Cusp Forms Associated with Elliptic Curves
144
where
P(Q)e (Re mo/
G(a,'Y) _
\
p (mod y)
.
If
Since p is a primitive character to conductor a = (y), it is clear that G(a, y) = 0 unless (a, y) = 1, in which case on changing Q to a,Q and observing that 1a12 = 1 (mod 4) we obtain G(a, y) = P(a)G(1, y) 
Finally we compute G(1, y) termbytermn as follows: 2
) + ile (Re i) + i2e (Re )
G(I, y) =e (R.e 'Y
+ i3e I Re
'Y
\
a3
)
'Y
=e4Iie4)el 1+ie(4)=2i(1i)=iy. Gathering together the above results, we arrive at (8.39). Next we show that f is automorphic of weight 2, i.e.
fl. = f
if T E ro(32).
This is obvious for r = (1 i) because f is defined by a Fourier series. Suppose z = (a d) with 0 < c = 0 (mod 32) and ad = 1 (mod c). We need to show that
z1f (z
(8.40)
ca) =zf (z+d)
We start from the righthand side as follows:
zf(zcd)= 4
P(a)ae
(_IaI2)
«ezli)
r
E P(Q)el
\
/3 (mod 2)
IQI2JF(zc ;Q,2 //
(note that a  Q (mod c) implies Ja12 = 1,012 (mod c)). Applying (8.37), we obtain
_ 2cz
aEZ(il
ae (
2l CP(Q)e (!/3!2 / 13 (mod
2)
+ 2Re a/3)
.
8.6. The autoinorphy of f
145
Here we have d1/312 + 2Re a/3  d113 + aa12  a1«12 (mod c); hence
z f (z
(8.41)
d) 2cz
c
(_z1`c
E ag(a, c)e
aEZ,i]

ala121
where g(a, c) is a kind of Gauss sum given by
g(a, c) _
p(Q)e I d IQ + aa12 I
\
p (mod 2)
43  aa)e (d 1012 J jI
(mo d 2 )
If (cx, 2) 0 1 then a  a (mod a), so changing /3 to /3 we infer that 9(a, c) = g(a, c), i.e. g(a, c) = 0. Suppose (a, 2) = 1; then (1 + i) 1,6, and so
1 g(a, c) = 2 E p((1 + i)/3  aa)e (2d IQ12 I
`
3 (mod 4)
/
If Re /3  I (mod 2), then the terms for /3 and /3+ s cancel out. If Im /3 1 (mod 2), then the terms for /3 and /3 + s cancel out. Therefore we are left with /3  0 (mod 2), which give (8.42)
g(a,c) _ 2p(aa)
e
0 (mod g)
(lfi,2) c
To evaluate the last suns we appeal to the following formula of Dirichlet (see(1.10)): a
(8.43)
e
(N 1 = (l +i)Nz
n (mod N)
which is valid for any N > 0 with 41N. Hence
E e ( alN / = ((1 +i)NS)2 = 2iN. l
13 (mod N)
For (d, N) = 1 one can find 6 E Zli] such that d  ±1612 (mod N), whence p(d) = ±1 and
e M) = 13 (mod N)
/l
e /3 (mod N)
1131
2
±2iN = 2ip(d)N.
146
8. Cusp Forms Associated with Elliptic Curves
Applying this for N = c/8, we get by (8.42)
g(a, c) = C p(a) Inserting this into (8.41), we obtain (8.40). Finally we claim that f is an eigenfunction of all the Hecke operators. Since the coefficients of f are derived from the Euler product (8.18) which coincides with (6.93) (the character is principal), one checks that Tn f = a(n) f for all n by using Proposition 6.3. Thus f is a newform.
Chapter 9
Spherical Functions Spherical fimctions belong to the theory of representations on symmetric spaces associated with specific Lie groups. The classical theory goes back to Legendre, Laplace, Jacobi (late eighteenth century), and the modern development began with Cartan and Weyl (1930's); other notable contributors are Gelfand and HarishChandra. For our purposes we need classical spherical functions in the context of a positive definite quadratic form.
9.1. Positive definite quadratic forms Throughout we use the following notation and simple facts:
A=(ai?) a real r x r matrix, to = (aji) the transpose matrix, A I = det A 54 0
nonsingularity,
A' = (a! .) the inverse matrix, 2.7
1A= A symmetry. Spectral Theorem. Any real, symmetric matrix A is orthogonally equivalent to a diagonal matrix, i.e. A = tUDU with'UU = I and D =
Here \l, ... , A, are the eigenvalues of A. 147
9. Spherical Functions
148
A matrix A E M(r) is positive definite if all its eigenvalues are positive. Equivalently
AA'
with lAjI>0 for all I
We shall write vectors x in R'' as column vectors so the transpose is the row vector tx = (x1, ... , x,.). Hence txx
= IxI2 =
x?.
With a matrix A = (aij) E M(r) we associate the quadratic form (9.1)
aijxixj.
A[x] = LxAx = ij
If A is positive definite, then so is its quadratic form, i.e. (9.2)
IxI2 < A[x] < IxI2
for all real x. A complex vector u ,E 0 is called isotropic for A if A[u] = 0.
(9.3)
For a positive definite matrix A = (aij) its inverse A1 = (aij) is also positive definite. We attach to A the Laplace operator (9.4)
AA = > atj i
j
8z dxi8xj
Moreover we shall consider the inner product with respect to A defined by (9.5)
(f,9)A =
f
f(x)(x)dx A
for functions f, g on the ellipsoid (9.6)
KA={xEIR':A[x]<, 1}.
We shall regard lie? as a Riemannian manifold with the matrix (of curvature zero) derived from the differential form given by
ds2=1y?j
9.1. Positive definite quadratic forms
149
in the rectangular coordinates yl, ... , yr. The volume element is
dy = fl dyj, and the Laplace operator is given by 0 = div grad = E
To simplify computations we shall often make a suitable change of variables
x (a diffeomorphism of W). The Jacobian of this transformation is a nonsingular matrix y
3 = (8y:/8xi). The arc element is expressed in new coordinates by
ds2 = I: gijdx=dxj
where G = (g,i)
4j
the volume element is given by
dy = g1/2 fl dxj where g = [GI,
i and the Laplacian takes the following form: g1/2
,
xd8g1/2i =i axJ
Let A be a symmetric, positive definite matrix and U be an orthogonal matrix which diagonalizes A. Then say,
B=
r
U = (6ij),
gives
Al
A=tU
U=tBB. Ar
Changing variables y = Bx, we transform the quadratic form A[x] into A[x] =
t(Bly)A(B1y) = ty(tB)`[AB1y
= tyy =
[y[2.
9. Spherical Functions
150
For this linear change of variables the Jacobian is 9 = B, and G tBB = A; hence IGl = JAI = 1B12,
ds2 = > aijdxidxj, t)
dy = IAI1/2 ll dxj,
A=Ea!. axiaxj
AA.
ij
The unit ball
K = {yElllr:lyl<11 is transformed to the ellipsoid ICA, and the inner product with respect to A on KA becomes (LOA = JAI1/2 fK f (x)g(x)dy
The boundary OK of the unit ball K is the sphere S''1
= {y E llt' : lyl = 1}.
The surface area of Sr1 is equal to 2ir''/2I'(r/2)1.
9.2. Space spherical functions A smooth function f on D C it is called a harmonic function if it satisfies the Laplace equation AA f = 0.
(9.7)
In dimension r = 1 the harmonic functions are linear functions. In higher dimension the basic examples are (for A = I) f (x) = log l xl
if r = 2
f(x)=lxl2^r ifr>3. These are the fundamental solutions for the Laplacian A (up to constant factors), and they are (must be) singular at the origin x = 0. However, if f is harmonic everywhere in Rr and is bounded by a polynomial, then f itself is a polynomial. This result is an analog of Liouville's principle for holomorphic functions. Clearly the homogeneous parts of f of every degree are also harmonic functions.
9.2. Space spherical functions
151
The harmonic functions which are homogeneous polynomials are called spherical functions or harmonic polynomials.
Theorem 9.1. Let f (x) be any homogeneous polynomial of degree v. Then the following assertions are equivalent: (i) f (x) is a spherical function, (ii) f (x) is orthogonal to any homogeneous polynomial of lower degree, (iii) f (x) = any constant, if v = 0, f (x) = any linear form, if v = 1, f (x) = any finite sum of forms (tuAx)" where u is an isotropic vector,
i.e. A[u] =0, if v,>2. Remark. At first glance the implication (ii)
(i) looks somewhat surprising. In a certain setting one could detect here a variational principle for the lowest eigenfunction of a selfadjoint operator in a Hilbert space. Our proof does not reveal this principle; it is more or less direct.
For the proof we make the change of variables y = Bx and v = Bu, which yields tuAx = tvy and tuAu = tvv. Therefore we can reduce the general case to the identity matrix. From now on A = I. Before starting the proof let us recall some facts from differential calculus
on W. Denote the differential forms
w? _ (1)jldxl ...
j ... dx,.
dx=dxt...dxr=dx?w? for each 1,<j,
where 71 is an (r  1)form on K. For 77 = f  w we get
di=d> where 6 denotes the differential operator (of Euler) (9.8)
Sf =
a xjdxi
Therefore for ri = f w Stokes' theorem gives (9.9)
f (bf + rf)dx =
fw.
9. Spherical ,Functions
152
Next, by Stokes' theorem for the form rl = (b f )w (or by the divergence theorem) one gets (9.10) fK
Af dx=fx(bf)w
where 0 is the Laplacian
Af
(9.11)
82f axe
If f is a homogeneous polynomial of degree v, then
bf = vf,
(9.12)
i.e. f is an eigenfunction of b with eigenvalue v (check this on monomials xi' ... xa, with al +... + a,. = v). Combining (9.9), (9.10) and (9.12), we conclude that (9.13)
fK
=v fx a
f=v(v+r)J
x
for any f which is a homogeneous polynomial in r variables of degree v. Finally let us record the following property of the Laplacian:
A(fg)=fAg+gOf+2
(9.14)
of ag ax; ax;
Now we are ready to prove Theorem 9.1 (for A = I). First we show that (iii) implies (i). This is obvious if v = 0, 1. Suppose v >, 2 and f (x) = (tux)" with tuu = Ju12 = 0. Then
Next we prove that (i) implies (ii) by induction on v. But first notice that if f is a spherical function, then so are its partial derivatives a f /axe because g commutes with 8/ax3 (all partial derivatives commute, and 0 has constant coefficients). Assuming that deg g < deg f and A f = 0, we proceed as follows:
f fgdx = cl JrI 0(7g)dx = cl f fLgdx t
Ic
= C2
li
0(1Og)dx = C2
f JA2gdx li
9.2. Space spherical functions
153
by (9.13), (9.14) and the induction hypothesis. We can continue this way until we get 0"g = 0, whence (f, g) = 0. It remains to show that (ii) implies (iii). Let us denote the space of all homogeneous polynomials of degree v by
P = {f E C[x] : f (ax) = a' f (x)}.
(9.15)
This has finite dimension over C; precisely, dim P" =
(9.16)
Cv+r  1
r1
The linear space spanned by all polynomials of type (tux)" with tuu = 0 is closed in P". Therefore the problem reduces to showing that if f E P" is orthogonal to all P. with µ < v and all polynomials of type
g(x) = (tux)" with tuu = 0
(9.17)
then f = 0. Since g and its partial derivatives satisfy (iii), they satisfy (i) and (ii). Using this information we infer the following:
22f 19 dx = cl J A(fg)dx = 2c, J ( E Nax ax j K
;
1C
i
dx
.i
by (9.13) and (9.14), since Og = 0 and g10 f , the last property being a consequence of the implication (i) times, we arrive at 2"c"
J
a"f
(ii) for g. Repeating this argument v
ax;, ... ax;,, ax;,
a1g .
ax;,,
dx = 2"c"vol (K)a
where o denotes the above sum of derivatives, which is a constant since deg f = degg = v. To compute a apply the formula (see (9.12))
of =2., xi
of ax;
successively to the resulting partial derivatives until we get
V! f = E x;, ... ..i,, a"f /ax;, ... ax;,.. .ii...;t,
9. Spherical Functions
154
Moreover, we have
avg/ax ... 8xj = v!uj, ... uj,,. From these two formulas it is plain that or = f (u). On the other hand, a = 0 because f 1g. Therefore we have shown that
f (u) = 0
if E u = 0.
This means that f (x) vanishes on the algebraic variety V = {x E C' : p(x) _ 0} defined by the polynomial
p(x) = Ex" By Hilbert's Nullstellensatz the ideal of V contains a power of f. Except when r = 1, the polynomial p(x) is either irreducible (if r 3 3) or factors into two coprime irreducible polynomials (if r = 2), so f (x) is divisible by p(x), say f (x) = p(x)h(x). By (9.13) we infer that h(x)h(x)dx = cl 10K h(x)(x)w
c10K f(x)h(x)w
Jx
= c2
J f(x)h(x)dx = 0
K
because p(x) = 1 on 6K and deg h < deg f. Hence h = 0 and f = 0. For r = 1 we have f (x) = axv, and if v > 2 then p(x) = x2 divides f (x), so the former argument applies. For r = 1 and v = 0, 1 there is nothing to prove. This completes the proof of Theorem 9.1.
Corollary 9.2. The space lv of harmonic polynomials of degree v > 2 has complex dimension
/1 v+r
dimW, = t
(9.18)
r1
_
v+r3
r1
Proof. By Theorem 9.1
EP,,: f1Pµ for all it
But P,,LP, if lc product), so
7v = {f E Pv : f 1Pv2, Pv4, ... }. We also have
(f, 9) =
J!i
f (x)9(x)dx = (J ti +IL+rdt) o`
JO
f (x)g(x)dx
9.3. The spherical functions reconsidered
155
where v = deg f, µ = deg g and r is the dimension. Hence
_
v+µ+r+3
v+p+rI1(f'Pg)
Rv = {f E P, : f 1Pv2}. Hence by (9.16) we get (9.18).
9.3. The spherical functions reconsidered A homogeneous polynomial f (x) of degree v is determined by its values on the unit ball. Namely, we have
f(x)=IxIvfCHI This formula establishes a onetoone correspondence between the space spherical functions and the surface spherical functions. The surface spherical functions are eigenfunctions of the Laplacian on the sphere
Si ={xER":IxI=1} with eigenvalues depending on the degree v and the dimension n (we changed
notation for the dimension from r to n to avoid confusion with the radius coordinate). We shall now briefly present the surface spherical functions from the point of view of representations of the special orthogonal group. The space X = R" is usually viewed as a homogeneous space of the additive group G = R" acting on itself by translations. A differential operator is Ginvariant when it has constant coefficients, and the joint eigenfunctions of all such operators are multiples of exponential functions. Therefore harmonic analysis on the flat space R' (curvature zero) reduces to the study of classical Fourier integrals. One can be more demanding by taking the whole group of orientation
preserving isometrics on ll8", say M(n), which is generated by the translations and rotations. Then R" acted on by the whole group M(n) is the euclidean space of dimension n. The group of rotations 0(n), which is the isotropy group of the origin, is represented by real orthogonal matrices
O(n)={U:tUU=I}.
9. Spherical Functions
156
Hence the euclidean space R" can be viewed as the space of cosets
R" = M(n)/O(n) which makes II8" a homogeneous space. Now an invariant differential operator is not only translation invariant but also rotation invariant. This additional requirement implies that such an operator is a polynomial in the Laplacian
a2
92
8x2
+ ... + ax2 n
i
with constant coefficients. In other words, the algebra of invariant differential operators is generated by A. The basic eigenfunctions of A are e('xy) = e(xiyl + ... + x:tyn), and harmonic analysis on II(" reduces again to the classical Fourier inversion. With the above point of view it is not surprising that the Fourier transform in l[8" commutes with rotations. In particular the Fourier transform of a
radial function, i.e. a function which depends on the distance
r(x) _
(:xi
2
is also a radial function (see (1.12)). Now we consider the sphere S' 1 C R' for n > 2. The special orthogonal group
SO(n) = {U E 0(n) : detU = 1} acts transitively on Sn1, and the isotropy group of each point can be identified with SO(n 1). To be specific, fix a point, say s = (0,... , 0, 1) E Sn1,
for which the isotropy group is the group of rotations around the x,taxis and is isomorphic to SO(n  1). Think of U in SO(n  1) as an element u (?) of SO(n). By means of this embedding the sphere can be regarded as the homogeneous space
S"1 = SO(n)/SO(n 1). As a Riemannian manifold S"1 has positive curvature. On convenient to work in the spherical polar coordinates x1 = rsinB"_1...sin 92 sin91 X2 = r sin 9,t_ 1 ... sin 02 cos 01 x,t_1 = rSin0,t_1 COS9n_2
xn = rcos9,t_1
S"1 it is
9.3. The spherical functions reconsidered
157
where 0 < r < oo, 0 < 91 < 21r and 0 < 0.7 < ir if 1 < j < n. The rotation invariant measure on Sn1 is given in terms of the angles 9j by dµ(9)
r (2)
fir
(sin 9j)jd9j. 1<,j
The Laplacian 0 on R" and the surface Laplacian 0' on Sn1 are related by
0 +r1 O'.
_
02
(9.21)
8r2+nr
Let P = C[x1, ... , xn] be the algebra of polynomials in n variables over the complex numbers. Clearly P is SO(n)invariant, and it splits into the direct sum of finite dimensional invariant subspaces
P
P
where P denotes the space of homogeneous polynomials of degree v. We have d1ln Pv
=
v+n
1
n1
because the monomials xi ... xn" with eel + ... + an = v make a basis for P. We endow P with a nondegenerate bilinear form (an Hermitian inner product) with respect to which the distinct monomials are orthogonal, Ai
a
r al!...an! if (a) =
(i3)
otherwise.
0
To be specific (though it will not be of much use), for any q, r E P we put.
(q,r) = (a(r)q)(0) where 8(r) is the differential operator a ar() r(ax1'
a
8x
Clearly the multiplication p , rp is adjoint to the differentiation q 4 8(r)q, i.e.
(q,rp) = (a(r)q, p) (it is sufficient and easy to check this for r(x) = xj, NO = 09/axe;). In particular, multiplication by r(x) = xi +... + x2 is adjoint to the Laplace operator A = a2/8xi + ... + a2/az a, i.e. 71
(q,?'p) = (Oq,p).
9. Spherical Functions
158
Observe that rP C P,,+2 and AP,, C
The kernel of 0 consists of harmonic functions on Rn, i.e. polynomials annihilated by A. We have the direct sum
7l = ker to = ®7l where H = H fl Pv is the space of harmonic polynomials of degree v. In general the kernel of a linear operator is equal to the orthogonal complement to the range of its adjoint, so we have
P,, = f'l,, ® r2P.,2 Hence by induction we get the orthogonal decomposition
P,, =
EBr21"/21 il
Here each subspace r2kxv2k is SO(n)invariant because r2 and A commute with rotations.
We denote by £ the space of functions in H restricted to the sphere Sn1, and call these the surface spherical functions (or simply spherical functions). By (9.21) it follows that a spherical function Y E £ is an eigenfunction of the surface Laplacian (9.22)
(A* + v(v + n  2)) Y = 0,
and since the eigenvalues v(v+n2) are distinct for different v, the spaces £v are mutually orthogonal. By the StoneWeierstrass theorem the space of all spherical functions (direct sum)
V
is dense in the space C(Sr1) of all smooth functions on Sn1 in the uniform norm by compactness. Hence we have the orthogonal decomposition L2(Srn1) =
® £,,.
By (9.21) and (9.22) it follows that the whole spectrum of 0* in L2(S11) consists of numbers of the type
A=v(v+n2) and that the Aeigenspace is £,,. Since 0' is rotation invariant, each eigenspace £ is SO(n)invariant. We shall show that the eigenspace representations are irreducible. Suppose £, = F 0 9 is an orthogonal decomposition
9.4. Harmonic analysis on the sphere
159
Figure 10. The sphere
into nonzero SO(n)invariant subspaces. By the SO(n)invariance of E there are two functions f E F, g E G which take value 1 at some point, say s = (0, ... , 0, 1), f (s) = g(s) = 1. We may assume that f, g are SO(n  1)invariant by averaging over SO(n  1). Such functions f,g depend only and the Laplace equation (9.22) for such functions on the angle 0 = becomes the ordinary, second order differential equation 2
Cd02 +(n2)cot0d0
+v(v+n2)) f =0.
This has a unique solution normalized by f (s) = 1/(the other linearly independent solution is ruled out since it is singular at 0 = 0). Therefore f = g, and this shows that the eigenspaces E,, are irreducible with respect to rotations.
9.4. Harmonic analysis on the sphere The case of three dimensions is particularly illuminating. We shall give an
account of this theory in quite explicit form. Our aim is to construct a complete, orthonormal system of spherical functions on S2. To this end it is natural to work in the spherical polar coordinates (see Figure 10) xl = r sin 0 cos cp,
0 < r < oo,
x2 = r sin 0 sin gyp,
0<0<7r,
x3=rcos0,
0
9. Spherical Functions
160
In the new coordinates we have ds2 = dx2 + dx22 + dxz3 dr2 + r2d92 + r2 sin2 9d 2 dµ = dxldx2dx3 = r2 sin 9drd8dcp,
If = r2(sin9)1 (Q(r2sin9i) +
(sing
) + (sin©)1 'Q
)
.
On the unit sphere S2 these formulas give (take r = 1) ds2 = d92 + sin2 9dcp2,
d1 = sin 9dOdp,
and the LaplaceBeltrami operator on S2 is
A` _ (sin 9)1 (sin
g)+(sin 9)`2
2
cP
2
2
a2 + cot 9 a9 + (sin o) 2
22 .
= The group of rotations SO(3) leaves ds, dit and 0' invariant. Applying 0 to a spherical function of degree v of the type f (x) = r"Y(9, cp),
we obtain the differential equation for Y = Y(9, cp), (9.23)
(sin9)1
(sin1) + (sin
+ v(v + 1)Y = 0. cP
In other words, Y is an eigenfunction of A` with eigenvalue v(v + 1). C is an eigenfunction of i , then its eigenvalue must be v(v + 1) for some integer v >, 0, and the function f (x) = r"Y(9, cp) is a spherical function of degree v. Therefore the surface spherical functions Y(9, cp) of degree v are just the eigenfunctions of the surface Laplacian A* with eigenvalue A = v(v + 1). The eigenfunction equation A*Y = AY can be solved by the method of separation of variables. We search for solutions of type
Conversely, one can show that if Y : S2
Y(9, cp) = P(cos 9)E(cp).
Inserting this into (9.23), we obtain two independent second order differential equations: d2E dcp2
(sin g)1
9
E
(sinOP(cosO)) + (v(v + 1)  p(sin 9)2)P(cos 9) = 0
9.4. Harmonic analysis on the sphere
161
where ate is an unknown parameter. Since E(w) must be periodic of period
2ir, the first equation has such solutions only for µ = m2, where m is an integer; they are linear combinations of E(W) = eti,np
(9.24)
Putting this value of µ into the second equation, and changing the variable 0 (recall that 0 < 0 < ir) to z = cos 0
(9.25)
(note that 1 < z < 1), we get. (9.26)
((1z2)P')'+(v(v+1)
1mz2)P=0.
First let us consider m = 0; then Y(0, cp) = Y(O) = P(cos 0) does not depend on cp (such functions are called zonal spherical functions). The equation (9.26) simplifies to ((1  z2)P')' + v(v + 1)P = 0.
(9.27)
It is not difficult to verify that the Legendre polynomial defined by U
(2"v!)1 dv (z2
P"(z) =
(9.28)
dz"
 1)v
satisfies (9.27). These polynomials have the generating power series 00 (9.29)
E P"(z)r" = (1  2rz +
r2)1/2,
v=o
has degree v. We have
The Legendre polynomial
(1)k (2v  2k)! 2k 2kk! (v  k)!(v  2k)!z
P"(z)
=2"
()2(z_1)(z+1)"3. O;i
V
1 3 2 2. P((z)=1, P1(z)=z, P2(z)=2z
9. Spherical Functions
162
Moreover, the polynomials satisfy a handful of recurrence relations; for example, for all v 3 1 we have (9.30)
(v + 1)P.,+1(z)  (2v +
0,
and they also satisfy differential recurrence relations, such as zv1P,,(z)
(9.31)
= (z"PY1(z))'
We already know that the P, (z) are orthogonal on the segment 1 <, z 5 1 because they come from spherical functions of different degree. However one can derive directly that (9.32)
r1 r 1
1
1
P,(z)P,(z)dz = (v +
Indeed, by (9.27) we have ((z2
(v  A) (V + p +
 1)(P'P1 
which gives (9.32) if v # µ by integration along 1 5 z 5 1. For v = p we use the recurrence relation (9.30) to get
v > 2, whence integrating and using orthogonality we get the recurrence
relation (2v + 1)
/ 1
I l
P,2(z)dz = (2v 
r
1) /
1
Pv_1(z)dz,
1
which proves (9.32) by induction (verify it directly for v = 1). By (9.32) the normalized functions (v+ form an orthonormal system in G2([1,1]) which is complete by the Weierstrass theorem.
More precisely, if f E G2((1, 1)) is piecewise smooth, then the spectral series 00
(9.33)
f (z) = E(v + 2)(f, P,.)P,.(z) V=o
converges to f (z) at every continuity point of f (z) and to 2 1f(z+0)+f(z0)] if z is a discontinuity. Note that the orthonormal system is not uniformly bounded with respect to the spectrum. Indeed, at z = 1 we have 1, whence
(v+2)ZP(1)=(4A)'
9.4. Harmonic analysis on the sphere
163
This property may have something to do with the fact that S2 is a Riemannian surface of positive curvature (equal to 1). The spectral expansion (9.33) can be written as the integral equation
f (z) = with kernel
f
K(z, w) f (w)dw
00
K(z, w) = >2(v + =o
Using the recurrence relation (9.31), one can compute the partial kernel K,, (z, w) which is obtained by restricting the sum to v < n. We obtain Kn(z, w) = 2 (Pn(z)P,,1(w)  Pn1(z)Pn(w)](z 
w)1
By orthogonality f1
Kn(z, w)dw = 1. Hence we derive an integral formula for the tail of the spectral series (9.33), namely
(v+ 2) (f,., P,.)P,.(z)
f (z) = v
///
+2
(Pn(z)Pn1 (w)  Pn1(z)Pn(w)]F(z, w)dw 11
where (z)  f (w)
zw This expression is very useful for establishing effective spectral approximations.
As an example of spectral series (9.33) we expand the characteristic function of a segment
f(z)=
if 1
1I
We get
f (") = 1+ a+ E[PY+1(a)  P v>O
i
9. Spherical Functions
164
Now we return to the equation (9.26) for any integer in. We consider in >, 0 only because the equation is the same for m. One can check that the socalled associated Legendre functions defined by (9.34)
z2),n/2d P ,,(Z)
P" '(Z) = (1 
satisfy (9.26). Note that P,m(z)  0 if m > v, because degP,, = v. For 0 5 m 5 v the associated Legendre functions are linearly independent. There is a simple integral representation ( 9 . 35)
P (cos 9 ) _
(Tri
i v)! i r
f
(cos 6 + i sin B cos
We conclude the above investigatiuus with the following (cf. [CHI, [Bocj and [Ter]). Theorem 9.3. The spectrum of the spherical Laplacian 0' consists of numbers of the type
A = v(v + 1),
(9.36)
v = 0,1, 2,... .
The Aeigenspace has dimension 2v + 1, and the functions y (O, W) = C,,,"', P,""`! (cos 9)eimw
(9.37)
with Im[ 5 v, where C",t=(2v+1(vm)!)
(9.38)
(v + m)!
4Tr
form a complete, orthonormal system of the Aeigenspace. All the functions (9.37) with Iml 5 v form a complete, orthonormal system in ,C2(S2, dµ) with respect to the measure dµ = sin 9dOdcp (a rotationally invariant measure). Every smooth function g : S2 > C has the spectral expansion
9(e, cp) _ EE
(9.39)
cP)
V>'ImI
where (9.40)
9;;'
n
27
91'v"dA 
fo
fo
9(e,
The series converges absolutely and uniformly.
W)Ym (9,
cp) sin 9d©dcp.
Chapter 10
Theta Functions 10.1. Introduction Given a symmetric, positive definite matrix A and a spherical function P with respect to A, we define the theta function (10.1)
P(m)e (A[mJz).
O(z)
Clearly 6(z) is holomorphic in the upper halfplane H because the series converges absolutely (its terms decay exponentially, because A[m] >> Jm12 and P(m) is a polynomial). One can generalize the theta function by introducing another complex variable, say v E C, as follows: (10.2)
P(m)e (A1m]z+B(xm)v)
O(z, v) _ fnEZ'
where x E C' and B(x,y) = txAy is the bilinear form associated with A; O(z, v) is called a Jacobi function. However we shall achieve our main goals by means of the special congruent theta functions (10.3)
O(z; h) =
P(nz)e
())
vn=_h (mod N)
for h E Z", where N is a positive integer. Note that O(z; 0) = N"O(z), where v = deg P. Throughout we assume that A is an integral matrix, i.e. (10.4)
A=(aij)with a;j =a1,EZ, 1<, i,j<, r, 165
10. Theta llinctions
166
and the same holds true for the matrix A' = NA1, but we do not require N to be the least positive integer with this property. Clearly N = IAI satisfies this condition; however a smaller number suffices quite often. Note that IAj]A'! = N'; hence we always have
N' M 0 (mod JAI)
(10.5)
so every prime factor of JAI is in N. Observe by symmetry that the quadratic form associated with A has even coefficients save for the diagonal, (10.6)
A[x] = txAx = 2 E aijxixj + > aiix?. i<.j
i
We shall say that A[x] is even if all its coefficients are even, i.e. if (10.7)
diag A  0 (mod 2).
However this assertion will not be needed for many results. Our objective is to show that A(z; h) and 6(z) are automorphic forms of weight
k=v+2
(10.8)
with respect to a congruence group of a certain level and for a multiplier system which depends on A and N (but not on P at all!). The theta functions are a vast source of automorphic forms. There are immediate applications to the theory of representations by quadratic forms and less immediate ones to analytic studies of Lfunctions for number fields. We shall present some of these topics in the subsequent chapters.
When the quadratic form A[x] has an odd number of variables, the multiplier system for the associated theta function is quite sophisticated because the weight k is not an integer. Needless to say, in this case one must be careful with multiplying the square roots of various complex numbers.
Let us recall that z9 is defined for any z E C` and S E C in Section 2.6. Observe the following rules:
zn+v =
(z1/2)s = zs/2, Z' = (1/z)s unless z < 0, (zlz2)m=z2 z2 , ifmE7L,
zuzv,
(cz)s = cszs, ws(z/w)s = zs
if c > 0, if z,w E H.
10.2. An inversion formula The automorphy of various theta functions will be derived from the following basic formula (a generalized Jacobi inversion).
10.2. An inversion formula
167
Proposition 10.1. Let A be a symmetric, positive definite matrix (not necessarily integral) and let P be a spherical function with respect to A of degree v. For z E H and z E Cr we have (10.9)
(i)k
EP(m+x)e (2A[m + x]z } =
Z
EP (m)e l \\
IAI
In particular (for x = 0)
A 1
m + rrtxJ
) ()k
(10.10)
9(z; A, P) =
0(z1; A1, P')
i"IAI1/2
Z
where P*(x) = P(A1x) is a spherical function for the matrix A'1. The formula (10.9) constitutes a far reaching generalization of (1.9), and its proof is based on the Poisson summation f (m)e(tmx).
f (m + x) = m
m
We also use the familiar Fourier integral (for u E R and z E 1111) (10.11)
J e(2y2zyu)dy= (z)112P(2z2)
Proof. We apply Poisson's formula to the function f (x) = e
(A[xJz).
To compute the Fourier transform
J(v) =
J
f (x)e (txv) dx
we change the variable x to y = Bx by the linear map B which diagonalizes
A as in Section 9.1. We have tBB = A, A[x] = tyy, v = tBU, txv = tyu, dx = IBIldy, IB12 = j (v)
IAI.
Hence
= IBI1 f e (1 tyyz  tyu dy , =IBI1TT f
=IBI
1(z)2e(_tuu)
10. Theta. Functions
168
by (10.11). Now we return to the original variables. Using the transformation tuu = t(tBiv)tBlv = = ttvAlv =A'Jv], we get tvPltBiv
f (v) = JAI "2
(Dr!2 e
(i1).
Hence Poisson's formula becomes
(10.12) >e (A1m+x]z)
= CAI
1/2 (zl
r/2
>2e in
in
CA2z1[m ]
+tmx 1\
This is a special case of (10.9) for P = 1. We shall derive (10.9) for any P by differentiating (10.12) with respect to x by means of a suitable operator. Without loss of generality we may assume (see Theorem 9.1) that P(x) = ('cAx)", where c is an isotropic vector if v >' 2 and c is arbitrary if v = 0, 1. Given this vector, one defines the linear differential operator
L=>cjO/ax;. We have LA[x] = 2tcAx and L2A[x] = 2A[c] = 0 if v > 2. Hence applying L" to (10.12) we obtain (divide throughout by (21riz)" and observe that z"(Z/z)r/2 = 2"(i/z)k)
>P(m+x)e C 1 A[m + x]z
(t
(!)
to
in
)"e
+ 2z
Here we insert (tcwn)" = P(Alm), completing the proof of (10.9).
Remark. If JAI = 1, then 9(z; A', P*) = 9(z; A, P) by changing m to Am in the summation. Therefore in this case (10.10) becomes (10.13)
9(1/z) = i"(iz)k9(z).
10.3. The congruent theta functions Let A, P, h, N be as in the first section. Note that (10.14)
9(z; h) = (1)"9(z; h).
Moreover, observe that 9(z; h) depends only on h (mod N). We shall consider the theta functions 9(z; h) exclusively for classes in f = {h (mod N) : Ah  0 (mod N)}. Note that grad A[x] = 2Ax  0 (mod 2N) if x E R.
Lemma 10.2. The set 11 is a finite abelian group (additive) of order (10.15)
mWi = CAI.
.10.3. The congruent theta functions
169
Proof. Think of A as a linear map A : RI _4 R'' which transforms the unit box 5 = [0, 1]' to the fundamental parallelogram P = A!3 of volume IAI. Since A is integral, P has integral vertices, so the number of integral vectors in P is equal to the volume; IP n Z'* I = IPI = IAI. Also the matrix N`IA
maps the box NB = [0, N]' to P, and the integral vectors h in NB with Ah  0 (mod N) correspond to the integral vectors in P. This completes the proof of (10.15). That 7l is a group is obvious.
For a diagonal matrix A = diag [a,,... , a,] the group f is the direct sum of Z/ajZ,1 j S r. Since the bilinear form B(x, y) = txAy is nondegenerate and 1 is isomorphic to its dual, it follows that every character on fl is given by (10.16)
O(i(h, l) = e(N2B(h,1)),
h E fi,
for some I E 1l. Note the symmetry 0/i (h, l) = zy(l, h). We have the orthogonality formula

(10.17)
if h = 0 (mod N)
( IAI
otherwise.
0
ILE%
Proposition 10.3. Let h E W. We have (10.18)
6(z + 2; h) = e
(1) 6(z; h),
and if diag A is even, then (10.19)
6(z + 1; h) = e
( 2N2 ) O(z; h).
Proof. Since A is symmetric we have (10.20)
A[x + y] = A[x] 4 21 xAy + A[y].
Hence, writing m = h + nN, we get A[m] = A[h] + 2N`nAh + N2A[n] = A[h]
(mod N2).
Moreover if diag A is even, then the above congruence holds true to the modulus 2N2 because A[n] is even (sec (10.6)). These congruencies lead to (10.18) and (10.19) immediately.
Next we shall establish a transformation formula for 6(z; Ii) with respect to the involution z > 1/z.
10. Theta Functions
170
Proposition 10.4. For any It E N we have
9(1/z; h) =
(10.21)
E?P(h,
l)6(z; 1).
IE1(
Proof. We apply (10.9) for x = hN1. The left side is just N"9(z; h). On the righthand side we change m to n = NA1m. The condition m E Zr is equivalent to n E Zr and An  0 (mod N). Moreover, under this change
we have P(Alm) = N"P(n), A1[m] = N2A[n] and tmx = N2tnAh. Therefore (10.9) becomes (multiply both sides by N" and change z  1/z)
9(1/z; h) = it l AI.1/2(iz)k
P(n)e
(I!!iz
A)
Ano (mod N)
We shall generalize (10.21) for some transformations T = (:) in SL2(Z) which depend on A and N. We assume that d# 0, because otherwise T = ± (1 1) and we are done. Changing r to T, if required, we may also assume that (10.22)
d > 0.
Next let us assume that one of the following two conditions holds:
b  c  0 (mod 2) diag A  0 (mod 2).
(10.23) (10.24)
We begin with the transformation
(a
1) = (d d) (1
ac).
Since dyz = b  (dz  c)1, we have
P(m)e
9(yz; h) = mh (mod N)
A[m] 2N2
b
1
(d  d(dz c) )
Note that e(bA[m]/2dN2) depends only on m (mod dN) (see (10.20), the condition h E 71 and (10.23)(10.24)). Therefore we can split the summation into residue classes to modulus dN, getting IA[g]
9(yz; h)
P(m)e
N2
g (mod dN) .9=h (mod N)
m=_g (mod N)
dA[m]
1
2(dN)2 dz  c
10.3. The congruent theta functions
171
Here the innermost sum is the theta function associated with the matrix dA and the residue class g (mod dN) evaluated at the point (dz  c)1. Since dAg  0 (mod dN), Proposition 10.4 is applicable to this sum, giving di )2
('IAg) P(m) J ml (mod dN)
e
I (mod dN)
e (j(dz  c)l
AI==O (mod N)
Note that I dAI = dr IAi and cA[m] _ cA[l] (mod 2dN2). Hence we deduce
that e(tix,
h) = (i(c  dz))2 2d
I AI
p(h,1) I (mod dN)
m)e (A[ m=I (mod dN)
\ .2N2]
z1 ///
AIO (mod N)
where
E
e ((bA[g] + 2tlAg  cA[l])/2dN2) .
9 (mod dN) g=h (mod N)
We shift g to g+cl so the new variable ranges over classes modulo dN which
are = h  cl (mod N), and in the exponential we get (bA[g] + 2adtlAg + acdA[g])/2dN2
by applying ad  be = 1. In the middle term 2adtlAg we can replace g by its class h  cl (mod N), getting cp(h,1) = e
(10.25)
=e
(2atlAh_acAEl)) acA[1]
2N2
cp(h  cl,0)
(ah,1)
This expression shows that cp(h, l) depends only on I (mod N), and we get
()(7z; h) = i'd /2IAI1/2(i(c  dz)) E W(h, h')e(z; h'). h'EN
Now we come to rz by changing z to 1/z (note that ry(1/z) = rz). On the righthand side we apply Proposition 10.4 (again!) to each e(1/z; h') and obtain (10.26)
e(rz; h) = i2'd
''12IAI1(cz
+ d)' E (D(h,1)e(z;1) tEN
10. Theta. Functions
172
where cp(h, h')V)(h',1).
41(h, 1) =
(10.27)
11,01
The automorphy factor in (10.26) came from the equality
(!(cz+d))k (z) k _ (cz+d)k. Check this multiplication rule separately for c > 0, c = 0, c < 0. To proceed further we need one of the following Lwo condiLions: (10.28)
c
(10.29)
c
0 (mod 2N) 0 (mod N) and diag NA1_ 0 (mod 2).
Observe that the hypothesis diag NA1  0 (mod 2) implies A[l] __ 0 (mod 2N) for any l with Al  0 (mod N). Indeed we have m = NIAl E Z'
and NA1[m]  0 (mod 2), whence A[l] = NIA1[m)  0 (mod 2N). Therefore under either of the above conditions the formula (10.25) simplifies to
cp(h, l) = /i(ah, l)cp(h, 0).
Hence by the orthogonality of characters (see (10.17)) we infer that (D(h,1) = tp(h, 0) E O(ah, h')?,l'(h', l) = cp(h, 0) IAA h'EN
if l  ah (mod N), and it vanishes otherwise. Hence there is only one terns on the right side of (10.26), and taking (10.14) into account we arrive
at (10.30)
e(Tz; h) = cp(h, 0)d^''12(cz + d)k®(z; ah).
It remains to compute cp(h, 0)

E g (mo d dN )
e ( 2A[9, )
2dN2 l
g,h (niod N)
Since ad = 1 + be  1 (mod N), we can write g = adh + xN with x ranging freely modulo d. We get cp(h, 0) = e(a2bdA[h]/2N2)G, where (10.31)
G = E e(bA[x]/2d) x (mod d)
10.3. The congruent theta functions
173
is a Gauss sum associated with the quadratic form In the exponential factor of cp(h, 0) we can replace ad by 1, because either ad =_ 1 2A[x].
(mod 2N), Alit] = 0 (mod N) in case of (10.28), or ad = 1 (mod N), Alit]  0 (mod 2N) in case of (10.29). Inserting this, we get
cp(h,0)=e(a2 2] ) G.
(10.32)
Before computing G we make a convenient transformation. We assume
that
d  I (mod 2).
(10.33)
Then (d, 2cIAI)  1, and changing x to 2cx cnodulo d we get (apply be = I (mod d)) .(2rA[Tl /d),
G=
(10.34)
x (rnod d)
Lemma 10.5. Let d be a positive integer with (d, 2clAI) = I. Then G  (LA
(10.35)
dl
(Zd
(2c)
y
d
Proof. The onedimensional case was considered in Lemma 4.8. We have z
e (d) _ \dl eddl/z
(10.36) x (rnod d)
if d > 0 and (d, 2c) = 1, where
_1)1/2 (10.37)
e
`t = {
d= i
I
1 (mod 4)
if d if d
1 (mod 4).
We shall exploit this result to evaluate G by a local diagonalization. First recall that a nonsingular symmetric matrix over a field of characteristic # 2 is equivalent to a diagonal matrix. Hence, applying a padic approximation
argument to A, one shows that there exist an integral matrix V and an integral diagonal matrix M = diag (ml, ... , m,.) such that
tVAV  M (mod d) with (d, I V I) = 1. Changing x to V y modulo d, we get A[x]  M[y] _ nt j yj (mod d) and
G=
ry,
e(2cnijyY /d) (rood d)
=
lnnl ... m,.l
\
J 1\
,
(2c) /I
by (10.36). Moreover we have nil ...nt,. = IMI ItVAVI = IAIIV'I2 (mod d), which leads to (10.35). Now we combine the conditions and collect our results into the following:
10. Theta Functions
174
Proposition 10.6. Let r = (c b) E SL2(Z) with d = 1 (mod 2). Suppose that any one of the following three conditions holds true:
0 (mod 2N), b  0 (mod 2). 0 (mod 2N), diag A = 0 (mod 2). 0 (mod N), diag A diag NA1  0 (mod 2). Then for any It E N we have
(i) c (ii) c (iii) c
(10.38)
(,r z; h) = e
(abA[h])'0(r)
(cz + d)'A(z; ah)
where
(10.39)
'0(r)=
(A) (Ed
d(v))".
Remark. Observe that for c = 0 the result was already established in Proposition 10.3. The original assumption d > 0 was relaxed in Proposition
10.6, since one can change r to T. This change does not alter the righthand side of (10.38) if one observes the convention (2.75), which extends the LegendreJacobi symbol (d) for 0 < d = 1 (mod 2) to all d = I (mod 2) by the rules (10.40)
\d)
Fcl
( d)
and p (10.41)
(d)
= {
if c
1
ifd=fl
0
otherwise.
0
Corollary 10.7. Let h E W. The congruent theta function 9(z; h) is an automorphic form for the principal congruence group I(4N) of weight k _ v + r/2 and the multiplier (10.42)
i. e. for r = (')
'0(rr) = Mr' (1 1) (mod 4N)
we have
d)'®(z; h). e(rz; h) _'0(r)(cz + Moreover, if the spherical function P is not constant, i.e. v > 0, then 6(z; h) is a cusp form.
(10.43)
Proof. The multiplier (10.42) is a simplified expression (10.39), since d  1
(mod 4N) and (A1) = 1 by (10.5). Then (10.43) is a simplification of (10.38). That ®(z; h) is holomorphic and vanishes at cusps if v > 0 can be seen from Propositions 10.3 and 10.4 (as in the proof of Theorem 10.8).
10.4. The autonlorphy of theta functions
175
10.4. The automorphy of theta functions For h = 0 we obtain O(z; 0) = N"O(z). Observe that v must be even; otherwise the theta function O(z) vanishes identically. The transformation formula (10.38) simplifies to (10.44)
O(rz) = 19(r)(cz + d)kO(z)
with 19(T) given by (10.39) for any T satisfying any one of the alternative conditions of Proposition 10.6. In particular, from the second group of these conditions we infer
Theorem 10.8. Let N be such that NA1 is integral, and let diag A be even. Then O(z) is an automorphic form for ro(2N) of weight k = v + r/2 and multiplier 19 given by (10.39), i.e. (10.44) holds true for any T E
r0(2N). If v > 0, then O(z) is a cusp form. Proof. We only need to show that for every cusp a of ro(2N) the function is holomorphic at infinity and vanishes as Im z + OIQa (z) = jpa 0o if v > 0. It is simpler to show these properties for all the congruence theta functions O(z; h) with h E. W. The point is that the whole linear space generated by these functions is preserved under the translation T = (1 1)
\1
by Proposition 10.3 and under the involution S = (1 1) by Proposition 10.4. Since T, S generate the modular group SL2(Z) and every cusp of ro(2N) is equivalent to oo with respect to SL2(Z), it suffices to show the holomorphy of each O(z; h) at oo. However this fact is obvious by the very definition of O(z; h). Moreover O(z; h) decays exponentially at oo if v > 0, as does O(z) at every cusp.
If r is even, we can do slightly better with respect to the level. Since e2 = ( 1), the multiplier system (10.39) reduces to the fixed real character (D\)
(10.45)
19(r) = XD(T) =
where D =
1),,12
JAI.
The weight is an integer k = v+r/2 r//2 (mod 2). Observe the consistency condition 19(1) = (1)k. We infer
Theorem 10.9. Let A be a symmetric, positive definite integral matrix of even rank r. Let N be a positive integer such that NA1 is also an integral matrix. Suppose both A and NA1 have even diagonal entries. Let P be a spherical function with respect to A of even degree v. Then the associated theta function O(z) is a modular form for ro(N) of weight k = v + r/2 and a multiplier z9(T) given by the character (10.45). Moreover O(z) is a cusp
form ifv>0.
10. Theta Functions
176
Proof. The automorphy of O(z) with respect to the group I'o(N) is already established by virtue of Proposition 10.6 (the third alternative) except for
transformations r = { a ) E i'o (N) with d even. If such a Tr exists, then 2 { c, which implies 2 [ N and 2 t A. In this case one can show that
D = (1)r/2 JAI  1
(10.46)
(mod 4)
(exploit that r and diag A are even). Hence by the quadratic reciprocity law XD(d) is a character to modulus D' = the squarefree kernel of D, so XD(d) is periodic in d (mod N) for D*IN by (10.15). Now applying the automorphy of O(z) for T,
(a d) (1 1)(c c4d)
at z' = z  1, we verify the automorphy for T at z (we have rz = r'z' and XD(c + d) = XD(d)). Finally, the holomorphy and vanishing at cusps are verified as in the proof of Theorem 10.8.
One should ask if the conditions of Theorem 10.9 can be satisfied with
N = 1, so that the theta function O(z) is a modular form for SL2(Z). In that case we must have i"(i)k = 1 by virtue of (10.13), which implies that A has rank r  0 (mod 8). That such matrices with diag A = diag A' = 0 (mod 2) exist was shown by H. Minkowski. As an example for r = 8 we have the Jacobi type matrix A (i.e. a tridiagonal matrix) with
diag A = 12,2,4,4,20,12,4,2) diag A = diag A = 11, 1, 3,5,3, 1, 1].
10.5. The standard theta function The simplest example of a theta function is given by the series 8(z) = 1: e(n2z)
(10.47)
nEz
which was first studied thoroughly by Jacobi. Besides automorphic transformations, Jacobi established the infinite product representation 8(z) _ fl(1  e(nz))(1  e((n + 1/2)z))2. I
10.5. The standard theta function
177
Therefore 0(z) does not vanish in H. There is a close connection between 0(z) and the Dedekind e
(24)
e(nz)),
namely 0(2z  1) = 772(x)/77(2x).
Both are automorphic forms of weight 1/2, but they have slightly different multiplier systems (they coincide on the group I'(24)). In Section 2.8 we have described the multiplier systems in terms of the Dedekind sums and the Jacobi symbol. Now we can deduce the automorphy transformation (2.73) from (10.38). Actually the method works for theta functions twisted by Dirichlet characters. Denote (10.49)
0x(z) =
>00 X(n)n"c(nlz) 00
where v = 0, 1 (there are no spherical functions in one variable of degree v > 1) and X is a character to modulus N such that (10.50)
otherwise 0.,(z) vanishes identically.
Theorem 10.10. For any r = (a d) E I'o(4N2) we have (10.51)
0,((rz) = X(d)Ed (a) (cz + d)"+1/20, (z).
Moreover, if X is a primitive character of conductor N, then (10.52)
0,(1/2Nz) =
i"E(X)(iz)v11/20, (z/2N)
where E(X) is the normalized Gauss sung (10.53)
E(X) =
N_112
X(h)e
(N)
h (niod N)
Remark. Observe that the involution (1 1) switches X to,i. If
is real,
then f(z) = 0.(2Nz) is an eigenfunction of the slash operator I (i "'1 ).
10. Theta. Functions
178
Proof. We split into residue classes n = It (mod N) to obtain
0.(z) = E X(h)e(2Nz;h)
(10.54)
h (mod N)
where 6(z; h) is the congruent theta function associated with the matrix
A = N and the spherical function P(x) = x". For r = (a d) we have 2Nr(z) = r'(2Nz), where r' _
(ca
2Nb).
If r E Fo(4N2), then r' satisfies the first alternative of Proposition 10.6, so (10.38) is applicable, giving 2N
e(2Nrz; h)  ©(r'(2Nz); h)  cd ( ) (cz + d)"+1/2e(2Nz; ah). Multiply this relation by X(h) and sum over It (mod N), getting (10.51) after changing h to A. To prove (10.53) we write (10.54) for 1/2Nz and apply (10.21) for each
h (mod N), getting
9Y(1/2Nz) = i"N1/2(iz)k
X(h)e ($) 6(z; l). h,1 (mod N)
Here the sum over It (mod N) is a Gauss sum which for primitive characters is equal to X(l)e(X)N1/2. Inserting this, we arrive at (10.52) by an appeal to (10.54) again. For the trivial character we have 9x(z) = 9(z), and Theorem 10.10 yields 2.73. Moreover, (10.52) becomes (10.55)
9(1/2z) = (iz)1/29(x/2)
which agrees with (1.9) for x = 0 and 29 = iz. For the real character x(n) _ (n), which is even and primitive of conductor N = 12, we have the following formula, due to Hecke: (10.56)
28. (z)
"0
n
e(n2z) = i(24z).
In this case v = 0, E(X) = 1 and (10.52) together with (10.56) yield (10.57)
1)(1/z) = (iz)1/271(z).
Chapter 11
Representations by Quadratic Forms 11.1. Introduction Let Q(x) be a positive definite quadratic form in r >, 2 variables. In Siegel's notation Q(x) = 2A(x], where A = (ai?) is a symmetric, positive definite matrix of rank r. Hence
Q(x) =
ai,jxixj +
aiixi
i<.j
Suppose A has integral entries which are even on the diagonal, i.e.
aii  0 (mod 2). Then Q(x) has integral coefficients. A central problem in the arithmetic theory of quadratic forms is solving the equation (11.1)
Q(m) = n
with m E Z' for a given positive integer n. In two variables this was studied by Fermat, Lagrange and Gauss. The theory of binary quadratic forms is special; it corresponds to the arithmetic of ideals in an imaginary quadratic number field. General cases began to be considered systematically in the late nineteenth century, and the theory matured in the hands of Minkowski, Hasse, Siegel, to mention a few original contributors. Minkowski created the geometry of numbers to answer the question of equivalence of quadratic forms. His methods of reduction of forms yield practical algorithms for the 179
11. i epresentatious by Quadratic Foals
180
determination of all solutions. Hasse's work is oriented towards the algebraic aspects. In particular, his localglobal principle produces intrinsic conditions for the existence of solutions to (11.1). Siegel developed analytic methods (first brought to the theory by Dirichlet) to find, among many other things, a formula for the number of representations. He is also credited with the extensive use and development of automorphic forms in this connection, such
as theta functions (cf. [Sie]). Other analytic tools, in particular the circle method of HardyRamanujan, considerably developed by Kloosterman, turned out to be alternatives to the theta function methods of Siegel. After the full number of representations is well estimated one next asks a deeper question about the equidistribution of representations. Most recent ideas for addressing such problems came from ergodic theory (due to Linnik) and from exponential sums methods. In this chapter we shall present in detail a few result, of analytic type. There are numerous surveys of the subject written by A. Malyshev, who was a champion in the classical theory of quadratic forms; cf. [Mal].
11.2. Siegel's mass formula Here we give a brief review of fundamental concepts and results in the theory of positive definite, integral quadratic forms. Occasionally we abuse notation by referring to the matrix representing a form rather than to the form itself.
The determinant of A is also called the determinant of Q(x) = 2A[x]; det Q = det A = JAS > 0. Not every positive integer is a determinant, since
JAI _ (1),/2 (mod 4) if r is even and IAA  0 (mod 2) if r is odd. We define the discriminant D = D(Q) E Z by D = (1)r/2 JAI =_ 0, 1 (mod 4)
if r is even
D = 2(1) 2 IAI
if r is odd.
Two quadratic forms Q1, Q2 (it is assumed that all forms under consideration have the same rank r >, 2) are equivalent if one can be obtained from the other by an invertible linear change of variables, i.e. if (11.2)
Al ='UA2U for some U E MM(Z).
Since (11.2) also holds for A1, A2 interchanged (with possibly different U in M,.(Z)), it is clear that both A1, A2 have the same determinant, and so JUG = ±1; in other words, U is a unimodular transformation. The equivalent forms form a class. The determinant is a class invariant. In general, for a given determinant the number of classes can be infinite, except for binary forms.
11.2. Siegel's mass formula
181
A quadratic form is equivalent to itself in a number of ways. Denote by (11.3)
O(A) = {U E Mr(Z) : LUAU = Al
the group of automorphs of A (this is also called the isotropy group of A with respect to the action of Mr(Z) on A given above). The group O(A) is always finite and is the same for equivalent forms; denote its order by
o(A) = #O(A).
(11.4)
For binary forms, o(A) depends only on the determinant IAl, but if r > 2 then o(A) varies from class to class. Obviously the representation number (11.5)
r(n,Q)_#{mEZ' :Q(m)=n}
depends only on the equivalence class of Q. Finding an exact and relatively simple formula for r(n, Q) is one objective; this problem has been a driving force throughout the history of quadratic forms. Only in very special cases does a satisfactory formula exist. For instance, if Q(x) = xi + x2 (determinant 4) we have
r(n, Q) = 4 > X(d)
(11.6)
din
where
(d) 
(4)
(d) =
1
if d = 1 (mod 4)
1
if d
1 (mod 4) is the character of the field Q(i) (this was known in a different setting by Fermat). Dirichlet generalized (11.6) to primitive quadratic forms of any determinant (11.7)
Q(x) = axi + bxl x2 + cx2,
(a, b, c) = 1.
Put D =  det Q = b2  4ac < 0 (the discriminant of Q), and (11.8)
XD(d) = (s).
Then for n > 0, (n., D) = 1 the character sum (11.9)
r(n.; D) = WD E \D(d) din
11. Representations by Quadratic Forms
182
gives the number of all representations of n by representatives of forms of all classes of discriminant D. Here WD stands for the number of automorphs:
WD=
6 ifD=3 4 ifD=4 2 ifD<4.
The Dirichlet formula solves the problem for discriminants D for which the class number h(D) is one. However there are only finitely many D with h(D) = 1, as can be seen from another formula of Dirichlet: h(D) = WD Since
DI._e I
2
L(1, XD)
< L(1, XD) « ID1e (the lower bound is ineffective, due to
Siegel), it follows that (11.12)
IDI2f << h(D) << IDI2 e.
Actually the Dirichlet formula (11.9) requires fewer classes than h(D), since certain forms do not represent n at all. If n > 0, (n, D) = 1, is represented by a primitive form (11.7) (assume without loss of generality that (a, 2D) = 1), then n must comply with various congruence conditions, for example Xp(n) = Xp(a)
for every pID, p odd.
Therefore the admissible classes must have the same values of Xp(a) for all pID, p # 2. For p = 2 one needs slightly different characters, namely By means of these characters Gauss was able X4 (n) = () n or x8 (n) _ (2). 'n
to classify equivalent classes into genera. For forms of arbitrary rank r >, 2 the Gauss theory of genera was beautifully generalized by Hasse. In modern fashion a genus of forms is created by padic equivalence or rational equivalence. We have several equivalent definitions. Two forms Qt, Q2 are in the same genus if they are equivalent over all padic rings Zp and over the real numbers (the latter assertion is redundant since both Ql, Q2 are positive definite). Actually it suffices to establish the equivalence over Zp for all p dividing 2IAdIIA2I Another characterization asserts that Qt, Q2 are in the same genus if they are rationally equivalent by matrices U having denominators coprime with any prescribed integer q; actually it suffices for q = 2IA1IIA2I. A third, quite appealing characterization goes as follows: two forms Qt, Q2 of the same determinants IAl I = IA21 are in the same germs if they are equivalent over Z to forms congruent modulo 81A1IIA2I.
11.2. Siegel's mass formula
183
The determinant is a genus invariant. The number of classes in a genus of a fixed determinant is finite. This number can be larger than one. As an example take Q1(xl,x2) = 5xi + 11x2 and Q2(xl,x2) = xi +55x2; both forms are in the same genus but not in the same class (the second form does not represent n = 5). Except when r = 2, 3, the genus is no longer characterized by the residue classes of a number n being represented. We return for a moment to the primitive binary quadratic forms. Suppose D is a fundamental discriminant. The equivalence classes form a finite abelian group (the composition of classes corresponds to the multiplication of ideals in the ring of integers of Q(./D)): its identity element is the class of the form
X  4x2
if D = 0 (mod 4),
X1+X1X2D41x2 if D = 1 (mod 4). The genus containing the identity class is called the principal genus. It was discovered by Gauss that the principal genus consists of squares of classes. The number of genera is equal to 2t1, where t is the number of distinct prime factors of D. Hence the number g(D) of classes in every genus satisfies (11.13)
2"(D)lg(D) = h(D).
Since 2W(D) << DIE, the estimate (11.12) for the class number h(D) remains valid for g(D). Therefore if IDI is large, every genus contains several classes, and we have strict inequality for the representation numbers
r(n, Q) < r(n, gen Q) = r(n; D). For certain n's one may establish estimates in the opposite direction. Indeed, using the structure of class group as described by Gauss, one may show that if for any class there exists a divisor of n represented by a form of that class, then n is represented by any class of forms in its genus (we tacitly assume it is represented by some form of discriminant D). In fact for such n's we have
r(n, Q) >> r(n, gen Q). Moreover, as the number of divisors of it in each class grows, then
r'(n, Q) ^' g(D)lr(n, gen Q) For such "good" n's the Dirichlet formula (11.9) at least solves the problem in the asymptotic sense for r(n, Q).
11. Representations by Qtmdratic Forms
184
C.L. Siegel found a wonderful generalization of the Dirichlet formulas (11.9), (11.11) for forms of any rank r >, 2. Since the number of automorphs varies over classes, even within a given genus, it is necessary to weight the representation numbers r(n, Q) with appropriate coefficients w(Q) = o(Q)'m(gen Q)1
m(gen Q) = E o(Qj)' Q, Egen Q
is called the genus mass. Observe that the total mass is
E w(Q;) = 1.
(11.16)
Q:Egen Q
Accordingly the averaged number of representations of n by forms of a given genus is defined as the weighted sum (11.17)
r(n, gen Q) _ E w(Q:)r(n, Q:) QiEgen Q
Siegel established the following formula: (11.18)
r(n, gen Q) = e,. fJ bp(n, Q) 6,. (n, Q) p
where e,. = 1/2 or 1, depending on whether to r = 2 or r > 2, and bp(n, Q) represents the density of solutions Q(x) = n in 7Lp. To define bp(n, Q) precisely, consider Q(x) as a map from 7Lp to 7Lp and take the limit
bp(n, Q) = lim
U+n
vol Q' (U) vol U
where U ranges over padic neighborhoods of n and the volumes are taken with respect to the Haar measures on 7L7, and Z. respectively. One can show
that the above ratio stabilizes when U falls into a sufficiently small ball centered at n. In other words, we have (11.19)
bp(n, Q) =
p^'1'1)#{m
E zr : Q(m) __ n
(mod pn)}
for sufficiently large a. More precisely, (11.19) holds for any a > 2 ordp8lAI.
For p = oo the real density bo,, (n, Q) is given by integrals with respect to Lebesgue measure on R' and R, and it is (11.20)
6,,(n, Q) = (2ir)1r (2) 1
n21IAI1/2
11.3. Representations by Eiseustein series and cusp forms
1.85
(recognize here the volume of the unit ball). Clearly the local densities bp(n, Q) are genus invariants. The product (11.18) converges absolutely if r > 4. For r = 2,3 the convergence is a subtle matter. We shall arrive at the Siegel product (11.18) for r > 4 from another direction when analyzing the singular series, which arises naturally in the circle method. But we do not prove the exact formula (11.18) in this way, only sharp approximations, however for each individual r(n,Q2)(see Theorem 11.2). The padic local densities bp(n, Q) will be expressed by Gauss sums.
11.3. Representations by Eisenstein series and cusp forms The representation numbers r(n,Q) are the Fmirier eoefficients of the theta function {11.21)
e(Q(m)z) _ >2 r(n, Q)e(nz).
®(z, Q) _
n=O
mEZr
On the other hand, we know by Theorem 10.8 that e(z, Q) is an automorphic
form for ro(2N) of weight k = r/2 and multiplier (11.22)
(LA) (Fd
(2c))"
(a
b
E ro(2N).
Here N is a positive integer such that NA1 is an integral matrix and might not be the minimal level. The space Mk(ro(2N), t9) of all such automorphic forms has finite dimension, and if we can write ®(z, Q) as a linear combination of some basic forms in Mk(ro(2N), t9), then by comparing the Fourier coefficients we obtain an exact formula for r(n, Q). However the question whether the resulting expression for r(n, Q) is appealing depends on our familiarity with these basic automorphic forms which make the theta function. In the next paragraph we expand 6(z, Q) into basic forms. From now on suppose r > 4, so k = r/2 > 2 and the space M,;(ro(2N), t9) is spanned by the Eisenstein series (one for each singular cusp with respect. to t9) and cusp forms. Hence (11.23)
e(z, Q) = E(z, Q) + F(z, Q)
where (11.24)
E(z, Q) _
(pn(Q)E.(z) n Singular
11. Representations by Quadratic Forms
186
is the unique linear combination of the Eisenstein series and F(z, Q) is a cusp form. To determine the coefficients cpa(Q), recall the Fourier expansions (3.15): 00
(11.25)
En1cb (z) = bab + > 77ab (n)e(nz)
where bb is the diagonal symbol. Hence V. (Q) is just the value of E(z, Q) at the cusp a; more precisely, (11.26)
coa(Q) _ li
nelna(z,Q)
The function E(z, Q) is called the Eisenstein series associated with the qua
dratic forn Q as well as with the theta function e(z, Q). How to find the values cpa(Q) in practice remains an open question. For the standard theta function e(z) associated with the quadratic form (11.27)
Q(x) = xi + ... + X2
one uses the group ro(4), which has three inequivalent cusps at a = oo, 0, 1/2, and one finds that cp °(Q) = 1, WO(Q) = ik and cpl/2(Q) = 0, so that (11.28)
E(z, Q) = E00(z) + ikEo(z).
If Q(x) is the sum of s 3 2 squares, we denote rs(n) = r(n, Q). This form has received much attention, and a lot is known about the representation numbers rs(n). Here are samples of results, gathered in two groups depending on the parity of s: If s = 2, the Dirichlet formula (11.9) yields r2(n) = 4 E X4 (n) din
where X4(d) = 1, 1, 0, depending on whether d  1, 1 and 0 or 2 modulo 4. Hence r2(n) # 0 if and only if every prime p  1 (mod 4) appears in n with even exponent. If s = 4, it was already known to Lagrange that every positive integer can be represented as the sum of four squares. Jacobi gave the formula r4(n) = 8(2 + (1)n) > d. dIn,2ld
11.3. Representations by Eisenstein series and cusp forms
187
If s = 6, we have
r6(n) = 16X4 (d) d2 4E X4(d)d2 din
din
If s = 8, we have r8(n) =
161:(1)ndd3
din
In the above cases rs(n) is exactly the Fourier coefficient of the corresponding Eisenstein series, but if s > 10 the cuspidal part is nontrivial. For s = 10 we have
rio(n) =
64
4
(d)d4 +
EdinX4 (d) d+ EdinX4
E xa
5
Nz=n
where z = a + bi runs over all Gaussian integers with norm n. For s = 12 we have
r12(n) = 8(_1)n1 E(1)d+n/dd5+4 din
r x4
Nlzz=n
where z = a + bi + cj + dk runs over all quaternions with norm n. The above formulas are due to Glaisher (1907). For an odd number of squares the formulas are more subtle. It was already known to Gauss that a positive integer n is representable as a sum
of three squares if and only if n # 4k(81  1). Moreover Gauss expressed the number r3 (n) of primitive representations (i.e. without common divisor different from 1) in terms of the class number h(n) of primitive positive binary quadratic forms of the determinant n, namely (11.29)
r (n) = 3
( 12h(n) if n = 1, 2 (mod 4) 8h(n) if n = 3 (mod 8).
1
In the other cases r3(n) = 0. Note that by Siegel's theorem (11.30)
n^E
« h(n) < 7112"';.
There are known explicit formulas for rs(n) for a few odd s >, 5. For example, it was shown recently by G.A. Lomadze that for nine squares
ra(n) = p9(n) +
7
(X1 xi+X +:c3=3n
X31 a
3
11. Representations by Quadratic Forms
188
where p9(n) is the Fourier coefficient of the corresponding Eisenstein series
(he computed p9(n) explicitly, but the expression is rather long to write here; see [Lom]).
For a general quadratic form Q the problem of finding the associated Eisenstein series E(z, Q) is not yet solved adequately. However, whenever E(z, Q) is known its Fourier coefficients, say p(n, Q), are relatively simple and usually give a very good approximation to r(n, Q), especially if the number of variables is large. An interesting case is for the quadratic forms of level one, i.e. Q(.s) _ 'A[x] with diag A' =_ 0 (mod 2).
diag A
(11.31)
For such a form N = 1, ;A; 1, A is equivaleiA Lu A"', and the number of variables is r  0 (mod 8); see the comments and an example after the proof of Theorem 10.9. By Theorem 10.9 the corresponding theta function is a
modular form for the modular group r0(1) = SL2(Z) of weight k = = 0 (mod 4). Therefore
e(z, Q) _ E0"r(n, Q)e(nz) = Ek(z) + F(z, Q) 0
where Ek(z) is the unique Eisenstein series whose Fourier expansion is (see (1.49)) ti
Ek(x) = 1 +
00 Y:akl(n)e(nz),
(( )r(k)
and
F(z, Q) _ >2 00 a(n, Q)e(nz) is a cusp form. Comparing the Fourier coefficients, we obtain k
(11.32)
r(n,Q) =
S(k)r(k)akt(n) +a(n,Q).
Here a(n, Q) is regarded as an error term; we have the bound (11.33)
a(n, Q) < nkz l r(n)
by Ramanujan's conjecture (Deligne's theorem). Note that the elementary bound O(nk/2) is sufficient. For k = z= 4,8 there are no cusp forms with respect to the modular group; therefore a(n, Q) = 0, and (11.32) becomes (11.34) (11.35)
r(n,Q) = 240a3 (n) if r = 8 r(n, Q) = 480x7 (n.) if r = 16.
11.3. Representations by Eisenstein series and cusp forms
189
As an example of a quadratic form of level one we have 8
Q(x) _
2
(Exi) 1
8
+ E x'  2x1x2  2x2x8. 1
Siegel has found two distinct quadratic forms in r = 24 variables of level
N = 1, say Q1 96 Q2. Thus 6(z, Q1)  6(z, Q2) = c0(z), where c # 0 is a constant; hence the Ramanujan function can be expressed by representation numbers for the two quadratic forms, cT(n) = r(n.Q1)  r(n, Q2).
We return to general Q. Although the Eisenstein part E(z, Q) is not satisfactorily understood, it is a relatively minor problem compared to that of the cuspidal part F(z, Q). Considerable research has been conducted on classes of quadratic forms for which the cuspidal part of the corresponding theta function can be determined explicitly by a special kind of cusp form. In particular one considers the following categories of forms:  Q is of Eisenstein type if the theta series coincides with the Eisenstein series; i.e. if the cuspidal part vanishes,
F(z, Q)  0.  Q is of Liouville type if the cuspidal part is a theta function for a quadratic form in two variables twisted by a spherical function, i.e.
F(z, Q) = 9(z, B, P) where rank B = 2 and P is a spherical function associated with B of degree necessarily k  1. We also allow linear combinations of such theta functions.  Q is of Mordell type if
F(z, Q) = 9(z, B, P) where rank B < r and necessarily z rank B + deg P = k. We also allow linear combinations of such theta functions.
Q is of Weil type if F(z, Q) is a cusp form associated with an elliptic curve over the rationals of conductor dividing the level. We also allow linear combinations of such cusp forms. Since the cusp form associated with an elliptic curve has weight k = 2, these quadratic forms are in r = 4 variables. For example, the quadratic form
Q(x) = xi2 + 4x22 + 4x.3 + 16x2
11. Representations by Quadratic Forms
190
is of Weil type. For p = 1 (mod 4) we have (11.36)
r(p, Q) = p + 1 

/u3
 u\
p u (mod p)
In order to generalize the notion of Weil type quadratic forms to any even number of variables r = 2k we must create in some way a cusp form of weight k out of a cusp form of weight 2 associated with an elliptic curve E/Q. Suppose E/Q has complex multiplication, and the Lfunction for E/Q is given by (up to local factors of bad reduction) 00
(11.37)
LE(s) = fl(1  apps)1(1  app
E AE(n)n'.
P
1
Define. 00
[[(1aklps)1(1s(p)anlps)1 `
(11.38) L(k) (s) _ p
E,\Fk)( 1
where e(p) = 1 except for k odd and A(p) = ap + ap = 0, in which case e(p) = 1 (note that e(p) is the character of the imaginary quadratic field which gives the ring of endomorphisms of E). For bad primes (those which divide the discriminant) the local factors can be chosen so that LEk) (s) is an entire function whose Mellin transform is a cusp form of some certain level and character, 00
(11.39)
fLk)(z) = j>(k)(n)e(nz) E sk(ro(q),x).
In view of this result L. A. Kogan says that (see (Mall)  Q is of WeilEichler type if F(z, Q) is a linear combination of cusp forms of type (11.39) of level q dividing N. For example, the quadratic form
Q(x) = xl +x? +x3+x22+4x2 +4x2 is of WeilEichler type. For p = 1 (mod 4) we have (11.40)
(u3_u))
r(p,Q)=4(p1)2+4
2
u (mod p)
The circle method was invented by Hardy and R.amanujan (1918) for the purpose of estimating the partition function p(n) (see Section 2.8). Shortly
11.4. The circle method after Kloostermani
191
afterwards Kloosterman (1926) made an essential refinement to estimate the number r(n, Q) of representations of n by a quadratic form Q(x) in four variables. He considered only diagonal forms (11.41)
Q(x) = a,x1 + a2x2 + a3x3 2 + a4x2,
but, of course, the method works for any positive definite quadric (see [Klo)). We begin by establishing a FourierKloosterman type decomposition of
the distribution b(n.) =
(11.42)
ifn=0 ifn#0.
1
{ 0
Proposition 11.1. Let C be a real number > 1. We have 1/cd
/
el
b(n)=2R.e
(11.43)
c<,C
\
fo
nxdx. C
Remark. For n = 0 our formula (11.43) gives the cute identity (try to prove this directly)
EE 2(cd)1
(11.44)
= 1.
c<,C
(c,d)=1
For the proof of (11.43) we may assume that C is a positive integer. Let f : R ' C be a periodic function of period 1. We shall evaluate the meanvalue
AM =
f 0
1
f (x)dx
by dissecting the interval with Farey's points of order C, a
a, 0
Let
< SC <
be adjacent points; they are determined by the conditions C < c + c' '< C,
ac
C < c + c' <, C,
ac"
1 (mod c) 1 (mod c).
Between the adjacent points there are mediants °°(a sim<°< pleminded rule for addition of rationals); they define the Farey segments of different length 71_°x,
M
ra) _ a'+a a+a" _
a
\c/  (c'+a' c+c"  (c
1
a
1
c(c+c')' c + c+c")j
11. Representations by Quadratic Forms
192
The Farey segments make a disjoint partition of the unit interval
UU M
C
1
+T C+
o
n)c
(a,c)=1
Hence by the periodicity of f we get
l'(f)
EY, 0a
f ( + x) dx c
o
(a,c)=1
:i
r1/cd
c
\
//
if
J
/
i C
d+xJdx.
(c,d)=1
Suppose that f has the symmetry f (x) = Y (x); then we have (11.45)
z(f) = 2ReEE
c
c
Changing the order of summation and integration, we write (11.46)
57
µ(f) = 2Re
c
r1/CC
J0
Kf(x,c)dx
where K f (x, c) is a kind of incomplete Kloosterman sum associated with f, (11.47)
Kf(x,c) _
f (x  d) C < d
.
Applying (11.45) for f (x) = e(nx), we obtain (11.43). Now let Q be a positive definite, integral quadratic form in r >, 4 variables and let n be a positive integer. In order to pick up the Fourier coefficient r(n,Q) of the theta function ®(z) = 9(z, Q) (see (11.21)) we apply (11.46) to f (x) = 9(z)e(nz), where z = x+iy with any y > 0 to be chosen later. We obtain (11.48)
r(n, Q) = 2Re.
rC
f
1/cC
T(e, n; x)e(n.z)dr
11.4. The circle method after Kloosterman
193
where
e(nJO c/
T(c,n;x)=
(11.49)
C
C
\\\
(c,d)=1,cdx<1
Next we split the theta function into congruence theta functions for residue classes h (mod c):
O (z  dlJ
= F e (_Q(h))
e(Q(nz)z). »i=h (mod c)
h (mod c)
Now we transform each congruence theta function by the Jacobi inversion formula (10.12) with x = h/c, getting r
e(Q(m)z) = IAI1/2c
/k
`
\z/
m=1t (mod c)
QC2x)
mEZr
 the J
where Q' (x) = A1 Ixj is the adjoint quadratic form. Combining these two 2 equations, we obtain
0 (z 
IAI1/2c(i)k
=
G
r E
(
(Q + m)J e(Q* (m)/c2z)
where G is the Gauss sum defined by (11.50)
GI
(Q + m)/) _
e
(
h (mod c)
thm)\
(Q(h) +
)
C
Inserting this into (11.49), we obtain .
(11.51)
/ T,n(c,n;x)e(Q*(n1)IC2z)
T(c,n;x) = IAI1/2cr ()'
TILE
Zr
where (11.52)
T,,,.(c, n; x) =
e
C
(n) C
(Q + m))
.
C
Observe that n.; x) is constant for x < 1/c(c+C); we write T,,, (c, n). This is a complete sum in r + 1 variables to modulus c, namely (11.53)
T,,,(c,n) _
*
d (mod c)
/
e (n
d1
\c
/d
G \\((Q+711)1 c
11. Representations by Quadratic Fornis
196
11.5. The singular series The infinite series (11.58) converges absolutely by virtue of (11.54); nevertheless in the literature it is called a singular series. We shall express a(n, Q) in terms of the local densities bp(n, Q). To this end notice that the sum over reduced classes d (mod c) in (11.59) is just the Ramanujan sum
e(d) _
(11.60) d (mod c)
(f) q gl(b,c)
q
Hence
li (9) q#{h (mod c) : Q(h) =_ n (mod q)},
qc(n, Q) qlc
c' gc(n, Q) = > µ (q) qlr#{m (mod q) : Q(m)
n (mod q)}
9lc
and
E c rgc(n, Q) = slr#{m (mod s) : Q(m) = n (mod s)}. cps
If every prime exponent of s is larger than that of 641A12, then the result does not depend on these exponents; it becomes the product of local densities (11.20):
c rgc(n,Q) = flsp(n,Q)
(11.61)
pas
cps
This shows that the complete product of local densities converges absolutely to the singular series
a(n,Q) = f op(n,Q)
(11.62)
P
Let p(n, Q) denote the main terns for r(n, Q) obtained by the circle method (the singular series with infinite valuation included), i.e. (11.63)
p(n, Q) =
(21r)''r(k)ins'1
JAI  1120,(n, Q)
(see Theorem 11.2). By (11.62), (11.20) and Siegel's formula (11.18) we find
that (11.64)
p(n, Q) = r(n, gen Q).
11.5. The singular series
197
Moreover Siegel proved that the p(n, Q) are the Fourier coefficients of the Eisenstein series E(z, Q) for Q (see (11.23)); therefore (11.65)
E(z, Q) _
p(n, Q)e(nz) 0 00
_ E r(n, gen Q)e(nz) 0
1:
w(Q2)e(z, Qs)
QiE gen Q
by (11.17). Hence we observe that E(z, Q) is genus invariant. Inserting (11.23) for each E) (z, Q;) in (11.65), we find that the Eisenstein series cancel because the total mass is 1 (see (11.16)), and we conclude that the weighted meanvalue of the cusp forms F(z, Qt) over the genus classes vanishes:
E w(Qi)F(z,Q;) = 0.
(11.66)
QiE gen Q
Next for primes p { 2IAI we shall compute bp(n, Q) explicitly using the formula (10.35) for Gauss sums, which translates to (11.67)
C \CQ) =
(!J)
)r
(()
in the notation of (11.50), provided (c, 2IAI) = 1. Hence
gc(n,Q) _
:s G (cQ) a (nd)
d (mod c)
r (11.68)
_ (AI) C
d (rood c)
c
e
(_). c
Further computations depend on the parity of r. First we consider r even; thus k = z is an integer > 2 and the discriminant of the quadratic form Q(x) = A [x] satisfies (11.69)
D = (1)r121A1 =O,1
Put
D (11.70)
9
(mod 4).
11. Representations by Quadratic Forms
194
The main contribution to the final formula (see Theorem 11.2) comes from m = 0. In this case To(c, n) is also denoted by gc(n, Q) whenever it is important to display the dependence on the quadratic form Q (cf. (11.59)). In any case we have the following estimate: Tm(c,n;x) K (c,n)ZCY+£
(11.54)
where e > 0 and the implied constant depends only on E and the quadratic form Q. For the proof of (11.54) one first completes the sum Tm(c, n; x) using a standard Fourier technique. Then one factors the complete sum into sums to moduli c1, c2 with clc2 = c, (cl, 21A1) = 1 and c2 having all prime factors in 21A1. In the sum to modulus cl one can transform the relevant Gauss sum (by completing the square) to one of the standard Gauss sums evaluated in Lemma 10.5. After inserting this evaluation one. is left with a single summation in d1 (mod cl ). Such a sum is either a Kloosterman sum if r is even or a Salie sum if r is odd. In both cases we have the same estimate (cl, n)1/2c1/2z(cl), due to Weil and Salie, respectively (see Chapter 4). For the other sum to modulus c2 one can proceed similarly, but the details are cumbersome. However one can be satisfied with an almost trivial bound for this sum, since we allow the implied constant to depend on the quadratic form Q. Multiplying the resulting estimates for complete sums to moduli cl and c2, then summing trivially over the frequencies which occurred in the process of completing the sum (11.52), we arrive at (11.54). Exercise. Following the above guideline, give a complete proof of (11.54). We shall use (11.54) to estimate T(c, n; x) except for To(c, n; x) = To(c, n)
in the range 0 < x < 1/c(c+C). Isolating this term, we deduce from (11.51)
that k
T(c,n;x) =
IAl1/2cr (!.)
To(c,n)
+ O((c, n)1/2C1/2+e(clzl)k E' exp(27ryQ*(m)/c21z12) mEZ*
where E' means that m = 0 is excluded from the summation if 0 < x < 1/c(c + C). We take C = n1/2,
(11.55)
y=7%1.
This choice implies clzly1/2 (C1 + Cy)y1/2 = 2 and clzly1/2 1/2Cy1/2 = 1/2 if 1/c(c+C) < x < 1/cC. Since Q*(m) 3 Im12/2A where A is the largest eigenvalue of A, we obtain trivially E / exp(21ryQ*(m)/c21 z12) << (clzly1/2)k mEZ*
11.4. The circle method after Kloosterman
195
(actually this estimate holds true with any positive exponent in place of k). Hence (11.56)
T(c, n; x) _ A
1/2c
(!) k To(c, n) + O((c,
n)1/2c1/2+enk/2).
Inserting this in (11.48), we get r(n, Q) = IAA1/2 E cTTo(c, n) C
ff 1/CC / i 1 k e(nz)dx +
0(nk;
+e).
I/cc
Here the integral is equal to (see (3.10))
00 (i) J
k
e(nz)dx =
(21r)kr(k)ink1
up to the error term O((cC)11). Summing over c < C, these error terms contribute no more than the error term already present (apply (11.54) for To (c, n)); therefore we have (27r)kr(k)1IAI1/2nk1
r(n, Q) =
` c rTo(c, n) + O(nza+e c
Finally, extending the summation to all of c (estimate the tail of the series by using (11.54)), we obtain
Theorem 11.2. If r > 4, then for any n > 0 we have (11.57)
(27r)AF(k)1nk1 JAI1/2a(n, Q)
r(n, Q) =
+ O(n2 a+e)
where u(n, Q) is given by the absolutely convergent series (11.58)
a(n, Q) _
crgc(n, Q)
and
(11.59)
gc(n, Q) = E* E e
(Q(h)  n))
d (mod c) h (mod c)
The implied constant depends on a and the quadratic form Q.
11. Representations by Quadratic Forms
198
We have EC = (ii), and the sum over reduced residue classes d (mod c) is
the Ramanujan sum (see (11.59)). Therefore, if (c, 2D) = 1,
9c(n, Q) = XD(C)C k E p (9
q.
gl(c,n)
Hence, if (s, 2D) = 1, ECrgr(n,Q) =
>2XD(C)Ck
gl(c,n)
cls
cls
µ (q) q
XD(q)glk 7T(1
gI(n,s)
 XD(p)p k).
PI 19
In particular, if every prime exponent of s is larger than that of n, we have (11.71)
> crgc(n, Q) = fl(1  XD(p)Tk) E cis
PIs
XD(q)glk.
qIn
Taking s = p` with v = 1 + ordp(n), we deduce by (11.61) and (11.71) that (11.72) Sp(n,Q) _ (1  XD(p)P k)(1

XD(p)plk)1(1
 XD(PY)p"
(1k))
for any p f 2D. In particular this shows that (11.73)
Sp(n, Q) 0 0 if p f 2D.
Another consequence of (11.61) together with (11.71) is that (11.74)
o(n, Q) = fi 6,(n, Q)L(k, X4D)1 E
X4D(d)dlk
din
p12D
where L(s, X4D) denotes the Dirichlet Lfunction. Observe that if (11.75)
Sp(n,Q)
0
for all p dividing 21A1,
then Sp(n, Q) is bounded from below and above by positive constants depending only on Q. The nonvanishing conditions (11.75) are equivalent to the existence of m E zr such that (11.76)
Q(m) = n (mod 271AJ3).
Assuming this congruence is solvable, we infer that a(n, Q) x 11 (1 + Pin
XD(p)p1k)
11.6. Eq uidistribu tion of integral points on ellipsoids
199
where the implied constants are absolute. This is bounded by constants if k > 2. For k = 2, i.e. r = 4, the product may vary slightly between (log log 3n) 1 and log log 3n. If r is odd, the exact computations of the local densities bp(n, Q) are more
involved. However, since r > 5, it is quite easy to estimate the series or(n, Q) by absolute constants. Indeed, by (11.68) we have trivially 1gc(n, Q)l < c2+1 if (c, 21AJ) = 1; hence
c 'I' = (1  23'2)((3/2) < 2,
E c''gc(n, Q) < (c,21A1)=1
21c
and the lower bound is 2  (1  23/2)((3/2) > 0. Therefore a(n,Q) x 1; provided (11.76) has integral solutions. By Theorem 11.2 we get (whatever the parity of r is)
Corollary 11.3. If r > 4, then any n sufficiently large for which the congruence (11.76) is solvable is represented by the form Q. The number of representations satisfies
r(n, Q) x nk1,
if r > 5,
r(n,Q)xnfl(1+XD(p)p 1), if r = 4. pin
11.6. Equidistribution of integral points on ellipsoids Let Q be a positive definite quadratic form with integer coefficients. If the number of variables r is greater than three, then any large integer n satisfying a finite number of necessary congruences is represented by Q in many ways (see Corollary 11.3). Thus it makes sense to ask how these representations
are distributed on the surface of the ellipsoid Q(x) = n. Let F be a nice domain (e.g. convex with smooth boundary) on the unit ellipsoid
6= {xER'':Q(x)=1} and put (11.77)
rF(n, Q) = {m E Z' : Q(m) = n,
Vnm
E F}.
One expects that, as n tends to infinity through the integers satisfying the relevant local conditions (the solvability of (11.76)), rF(n, Q)  BFI r(n, Q)
11. Representations by Quadratic Forms
200
where BFI denotes the surface area of F. In other words, we say the integral points on Q(x) = n are equidistributed with respect to the invariant measure on E. Applying a suitable approximation, this property can be stated as (11.78)
r f(n, Q) 
(
f (x)dx ) r(n, Q) E
where (11.79)
rf(n,Q) _
f+
)
for any smooth function f : E . C. Actually one can restrict the test functions to the spherical functions (with respect to the matrix A) from a complete orthonormal system on 6 (see Chapter 9). If f is a constant, then (11.78) becomes an identity. If f is a harmonic polynomial of degree v > 0, then n lir f (n, Q) is just the Fourier coefficient of a cusp form (see Theorem 10.8) 00
(11.80)
e1(z) _ E n2 r f(n, Q)e(nz).
This has weight k = v + 2, and a multiplier z9 on the group I'o(2N) (z9 becomes trivial on a congruence subgroup). Proposition 5.5 yields (11.81)
rf(n,Q) << r41+e
Hence we obtain
Proposition 11.4. Let r > 4. If f is a linear combination of spherical functions, then (11.82)
rf(n,Q) _
(ff(x)dx)r(nQ)+o(n5).
Here r(n, Q) can be replaced by the complete singular series p(n, Q) (see (11.63), (11.62), (11.57)).
If r >, 4, then the error term in (11.82) has lower order of magnitude compared with r(n, Q) (see Corollary 11.3), so (11.82) implies (11.78), proving
11.6. Equidistribution of integral points on ellipsoids
201
Theorem 11.5. If r > 4, then the vectors m E Z" with Q(m) = n are asymptotically equidistributed as n  oo over integers satisfying the congruence (11.76).
For ternary forms (r = 3) Proposition 11.4 is still valid, but the result is (just) too weak to imply the equidistribution. First we need estimates for the total number of representations Q(m) = n. The circle method does not work directly as for Corollary 11.3; however there are other ways to estimate r(n, Q). If n is squarefree with (n, 21AI) = 1, one can show that (11.83)
ni/2E << r(n, Q) << n1/2+e
provided n is represented by Q modulo 271A13, i.e. (11.76) is solvable. The upper bound is obvious, and the lower bound can be esta.hlishedl by means of the theory of binary quadratic forms discussed in Section 11.2 using some
ideas of Linnik from ergodic theory. Since the details are formidable, we leave (11.83) as a hypothesis. Next we need a bound for the number r f(n, Q) of representations weighted by harmonic polynomials f $ constant. In view
of (11.83) the standard estimate (11.81) for r = 3 is just insufficient, but Theorem 5.6 is applicable (for a cusp form a f(z) of weight k = v + 2 % 2)+ giving (11.84)
r f(n, Q) << n2 222+e.
By (11.83) and (11.84), applying the same principles of the equidistribution theory as for r >, 4, we obtain
Theorem 11.6. Let Q be a ternary, positive definite quadratic form with integral coefficients. Then the vectors m E 7L3 on the ellipsoid Q(m) = n are asymptotically equidistributed as n ' oo over squarefree numbers satisfying (11.83).
This result is due to W. Duke [Dul] (actually he makes slightly weaker assumptions about n).
Chapter 12
Automorphic Forms Associated with Number Fields
In this chapter we shall illustrate with numerous examples the unproven principle that the Lfunctions which are encountered in algebraic number theory are derived from automorphic forms of various kinds. To accomplish this we make use of the HeckeWeil converse theorems, which transfer the problem to knowledge of a large collection of adequate functional equations. Another way of establishing the automorphy is by expressing the suspected
form in terms of theta functions. The latter method is constructive, but it is less general.
12.1. Automorphic forms attached to Dirichlet Lfunctions We begin with the Riemann zetafunction 00
Ens = fl(1
((S) = 1

pa)1
p
This alone does not correspond to an automorphic form, because it has the wrong gamma factor in its functional equation: (12.1)
7r
21P (2) ((S) = 7f'2r (1
2
s) S(1 s). 203
12. Autotnoiphic Forms Associated with Number Fields
204
Moreover, the Euler product for S(s) has local factors of degree 1 at each prime while the automorphic Lfunctions have local factors of degree 2 at almost all places. These observations suggest that one should consider the product of two zetafunctions. Put L(s) = t;(s)((s  k + 1)
(12.2)
where k is an even integer > 2. Applying (12.1) for S(s) and ((s  k + 1), we find that the complete Lfunction A(s) = (21r)9r(s)L(s)
(12.3)
satisfies
A(s) = ikA(k  s).
(12.4)
Here to check that the gamma factors agree one needs the recurrence formula
r(s + 1) = sr(s)
(12.5)
and the duplication formula
r G) r
(12.6)
7r1/22191'(s).
(s 2
By Hecke's Theorem 7.7 there exists a modular form of weight k for the modular group whose Fourier coefficients are the Dirichlet coefficients of L(s). Actually it is easy to find that correspondence. Namely, we have 00
L(s) = E Ork1(n)ns; 1
therefore L(s) is the Lfunction associated with the Eisenstein series (see (1.49)) x
00
Ek(z) = 1 + ((k)r(k)
ak.t (n)e(nz).
Next we generalize the above construction to the Dirichlet Lfunctions 00
L(s, X) _
X(n)n9 = fl(1
X(p)ps)
n
For a primitive character X (mod q) with X(1) = (1)", u = 0, 1, we have the functional equation g
(12.7)
(! 2rl s 2ulL(s>X)=e l7r/
1s 2
r(12+u\L(1s,
12.1. Automorphic forms attached to Dirichlet Lfunctions
205
where eY = iuT(X)q'/2 and r(X) is the Gauss sum. Let X1 (mod ql) and X2 (mod q2) be primitive characters of conductors ql,q2 respectively, not both trivial, i.e. q = qjq2 > 1. Then X = X1X2 is a character to modulus q, not necessarily primitive. Let k be a positive integer such that X(1) _ (1)k. We consider the product of Lfunctions 00 anns
L(s, X1)L(s  k + 1,X2) _ >2 where
1.
o, = an(X1,X2)
X1(a)X2(d)dk
ad=n
Put 00
(12.8)
f(z) = fx,,xz(z) =>ane(Tz) 0
where ao = 0 unless k = 1 and q2 = 1, in which case ao is defined by ao =
1
L(0, X)
2
Theorem 12.1. The function (12.8) is an automorphic forma of weight k for the group r0(q) and character X (mod q), i.e. (12.9)
f E M .(ro(q), X)
Proof. Consider also the function 00
g(z) _ E bne(nz) 0
with b,, = bn(X1, X2) = wan(X2, X1), where (12.10)
w = (1)kT(X1)T(X2)q112(g2lg1)2
Put
2 ) r(s) E s
of (s) =
00
J A9(S) =
) r(s)
AY
00
> 1
ans
12. Automorphic Forms Associated with Number Fields
206
By (12.5), (12.6) and (12.7) one derives the functional equation (7.29). Furthermore one verifies the holomorphy of (7.27) and (7.28) (if ao or bo does not vanish one uses ((0) = 1/2). Therefore the assertion (A2) in Theorem 7.3 is completely satisfied, and consequently (A1) holds, i.e. 9 = fl,,,
with w = (q
1 )
.
Next let 0 (mod r) be a primitive character of conductor prime to q. Consider
Lf(s,
00
t(n)ann
.4
= L(s,'+GX1)L(s  k + 1,'WX2)
and
L9(s, 4h) _ 00E O'(n)bnns = wL(s,''X2)L(s  k + 1, 0X1
Put Af(s, ali) =
2
r(s)L f(s, V))
2r B
A9(s, G) =
(s) L9(s, VG)
where N = qr2. Using (12.5), (12.6), (12.7), one derives the functional equation (7.37) with w(O) given by (7.32). To get exactly w(v)) one needs X(r)'1I1(q)7_(X1)T(X2)7_(1p)2. There(12.10) and the formula r(V)Xj)r(?kX2) = fore the pair of functions f, g satisfies all the hypotheses of Theorem 7.8, and consequently we have (12.9). Note that (12.8) is an eigenfunction of all the Hecke operators; more precisely, Tn f = an f if n > 1; however f is not a cusp form, even if ao = 0. In fact, given X (mod q) with X(1) = (1)k, one can show that all the functions ff1iX2(dz) with Xl (mod ql), X2 (mod q2) (both characters are primitive such that X1X2 = X) and dqlq2lq span the whole subspace ek(ro(q), x), which is the orthogonal complement to Sk(ro(q), X) in MA(ro(q), x). We call ek (ro(q), x) the space of Eisenstein series, because if k > 2 it is indeed spanned by the Eisenstein series associated with cusps of ro(q) (the cases k = 1, 2 are somewhat delicate).
12.2. Hecke Lfunctions with Grossencharacters The next generation of Dirichlet Lfunctions emerges from characters on ideals in a number field. Let K/Q be a finite extension of rationals of degree
12.2. Hecke Lfunctions with Grossencharacters
207
n = [K : Q]. There are exactly n isomorphisms of K into the field of complex numbers
a:K>K"CC x+X". Among these there are r1 real embeddings K° C R and 2r2 complex embeddings, pairwise complex conjugate, where rl + 2r2 = n. We say that K is totally real if r1 = n or totally imaginary if r1 = 0. Let 0 C K denote the ring of integers, so 0 is a free Zmodule of rank n:
where {w ...... ,,nn} is an integral basis. This gives us the discriminant D = det(Tk w=w,) E Z
where Tr : K Q denotes the trace. Furthermore we introduce the different D C 0; its inverse is given by the fractional ideal
a1={xEK:TlxOCZ}. The prime ideals p C 0 which divide the different also divide the discriminant. Prime divisors of D, and only they, are ramified in the extension K/Q. We have sign D = (1)r2.
The group of units, say U C 0, is the direct product of a finite cyclic group
E C K of roots of unity and a free abelian group of rank r = Ti + r2  1, i.e. every unit can be written uniquely as the product
with C E E and n1, ... , n,. E 7G, where {E1 i ... , e,} is a fixed system of free generators, called fundamental units of K. Having a system of fundamental units, we define the regulator of K by R = j det(log Ie"
where aj ranges over r = Ti + r2  1 isomorphisms no two of which are complex conjugate (R does not depend on the choice of fundamental units nor the choice of the isomorphisms, and it does not matter which a is missing
in the determinant for R).
12. Automorphic Forms Associated with Number Fields
208
Let I be the group of fractional ideals # 0,
I = { at : al, 02 C O, ala2 # 0} a2
and P C I the subgroup of principal ideals
P={(a)=aO:aEK*}. Then R = I/P is the class group, and its order h = [I : P] < oo is called the class number of K. For any integral ideal m C 0 we consider the groups
Im = {a E I : (a,m) = 1}
Pm={(a)EP:a1 (mod m)). Here (a,m) = 1 means a = al/a2, al,a2 C 0, (ala2im) = 1; and a  1 (mod m) means a = al/a2, al, a2 E 0, (ala2, m) = 1, al  a2 (mod m), i.e. al  a2 E m. Clearly Pm C Im, and hm = [Im : Pm] is finite. The factor group Im/Pm is called the ray class group. In what follows a ranges over a fixed system of isomorphisms having exactly one isomorphism from each pair of complex conjugates. For any a from the system we take numbers u0, v, subject to the following restrictions: uo = 0, 1 uo E Z
if a is real, if a is complex,
v, E R
such that E vQ = 0.
Let S' = {z E C : Izl = 1}. Define the homomorphism G. : K*  S' by the product [J(aC/Ia°I)°'Ia°I'vv.
.(a) = Cr
The condition E vQ = 0 is introduced for normalization; it implies that &," is trivial on the multiplicative group of positive rational numbers. However 1;. may not be trivial on the group U of units of K, so oo is not a function of principal ideals (a) E P. Let m be an integral ideal, small enough, such that the group (12.11)
Um={riEU:771 (modm)}
is in the kernel of Cm. Then &,o can be regarded as a function on Pm, and we call m a modulus for Coo.
12.2. Hecke Lfunctions with Grossencharacters
209
A group homomorphisin : I,,, p S' is said to be a character to modulus m if it coincides with ,,, on P,,,, i.e. (12.12)
(a) = &(a) if a = (a) with a = 1
(mod m).
Of course if C is a character to modulus m, then it is also a character to any modulus n c m. Moreover, given C (mod m), there may exist a character C* (mod m*) with m c m* such that C*(a) = C(a) on I,,,. The largest ideal m* with this property (which is the greatest common factor of all such ideals) is called the conductor of C. If m* = m, then is called primitive. For any character C (mod m) there exists a unique primitive character C* (mod m*)
such that e*(a) _ C(a) if (a,m) = 1. Any Hecke character extends to C : I  C by setting
Im > S'
(a) = 0 if (a, m) # 1.
(12.13)
These socalled Grossencharacters were introduced by E. Hecke (191718) (cf. [Hel]). If all uo and vQ are zero then is a class character; that is, is induced by a character on the ray class group Im/Pm. The Lfunction associated with a Hecke character (mod m) is defined by (12.14) o96aco
where a ranges over nonzero integral ideals and N : I > Q* is the norm. In the case of integral ideals Na is just the number of residue classes modulo a:
No = #(O/a),
(12.15)
and it extends to fractional ideals by multiplicativity. The series (12.14) converges absolutely in Re s > 1, and it has the Euler product over prime ideals [J(1 
(12.16)
4(P)(NP)s)1
P
For the trivial character we get the Dedekind zetafunction (12.17)
E(Na)s = 11(1
(x(s) = a

(Np)s)'
P
By a method similar to that applied by Riemazm to CQ(s) = ((s) one proves the following theorem.
210
12. Automorphic Forms Associated with Number Fields
Theorem 12.2. The Junction (K (s) can be holomorphically continued over the whole splane except for a simple pole at s = 1 with residue (12.18)
ress=1 (K (s) = 2'''
(27r)'ZIDI1/2hRw1
where D is the discriminant, R is the regulator (we take R = 1 if rl+r2 = 1) and w is the number of roots of unity in K. Moreover the complete zetafunction (12.19)
AK(s) =
2r27r n/2IDI1/2r
(2)r` r(s)r2(K(S)
satisfies the functional equation (12.20)
AK(s) = AK(1  s),
and s(s  1)AK(s) is bounded in vertical strips.
Hecke generalized this result for L(s,1;) with a primitive character (mod m). To this end one needs Gauss sums. Put for a E K*.
!;m(a) =
Note that ,,, (a) is periodic of period m, i.e. £. (a) = lm(b)
if a, b E 0, a  b (mod m).
Choose an integral ideal r prime to m such that rdm is principal, say tom = (b) with b E 0, and define the Gauss sum by (12.21)
(Tb)
ins(a)e(TY). aEr/rm
One can show that W(Z;) does not depend on the choice of r and b. For primitive (12.22)
IW
I2 = Nm.
If 1 (mod ml) and 1;2 (mod m2) are primitive with (ml,m2) = 1, then (12.23)
W(41e2) = 41(m2)42(m1)W(fl)W(42)
12.3. Automorphic forms associated with quadratic fields
Theorem 12.3 (Hecke). Let
211
(mod m) be primitive, nontrivial. Put
(12.24) (12
A(s,C) = (2r'(21r)"IDINm)2 jir
(luol+no(s+iva)) )
where no = 1 if a is real and nQ = 2 if a is complex. The function A(s,t;) is entire and bounded in vertical strips, and it satisfies the functional equation (12.25)
A(s, t;) = w(C)A(1  s, t;)
where (12.26)
w(l;) = iuW(6)(Nm)1/2,
u=
uo
In the simplest case K = Q the Hecke Lfunctions coincide with the classical Lfunctions for Dirichlet characters. A character X (mod q) extends to fractional ideals of Q by
b with a, b E N.
X(a) = X(a)X(b) if a =
This makes X a Hecke character of modulus q with u = 0, 1 chosen so that X(1) = (1)u. Any Hecke character of Q is obtained in the above way. Of course, the functional equation (12.25) in this case is just (12.7).
12.3. Automorphic forms associated with quadratic fields Our primary objective is to construct cusp forms from Hecke characters (mod m) of a number field K. First notice that if L(s, ) corresponds to the Lfunction of a cusp form, say f E Sk(ro(N), X), then (12.27)
Lf(s) = L (s
 h 2
1,t;)
because the center points for the relevant functional equations (12.25) and (7.37) must agree. Moreover K must be a quadratic field, because the number of gamma factors in the functional equations is two. Next, examining the parameters in the arguments of the gamma factors, we infer that Z;00 must have all its imaginary exponents ivv, vanish, and in the case of a real field we also must have u = u1 + u2 = 1, i.e. either ul = 0, u2 = 1 or ul = 1, u2 = 0 (because a different parity is required by the duplication formula (12.6) to make one gamma function out of two). Therefore, if K = Q(V' ) is imaginary (so D < 0), we must have (12.28)
(a) =
a
(Tall
z ,
a E K',
212
12. Automorphic Forms Associated with Number Fields
for some integer u, and if K = Q(/D) is real (so D > 0) we must have either too(a) =
(12.29)
IaI,
a E K',
or
a E K',
!;te(a) = la;I,
(12.30)
where a' denotes the conjugate of a over Q.
Before stating the main results we recall a few basic facts about the quadratic field K = Q('/). We assume that D is the discriminant of K. The field character is given by the Kronecker symbol XD(n) =
D) n
Note that XD(l) = sign D = 1, 1 if K is real or imaginary, respectively. Moreover the values of XD at primes characterize the factorization into prime
ideals in K, namely p = p2
p = pp' p=p
if XD(p) = 0 (ramified prime), if XD(P) = 1 (split prime), if XD(P) = 1(inert prime).
The norm of an ideal is Na = aa', where a' denotes the conjugate (a' = a if K is imaginary). If ' (mod r) is a Dirichlet character on Z, it gives rise to a character on the fractional ideals of K by composing with the norm
(0 o N)(a) = O(Na),
a E I.
If /i is primitive of conductor r with (r, D) = 1, then 0 o N is a primitive Hecke character. We need to know the Gauss sums for XD and 0 o N.
Lemma 12.4. Let K = Q(v), where D is the discriminant. We have (12.31)
( D1/2 r(XD) = S l ilDI1/2
ifD > 0 if D < 0.
If i (mod r) is primitive with (r, D) = 1, then (12.32)
W(+p o N) = XD(r)'tlb(I DI)r(P)2
12.3. Automocphic forms associated with quadratic fields
213
Proof (sketch). One can give a direct argument, but it is simpler to derive these results from the functional equations stated earlier. To check (12.31) we appeal to a factorization of the Dedekind (12.33)
(K (S) = ((s)L(s, XD).
Hence, comparing the functional equations (12.20), (12.1) and (12.7), we deduce (12.31). To check (12.32) we appeal to the factorization
o N) = L(s, ,)L(s,'OXD).
L(s,
(12.34)
Hence, comparing the functional equations (12.25) and (12.7), we deduce (12.32).
Now we are ready to present the main results. For clarity we state the imaginary field and the real field cases separately.
Theorem 12.5. Let K = Q(fD) be an imaginary quadratic field with discriminant D < 0 and ( (mod m) a Hecke character such that (12.35)
((a)) =
(
if a = 1 (mod m m)
jai)
where u is a nonnegative integer. Then. (12.36)
f (z) _
((a) (Na) 2 e(zNa) E MR(ro(N), X)
where k = it + 1, N = IDINm and X (mod N) is the Dirichiet character given by (12.37)
X(n) = XD(n)6((n))
if n E Z.
Moreover, f is a cusp form if u > 0.
Remark. We assume that u > 0 by conjugating ( if necessary.
Proof (sketch). Note that ( (mod m) is not required to be primitive; nevertheless we only sketch the arguments for ( primitive. Consider another function (12.38)
g(z) = C
Z(a)(Na) Z e(zNa)
214
where C =
L (s  2,
12. Automorphic Forms Associated with Number Fields i2u'W(1)(Nm)1/2. By definition of f and g we have L f(s) =
and Lg(s) = CL (s  2, '). To these we attach the factor
(vW/21r)9r(s) to make the complete Lfunctions
Af(s) =
Sr(s)L (  2
21r
3
Ag(s) =
2 r(s)CL
(s
2,
and verify by (12.26) that A f (s) = i' Ag (k  s) (recall that u = k 1). Hence by the converse Theorem 7.3 it follows that
9=flwiv)
WN=(N
1).
Next let 0 (mod p) be a primitive Dirichlet character of conductor p t N. Then the twisted Lfunctions are given by L f (s, i/') = L (s  2, ,0 o N) and Lg (s, ?P) = CL (s  2 , £ . o N) . We set Af(s,0) =
(P)8r(s)L(soN) S
A9(s, i) =
p22
r(s)CL (s  2 ,
tG o N)
and verify by (12.25) that Af(s, 0) = ikw(,O)A9(k  s, with the appropriate root number )2p1 w(O) = X(p)V,(N)T(' (to get exactly this number, use (12.37), (12.32), (12.23) and (12.22)). Hence
by the converse Theorem 7.8 we obtain (12.36). If it > 0, then L1(s) = L (s  2, C) converges absolutely in Re s > 2 + 1, where 2 + 1 = 1 < k, so f is a cusp form.
Remark. If 6 (mod m) is primitive, then f given by the Fourier series (12.36) and g = fl,,, given by the series (12.38) yield Lfunctions with adequate Euler products (i.e. of type (6.93)); therefore f is a newform, namely an eigenfunction of all the Hecke operators T,),( with eigenvalues an = n a E 6(a), Na=n
and f is also an eigenfunction of the operator W = KW with eigenvalue C= i2k'W()(Nm)1/2.
12.4. Class group Lfunctions reconsidered
215
Theorem 12.6. Let K = Q(v' ) be a real quadratic field with discriminant D > 0 and l; (mod m) a Hecke character such that (12.39)
i;((a)) = lal
if a = 1
(mod m)

ifa  1
(mod m)
or (12. 40)
((a)) =
Via'
where a' denotes the conjugate over Q. Then (12.41)
f (z) _
l;(a)e(zNa) E S1(T'o(N), X)
where N = DNm and the character X (mod N) is defined as in (12.37).
Proof (hint). This follows from the converse Theorem 7.8 by arguments similar to those used in the proof of Theorem 12.5. Note that a Hecke character on a real quadratic field always yields a cusp form of weight k = 1. If (mod m) is primitive, then f given by the Fourier series (12.41) is a newform with Hecke eigenvalues (12.42)
a._ E (a) Na. n
12.4. Class group Lfunctions reconsidered Our arguments used for Theorem 12.5 and Theorem 12.6 were sketchy and the proofs were not really complete, since we appealed to Hecke's general results about analytic continuation and functional equations for Lfunctions
attached to Grossencharacters. Our purpose was merely to illustrate an application of the HeckeWeil converse theorems. Now we take a direct approach to show that the automorphic forms associated with Lfunctions of a quadratic field correspond to theta functions. For simplicity we confine
the demonstration to characters of conductor m = 0, so such a character is primitive. Actually we only consider the class group characters. For an imaginary field this means u = 0 in (12.35). However if K is real and every unit of K has norm 1, then neither (12.39) nor (12.40) defines a function on principal ideals. In this case in order to speak of a class group character one needs a more subtle definition of ideal classes.
12. Automorphic Forms Associated with Number Fields
216
Let K = Q(') be any quadratic field of discriminant D, positive or negative. Two ideals a, b E I are said to be equivalent in the narrow sense if
b=(a)awith c K, Na>0.
This is a new concept only when K is real and all its units have norm 1. Here we may replace Na > 0 by the condition a > 0, which means a > 0 and a' > 0 in case of a real field and a 0 in case of an imaginary field. We put
P+={(a):aEK,a>0}.
Then R+ = I/P+ is the class group of narrow classes and h+ = [I : P+] is the narrow class number, h+ 2h if K is real and Ne = 1 otherwise where E is the fundamental unit. By class character we mean a group homomorphism X S'. With X we associate the Lfunction Sl h
(12.43)
Ltt(s, X) = E X(a)(No)8 CL
where a ranges over nonzero integral ideals. We split this series into narrow classes, getting
LK(s, X) = E X(A)Cx(s, A) AEN
where (K (s, A) is the zetafunction of the class A, (12.44)
A) = E(No)s. aEA
It turns out that every class zetafunction (K (s, A) has meromorphic continuation over the splane and satisfies a functional equation of the same type. Hence the whole function LK(s, X) inherits these properties. From now on we give further details only for an imaginary quadratic
field K = Q(v/D) of discriminant D < 0. Let w = #U be the number of units of K. For every class A E f we introduce the theta function (12.45)
eA(z) = w1 + E e(zNa), aEA
and for any character X E fl we put (12.46)
fx(z) = E X(A)eA(z) AEf
= w1hb(X) + EX(a)e(zNa) a
12.4. Class group Lfunctions reconsidered
217
where 5(X) = 1 if X is trivial and 6(X) = 0 otherwise. Every class A contains an integral primitive ideal (i.e. not divisible by a rational integer > 1). Every primitive ideal can be written as (12.47)
a= [ai b +2 with a> 0, b2  4ac = D, (a, b, c) = 1. j
The above notation means a is a free Zmodule,
a  aZ + Note that b+
as = a,
Z.
2
E 0 and a = Na: indeed, we obtain a,
Z
b*"D 2 = a 2,a
b+bfD ,a 2
2
c
= [a2, ab, ac) = a[a, b, cJ = a0.
With the generators of a we associate the quadratic form
con(x) = axi +bxlx2 +cx2 =
2A[x]
where A = C 2y c ) . This establishes a onetoone correspondence between
the ideal classes A E 11 and the equivalence classes of primitive binary quadratic forms cpn of discriminant D. We choose fD = iv/I'Dl so that za
b+ fD =
2a
E H.
Then the inverse ideal a1 is a free Zmodule generated by 1 and a1
= [1, zaJ = Z ±
bVD2a
Z.
Now, given a class A which contains a, we can write
®A(z) = w1 + E e(zNb). bn
Here the equivalence b  a means b = (a)a with a E a1, a # 0, i.e. a = m + n.2a with m, n E Z, not both zero. As m, n range over the integers
12. Automoiphic Forms Associated with Number Fields
218
every ideal b  a is covered exactly w times. Moreover we have Nb = I al22a = am2 + bmn + cn2, whence (12.48)
9A(z) =
w1
e(coA(m,n)z) (m,n)EZ2
b This is indeed the theta function associated with the matrix A = r 2a b 2c)
Note that DA1 = ( ? b 2a) Therefore by Theorem 10.9 eA E .M1(ro(IDI),XD)
(12.49)
(observe the consistency condition XD(1) = 1). Furthermore, by the Jacobi inversion formula (10.10) we get (12.50)
9.4(z) =
ID11/22z19,AI(1/IDIz)
where A1 denotes the inverse class to A. Averaging over the ideal classes A E 1, each one weighted by X(A), we extend the above automorphy relations to fx. These show that
fx E M,(ro(IDI),XD)
(12.51)
f
(12.52)
xl(IDI
respectively (note that fx
1)
= ifx
= fz)
Put (12.53)
AK(s,X) =
(viTi)5r(s)L(sx),
and observe that the functions f = fx and g = f XI (
1
= i fx have
IDr
the Fourier series (7.20) and (7.21) with constant terms ao = w1hb(X) and bo = iwlhb(X) respectively (to compute the constant terms, use the asymptotics for s 4 1 from the end of Section 7.2). Therefore (A2) of Theorem 7.3 asserts that the function (12.54)
AK(S, X) + wlhb(X)(s(1  s))'
is entire and bounded in vertical strips, and AIj(s, X) satisfies the functional equation (12.55)
AK (s, X) = AI{(1  s, X).
12.5. Lfunctions for genus characters
219
Actually we have the following integral representation: s))1
Atc(s, X) + w1hb(X)(s(1 
=
f
00
(ys1 + ys)(E X(a) exp(27ryNa/
I DI ))dy
a
from which the above assertions follow at once. As a byproduct we derive from (12.53), (12.54) and (12.33) the Dirichlet class number formula (examine the residue at s = 1): (12.56)
h=
w
2,DI L(1, XD)
If K = Q(/D) is real (D > 0), the above analysis fails since the group of units is infinite while the quadratic forms corresponding to ideal classes are indefinite. The latter topics lie beyond the scope of this book, so we don't proceed beyond what has been stated about LK (s, X) in Theorem 12.2 (if X is trivial) and Theorem 12.3 (if X is nontrivial).
12.5. Lfunctions for genus characters Next we shall examine LK(s, X) closely for special class group characters (for
either imaginary or real quadratic fields), the genus characters. The genus theory was created by Gauss in the context of binary quadratic forms. Here we recall its content in terms of ideals. A discriminant of a quadratic field is said to be a prime discriminant if it has only one prime factor, so it must be one of the following type: (12.57)
4, ±8, ±p = 1 (mod 4).
The product of coprime discriminants is again a discriminant. Every discriminant can be written uniquely as a product of prime discriminants, say D = P1 ... Pt. Hence D can be arranged as a product of two discriminants D = D1D2
(12.58)
in 2t1 distinct ways if we allow interchanging D1 with D2 (here and here
after, t denotes the number of distinct prime factors of D). For any such decomposition we define a character XD,,D2 on ideals by setting (12.59)
XDi,D2(p) =
YD,(Np) X D2 (Np)
if p f D1 if p $ D2
220
12. Automorphic Forms Associated with Number Fields
(recall that Xd(n) = (4) is the Kronecker symbol). This is well defined on prime ideals because (12.60)
XD(Na) = 1
if (a, D) = 1,
and we extend XD1,D2 to all fractional ideals by multiplicativity. Therefore
XD,,D2 : I > {±l} so XDl,D2 has order two, except for the trivial character which corresponds to the trivial factorizations D = D 1 = 1 D. Every such XD,,D2 is called the genus character of discriminant D; these are different for distinct factorizations (12.58), so we have exactly 2`1 genus characters. The genus characters are the narrow class group characters, XD1,D2 E H+, i.e. (12.61)
XD1,D2(a) = 1
if a = (a),a > 0.
Theorem 12.7 (Kronecker). The L Junction of K = Q(// ) associated with the genus character XD1,D2 factors into the Dirichlet L functions, (12.62)
LK(s, XD1,D2)
= L(s, XD1)L(s, XD2)
Proof (hint). For the trivial character (D1 = 1 or D2 = 1) this is the factorization (12.33) for the Dedekind zetafunction. In general one can verify the Kronecker factorization (12.62) easily by examining the local factors
in the corresponding Euler products and using the law of factorization of primes in K expressed in terms of values XD(p) = 0, ±1. One shows that every real character of x+ is a genus character, and they all form a group of order 2t1 which is isomorphic to (Z/2Z)t1
Now let us describe the dual side of genus characters. We say that two nonzero ideals a, b E I are in the same genus if (12.63)
X(a) = X(b)
for all genus characters.
The ideals with x(a) = 1 for all genus characters form the principal genus (a subgroup of I). The same definitions apply to the narrow classes. We denote the subgroup of principal genus classes by (12.64)
9 = {A EH' : X(A) = I for all genus characters}.
The factor group F = ?l+/9 is called the genus group. By duality F is isomorphic to
(Z/2Z)t1
12.5. Lfunctions for genus characters
221
It was proved by Gauss that the principal genus consists of squares of classes
9 = {A2 : A E fl+}.
(12.65)
Hence two nonzero ideals a, b belong to the same genus if and only if (12.66)
a = bc2
for some c E I.
One can also show that a, b are in the same genus if and only if
Na = NbNy for some y r= K.
(12.67)
A class A E x+ is said to be ambiguous if A = A1, i.e. A2 = 1. The group of ambiguous classes (12.68)
D= JA Efl :A2=1}
is isomorphic to the genus group .7 = 11+/9 (since V is the kernel of the homomorphism A H A2). Ambiguous classes are represented by ideal divisors of V/D; indeed if (12.69)
fD = 0102, (01,02) = 1,
then Cl(D1) = C102) and C1(D1)C1(D2) = 1.
From now on we consider a real quadratic field K = Q(v) of discriminant D. Much of the material presented below can be found in [Asa], (He2] and [Miy]. Though D > 0, it does not mean that (12.58) has positive factors. For any positive MID, consider matrices of type (12.70)
1
WAI _ VrM
(a b) c
dJ
where a, b, c, d E Z, MIa, Mid, Dic, ad  be = M. From the determinant it follows that (M, DIM) = 1, i.e. D = MN with (M, N) = 1. In particular, for M = 1 we obtain all elements of the group ro(D). Each WA! normalizes I'o(D), i.e. Wnfro(D)W,II = ro(D), and W.421 E ro(D). The group of all such matrices, (12.71)
r*(D) = {WAJ : MN = D, (M, N) = 1},
is the normalizes of ro(D), and the factor group
4)(D) = r*(D)/ro(D)
222
12. Automorphic Forms Associated with Number Fields
is isomorphic to (7G/2Z)t. The elements of '(D), that is the classes of matrices W& f modulo ro(D), correspond to the ordered factorizations D = D1D2 into discriminants (take M = ID1I). We may regard fi(D) as a group
acting on M/ro(D). By the action of t(D) any cusp for ro(D) can be brought to any other cusp; for example, WMOO = a/c is equivalent to the cusp a = 1/N. acts on automorphic forms (by the slash operator) The group
4P(D) : Mk(ro(D), XD)  Mk(ro(D), XD) (D(D) : Sk(ro(D), XD)  S1.(ro(D), XD) decomposing each of these spaces into 2t eigenspaces of type
Jtiik) = { f E Mk : f (z) = Y a(n)e(nz) with a(n) = 0 if Xp(n) = bp for all pID} Ilere Xp(n) _ (n) is the Kronecker symbol associated with the prime divisor p of D and P = ±p, ±8,4 is the corresponding prime discriminant. To each such prime we assign a symbol by which takes two values by == ±1. Out of this we form the vector symbol (b) = (... , 5,,,...) with plD. Denote
P(b) _ f by = ±1. pI D
Any (b) with P(b) = 1 corresponds to the genus of classes in K = Q(v' which contains an ideal a such that (12.72)
)
Xp(Na) = by for all pID.
Therefore it is natural to single out the spaces
MA = ®Mk),
Mk = ® M.
P(6)=1
Y(b)=1
We have a direct sum decomposition Mk = M+ ® M. The space M+ is particularly interesting; it consists of forms whose Fourier coefficients are supported on integers n > 0 with XD(n) = 0, 1, that is,
Mk = f E Mk : f (z) _
a(n)e(nz) YD(n)#'1
The above decomposition of MR into 211 linear spaces Mk) (one for each genus of K) can be examined further in terms of Eisenstein series. It turns
12.5. Lfunctions for genus characters
223
out that the codimension of MjjI fl Sk(ro(D), XD) in M(') is 1, and that MA a)
contains one Eisenstein series. More precisely it is a specific linear com
bination of standard Eisenstein series which depends uniquely on the ideal a having the genus invariants (12.72). For example if k = 2, we have the following 2' standard Eisenstein series in M2(ro(D),XD), each one corresponding to an ordered factorization D = D1D2 into discriminants, namely EDI,D2(Z)
1
00 >dXD,(d)XD2(n/d) e(nz) = 2 L(1,XD)eD,,D2 +E
n=1
dIn
where eD,,D2 = 0 except for ED,1 = 1. Then, out of these one can construct an Eisenstein series in .M2bl for any 6 with P(6) = 1 as followsE(b) (z)
=
(ri 6p
ED1,D2 (Z)
D1 D2=D
_
L(1, XD)+E f n=1
ntn2=n
n1XD,(n1)XD2(n2Na)
e(nz)
D, D2=D
where a is an ideal satisfying the genus conditions (12.72) and Na is the norm.
Another benefit from the action of the normalizer r(D) on the space Mk(ro(D), XD) is that one can explicitly link the Fourier coefficients of an automorphic form at any desired cusp to these at the cusp oo. More precisely, suppose
a(n) e(nz) E Sk(ro(D), XD)
f (z)
is a newform, normalized by a(1) = 1. Given a decomposition D = D1D2 into discriminants, define twisted coefficients b(n) by the following requirements:
b(n) = XD1(n)a(n) if (n, D1) = 1 b(n) = XD2(n)a(n) if (n, D2) = 1 b(mn) = b(m)b(n) if (m, n) = 1.
Since a(n) = XD(n)a(n) if (n, D) = 1 (see (6.57)), one shows that the above requirements are consistent and define b(n) uniquely. The resulting twisted series
00
g(z)
b(n)e(nz) E Sk(ro(D), XD) 1
224
12. Automorphic Forms Associated with Number Fields
is also a newform, and it is related to f by (12.73)
WAI f = ANIg
for M = ID, I, where aas can be expressed by Gauss sums. For example, if M = p is prime then k12.
(12.74)
P = T(XP)a(p)p
For M = D this agrees with Theorem 6.29 (the proofs are similar).
12.6. Automorphic forms of weight one A lot of fascinating connections between number fields and automorphic forms have been revealed by Langlands' interpretation of Artin's Lfunctions. We shall give a glimpse of some results (without proof) in the context of two dimensional Galois representations visavis modular forms of weight one. References for these and more results can be found in the survey article by JP. Serre [Ser]. We begin by recalling well known facts from arithmetic in number fields.
Suppose L/K is a Galois extension of degree n = (L : K] and the Galois group G = Gal(L/K). Let OK C OL be the rings of integers in K and L respectively, so n = (Of, : OK]. The central problem is to determine the way in which a prime ideal p C OK factors into prime ideals 3 C OL. We have
p = Mi ...F'9)e where E31 i
... , X39 C OL are all distinct prime ideals, mutually conjugate,
say to 3, each one of degree
f = (OL/T : OK /p],
efg = n.
The absolute norms are related by NT = (Np)f. Consider the decomposition group
G93={aEG:aT =T} and the inertia subgroup
Iq={aEG:ax=x (inodg3)forallxEOL}. To conjugate ideals there correspond conjugate groups. The decomposition group GT acts on the residue class field OL/g3 leaving OK/P fixed; hence we have a natural map
Gp a Gal((OL/q3)/(OK/p))
J2.6. Automoiphic forms of weight one
225
whose kernel is the inertia group IT, so f = (Gq3 : Iqj]. Suppose p is unramified, i.e. e = 1 (there are only finitely many ramified
p in the extension L/K, namely the prime factors of the discriminant), so IT = 1, (GpJ = f and fg = n. For such T the Frobenius generator of the Galois group of the residue class fields extension can be lifted uniquely to the Frobenius automorphism in Gp; we denote such automorphism by aT = (3, L/K); it is characterized by the property 2°T _ xNp
(mod T) for all x E OL.
The Frobenius ap depends on T rather than p, but it is in the conjugacy class ap = {aaTO 1 : a c G},
which we also denote by ap = (p, L/K). In particular, ap is a single element if G is abelian. Very important problems concern the prime ideals of K which split completely in L, i.e. such that
p = Ti... T., a=f = 1. These are characterized as having the trivial Frobenius automorphism, ap =
ap = 1. According to Bauer's theorem the set Sp(L/K) of primes of K which split completely in L determines the Galois extension L/K uniquely. Therefore it becomes an attractive proposition to describe the set Sp(L/K)
in some kind of arithmetical terms. For K = Q by Cebotarev's theorem Sp(L/Q) has density 1/n. For abelian extensions L/Q the class field theory asserts that a prime number (ideal) splits completely if and only if it belongs to certain arithmetic progressions modulo the discriminant. For example, if
L = Q(v), then (DP )
Sp(L/Q) = {p :
1}.
If the extension L/Q is not abelian, the law for splitting primes is no longer governed by arithmetic progressions. However a great number of nonabelian extensions have been described by arithmetic properties of Fourier coefficients of certain automorphic forms. Much of this research is conducted in connection with Artin's Lfunctions.
Let L/K be a Galois extension, G = Gal(L/K) and p : G a representation. For each p C K unramified in L/K, define the local Lfunction by Lp(s,p) = det(I 
p(ap)(Np)s)1
226
12. Automoiphic Forms Associated with Number Fields
where op is the Frobenius conjugacy class (note that Lp(s, p) depends on p but not on primes in L which lie above p, because the characteristic polynomial depends only on the conjugacy class of p). If a(1), ... , ap"''1 are the eigenvalues of p(op), then m
Lp(s, p) = fI(1  a,, (Np)`9)+1. i=1
For the ramified primes (a finite number of them) the local Lfunctions can
be defined in a similar fashion. Then the Artin Lfunction is formed by multiplying all the local Lfunctions,
L(s,p) = JJ Lp(s, p). p
Example. The onedimensional representations correspond to the Dirich
let characters. To this end take K = Q C Q(() = L, where S is the Nth root of unity. This is a cyclic extension of degree n = cp(N); its Galois group is isomorphic to the group of primitive residue classes mod N:
(Z/NZ)*  Gal (L/K)
a Dirichlet character X (mod N) one associates the representation
p : Gal (L/Q) F GL1(G) = G* oa F+ x(a).
For p } N the Frobenius automorphism op E G is the one which maps ( to (P, and we have p(op) = X(p). In this case the Artin Lfunction attached to p coincides with the Dirichlet Lfunction for X; L(s, p) = L(s, X). From now on L/Q is Galois with G = Gal(L/Q). There exists a continuous representation p : Gal(Q/Q)  GL,n(G) whose kernel is Gal(Q/L); this yields an injective homomorphism p : G + GL.,,F.(G). Therefore the set of splitting primes is given by
Sp(L/Q) = {p : p(up) = I}, and the local Lfunctions at such primes are Lp(s, p) = (1p') '. Writing the Artin Lfunction as a Dirichlet series 00
L(s, p) = 1
ann _s
12.6. Automorphic forms of weight one
227
we see that p splits completely in L/Q if ap = m (i.e. ap takes the maximum value). Analytic properties of Artin Lfunctions are largely intractable. Artin conjectured that L(s, p) are entire functions except for some explicitly given representations (L(s, p) is expected to be entire if p is irreducible of dimen
sion > 1) and that L(s, p) satisfies a suitable functional equation which connects values at s and 1  s. Langlands has deeper vision; briefly the celebrated "Langlands program" predicts that L(s, p) coincides with the Lfunction attached to some automorphic representation on GL,,,, where m is the degree of p. If this is true, then the Artin conjecture would follow in a straightforward manner. What we know at present is that L(s, p) is meromorphic on the whole splane (due to R. Brauer), since L(s, p) can be represented as a quotient of products of zetafunctions corresponding to Hecke's characters in various fields. But the holomorphy question in general is not resolved. Yet, Artin succeeded in proving his conjecture for onedimensional representations of abelian extensions as part of the "reciprocity law". From this perspective Langlands' program offers an attractive direction towards nonabelian class field theory. From now on we shall discuss various cases of twodimensional Galois representations. Such representations (according to the Langlands program) should correspond to classical automorphic forms of weight one. Indeed, this correspondence has already been established in the direction from automorphic forms to number fields by P. Deligne and JP. Serre.
Theorem (DeligneSerre, 1974). Let f be a normalized cusp newform on lo(N) of weight k = 1 and character X (mod N) with X(1) = 1. Then there exist a Galois extension L/Q and an irreducible twodimensional
representation p : Gal(L/Q)  GL2(G) such that L f(s) = L(s, p). R.emark. In the above result the level N is the conductor of p, and the character X (mod N) corresponds (via class field theory) to the onedimensional representation e = det p. This representation is odd, i.e. E(c) = 1, where c is the complex conjugation automorphism (think of p : Gal(Q/Q) > GL2(G)
and e : Gal(4/Q)  G`). As an immediate consequence of the DeligneSerre theorem one gets the Ramanujan conjecture for the Fourier coefficients of cusp forms in the space Si(ro(N),X). Namely, one obtains the estimate Ia(p)I S 2 for prime coefficients of 00
f (z) _
a(n)e(nz).
228
12. Automorphic Forms Associated with Number Fields
We assume that f (z) is normalized so that a(1) = 1. If
g(z) _
b(n)e(nz)
is a normalized newform in S1(ro(N), X) other than f , then Serre showed
that density {p : a(p) = b(p)} < 7/8,
and 7/8 is attained. Much is known about the Artin conjecture for two dimensional representations. Suppose
p : G  GL2(C)  PGL2(G) = GL2(C)/(C)* is continuous, i.e. its image is finite. On the other hand all finite subgroups
of PGL2(C) were described by Klein; they are: C cyclic of order n; D,, dihedral of order 2n, n > 2; A4 tetrahedral; S4 octahedral; A5 icosahedral. The cyclic groups appear for reducible representations; the corresponding Artin Lfunction is given as the product of two Lfunctions for Dirichlet characters and the corresponding automorphic form is an Eisenstein series. The remaining subgroups are in the image of irreducible representations. Let us recall that a finite group D is called dihedral if it has a subgroup
H of index 2 and an element x ¢ H such that x2 = 1 and xyx1 = Y1 for all y E H. In particular, D,, can be viewed as the group of rotations and reflections of the plane which preserve a regular polygon with n vertices.
Thus D is generated by a rotation r of angle 27r/n and a reflection s, and the only relations between these generators are r" = 1, s2 = 1, srs = r1. The Artin Lfunction for a dihedral representation coincides with the Hecke Lfunction for some Grossencharacter; hence the Artin and Langlands conjectures are both true. For tetrahedral representations the Artin conjecture was established by Langlands, and then his ideas were adopted by Tunnell for all octahedral representations. However the icosahedral representations still resist any treatment, except for a few examples established by J. P. Buhler. Finally we make a few comments about the dimension of S1(r0 (N), X).
Serre has established a surprising fact: a newform f E S1(ro(N),x) has almost all Fourier coefficients zero (in the sense of density). Hence it seems there are not as many cusp forms of weight one as for weight > 2. Serre has examined the space S1(I'o(p), ,yp) for prime level and character )(1,(n) _ ().
Since XN(1) = 1, we must have p = 3 (mod 4). Applying the Selberg
12.6. Automorphic forms of weight one
229
trace formula, one can showed that (private communication by P. Sarnak) (12.75)
dimS1(ro(p), Xp) <<
l gp
This is already an interesting estimate, because the fundamental domain ro(p)/III has volume 1(p + 1), which is larger than the bound (12.75) by the factor log p. There is a reason for such a good bound, namely the fact that the forms y1/2 f (z) lie at the bottom of the continuous spectrum of the Laplace operator 2
2
8x2 + 8y2)  Zy 8x
y2
Presumably the estimate (12.75) can be improved further, but not simply by manipulating the choice of test functions in the trace formula. A serious weakness of the trace formula in this respect is that Si (Po(p), Xp) is a subspace for the lowest eigenvalue AI = 1/4 which is not isolated, so any reasonable test function picks up a large chunk of eigenforms which are not holomorphic forms. This said, in order to get an accurate estimate for dim Si (170(p), Xp) one
must exploit the very arithmetical structure of holomorphic cusp forms of weight one. Serre established the formula (12.76)
dimS1 (to (p), Xp) =
h

1
2
+ 2s + 4a
), s is the number of quartic,
where h is the class number of K = Q(
normal fields with discriminant p (any such a field has the Galois group S4) and a is the number of quintic, nonreal fields with discriminant p2 whose normal closure has Galois group A5. He also conjectured that (12.77)
2s+4a<
whence (12.78)
dims1(ro(p), Xp) = h
2
i + OW).
Here the h21 linearly independent cusp forms in Si(Po(p), Xp) are actually theta functions constructed from the class group characters of K, namely
X(a)e(zNa)
f.<(z) _ O54aCO,c
for any X E ?l, X # 1 (see (12.46)). Note that X are complex characters (by
the genus theory, because K has prime discriminant), and f, = fc. There
230
12. Automorphic Forms Associated with Number Fields
are no other linear relations between the fx. Recall that the class number is bounded by p1/2e << h «p"2log p.
While Serre's conjecture seems to be out of reach, a striking breakthrough was recently made by W. Duke [Du2), who succeeded in showing that (12.79)
dimSi (I'o(p), x) <<
p12L' (log P)4.
His arguments combine arithmetical and analytic ideas beautifully.
Chapter 13
Convolution Lfunctions
13.1. Introduction We have already experienced how fruitful the idea of twisting automorphic
forms by Dirichlet characters is. One can regard the Dirichlet characters X (mod q) as a kind of automorphic forms on GLI, and furthermore one can regard the Hecke Grossencharacters (mod m) for a number field K of degree n as automorphic forms on GLn. Hence a question arises: what does one obtain by twisting two genuine automorphic forms? A quick guess is that one should get an automorphic form on some higher rank group. We cannot address these advanced topics in this book; however, we shall be able to make convincing observations in the context of relevant Lfunctions. To illustrate the general picture we consider two Lseries L l (s) =
L9(s) =
I:b(n)ns
1
which converge absolutely in some right halfplane. With these we form the convolution Lseries by multiplying the coefficients,
L(f ®g, s) _
a(n)b(n)n'. 231
13. Convolution Lfunctions
232
For example, if f, g are Eisenstein series whose Lfunctions are 00
Lf(s) = E aQ(n)ns = ((s)((s  a) 1
00 Eab(n)ns
L9(s) =
= ((s)((s  b),
1
then we get 00
L(f (& g, s) = > aa(n)ab(n)ns = ((s)((s  a)((s  b)((s  a  b)((2s  a  b)1.
This formula was proved by Ramanujan. Rankin and Selberg considered the convolution Lfunctions attached to arbitrary automorphic forms f, g. It is easy to show that if L1(s) and La(s) have Euler products of type Lf(s) = Ti(1  apps)1(1

QPPs)1
P
La(s) = [J(1  7pp)'(1  6PPs P
then L(f 0 g, s) also has an Euler product of type L(f O g, s) = fJ(1  aPNP YP5Pp 2s)
(13.1)
P apypp8)1(1
X

aP6PP~e)1(1
Qprypp 8)'(1  (3P6PP8)1.
P
Our main goal is to establish the analytic continuation of L(f 0 g, s) and some kind of functional equation.
13.2. RankinSelberg integrals We begin with two automorphic forms f E Mk(r,19), g E M1(r,21) for a group r and multiplier systems 29, n of weights k,1, respectively. We assume that one of these forms, say f, is a cusp form (to avoid problems of convergence of certain integrals). Let a be a cusp for r which is singular with respect to both multipliers. Then f and g have Fourier expansions at a of type (see (2.64)) f
fa(n)e(nz),
(z) _ 1
00
go(n)e(nz).
910, (z) _ 0
233
13.2. R.anlcinSelberg integrals
For n >, 1 we normalize the coefficients by setting Ik fa(n) = (41rn)
1. (n)
ga(n) = (47rn) 2tt ga(n), and we define the convolution Lfunction by
L0(f 0
(13.2)
,
s) _
fa(n)9a(n)ns
00
Note that AO0, glO0 are autoinorphic forms for the conjugate group aQ Tor. and the multiplier systems 19'a, 77'd given by (2.55). Although f. (n), ga(n) depend on the choice of the scaling matrix a0', the product fa(n)ga(n) does not, and La(f ®g, s) depends only on the equivalence class of cusps. We have
f
1
fiO0(z)giOp(z)dx
0" = E.fa(n)9a(n)e
Avrny
1
Hence using the gamma integral 00
r(s) = f e
yldy
we get
/
(47r)9r l s +
k+1 2
 1) La(f (9g, s) =
ff °O
0
ysli(z)d;iz
0
where
h(z) = Y'2
f1Qa(z)g1O0(z)
By changing the variable z  o a z we can write the above integral as 1
fH
Jsh(z)dpz = fa\H(Im 0'Q 1z)h(a 1z)dµz
f
(Im a.1ryz)sh(aa 1'yz)dµz. \H 7Ero\r
We have flan (a0' 1`yz) = 700 (an
,yz)'.f
(7z)
=700(0'. 7z),77(z)"9('y).f(z) = 19(ry)LLk(0'a, 0'a 17)7ad I.y(z)''f(z)
13. Convolution Lfunctions
234
by (2.50), (2.60) and (2.49). A similar formula holds for g. Hence by (2.7) we get h(aa 17z) = X.('y, z)y s f (z)g(z)
where y(z)I)kt
(13.3)
Xa(7,z) _
17)
Summing over 7, we create the Eisenstein series
EQ(z,s) = E XQ(7,.z)(Im as 17z)s, 7Er,\r
(13.4)
and we obtain
Proposition 13.1. If Re s is sufficiently large, then (13.5)
(47r)9r (s + k
l  1 1 LQ(f ®9, s)
r\ 2y 2 f (z)9(z)EQ(z, s)dµz. The result simplifies a bit if k = 1. Namely, we get (13.6)
(47r)'F(s + k  1)Ln(f (9 g, s) = J \H yk f (z)g(z)EE(z, s,19f)dµz
where (13.7)
E t9(7)i ey)(Im as "Yz)s.
EE(z, s,
At the cusp a = oo with a. = 1 for any k, l the Eisenstein series becomes (13.8)
E(z, S) =
E t9(7)f(7)(9ry(z)/I.iry(z)I )t'l(Im 7z)8 yEr.\r
For t9 = 1 and 17 = Ed (d) (the multiplier of the standard theta function (2.72) of weight 1 = 1/2) on the group ro(4) we have (13.9)
=
2
ed (C) (c,d)=1 c=_0 (mod 4)
cz + d
(lcz+dI)
k
s
y
(Icz
I
dI2)
13.3. Selberg's theory of Eisenstein series
235
The integral representation (13.5) for the RankinSelberg Lfunction is a close analogue of (7.4) for the Rieniann zetafunction. In a similar fashion (13.5) will yield analytic properties of L. (f ®g, s) from those of Eg(z, s). Since f is a cusp form and g has at most polynomial growth, the associate measure (13.10)
µ19(z) = yk f (z)g(z)dµz
has exponential decay at the cusps. Hence the integral (13.5) converges absolutely and uniformly with respect to s on compacta on which the Eisenstein series E0(z, s) can be analytically continued.
13.3. Selberg's theory of Eisenstein series A general theory of Eisenstein series on GL2 was completely developed by A. Selberg in the 1950's and subsequently generalized to higher rank groups by R. Langlands in the 1960's. The real analytic Eisenstein series are indispensable for the spectral theory of automorphic forms; they are building
blocks for the subspace of continuous spectrum. At the root of this lies the problem of analytic continuation over the whole complex splane. For a congruence group the desired properties of Eisenstein series follow straightforwardly from explicit Fourier expansions (we shall rely on them), so the real difficulties are encountered for other groups. Although we don't need
to be general, it is worthwhile to establish some properties (such as the functional equations) in Selberg's original way. We restrict the presentation to Eisenstein series of weight zero (such as (13.7)), i.e. (13.11)
X('r)(Im a lyz)s
E0(z, s, X) =
rEr,\r
where X is a multiplier system of weight zero (X : r * C* is a group homomorphism) singular at cusp a. The series (13.11) converges absolutely in Re s > 1 by virtue of Proposition 2.9. The Eisenstein series is in the space A(r, x) of functions f : H > C which satisfy the automorphy equation (13.12)
f (yz) = X(ry) f (z)
if 'y E r.
Though Ea(z, s, X) is not holomorphic in z E 1111, it is an eigenfunction of the
Laplace operator
r
+02
L = y2 r ,9x2 2
13. Convolution Lfunctions
236
namely (13.13)
(A + s(1  s))E0(z, s, x) = 0.
This is clear because A commutes with the action of any r E GL2(R) on H, the power function (Im z)3 satisfies (13.13), and the Eisenstein series is built from (Im rz)s. We denote by A, (17, x) the space of functions f : 1H[ > C satisfying (13.12) and (A + s(1  s))f = 0.
(13.14)
Note that s  s(1  s) is a double cover of C since s and 1  s yield the sauce eigenvalue A = s(1  s), except for s = 1/4 which covers A = 1/4 once. The Laplace equation (13.14) replaces the CauchyRiemann equations
for holomorphic functions. Of course, we require f to be at least twice differentiable; actually, since i is an elliptic operator, its eigenfunctions are realanalytic. By arguments similar to those employed for the holomorphic Eisenstein series (3.14) we derive the Fourier expansion of Ea(z, s, x) at cusp b (singular) of the following type: X)yl_s
Ea(obz, s, X) = 5abY8 +'Pab(S,
+
(13.15)
'Pab(n, s, X)W8(nz), n,EO
where (13.16)
lPab (S, x) _ lr1/2 r(
E r(s) /2)c>o
C2s S
b(0, 0; c),
(13.17) WO (n, s, x) =
lnls1
r(s)
>
c2s Sab(0,
n; c),
c>0
Sab(0, n; c) are (KloostermanGaussRamanujan sums) given by (3.13) and (13.18)
W8(z) = 21y11/2Ks1/2(21
lyl)e2,Ti3
is the Whittaker function on z E C\IR (an analogue of the exponential function e(z)). The Bessel function 2
f
0
has the integral representation
13.3. Selberg's theory of Eisenstein series
237
which shows that Kv(y) is entire and even in v and decays exponentially in y; more precisely, one can prove the approximate formula
K(y) _ (i;)
e1{1
+ O(lvl2y1)}
uniformly for Re v > 1/2, y > 0. Hence WS(z)  e(z)
as y + oo.
As an example, take the modular group r = SL2(Z). There is only one cusp a = b = oo (up to equivalence), and the multiplier is trivial. We have S(0, 0; c) = cp(c), whence (13.19)
W(S) _ 7r
1/2r(s  1/2) ((2s  1) r(s)
((2s)
where ((s) is the Riemann zetafunction. By the functional equation (12.1) we can write cp(s) symmetrically: (13.20)
cp(s) = 0(1  s)0(s) 1
where 0(s) = 7r_.Sr(s)((2s).
(13.21)
The other Fourier coefficients of E(z, s) are given by (13.22)
co(n, s) =
0(s)1Inl
1/2
E (a/d)s1/2
ad=InI
In particular, for s = k  1/2, where k is a positive integer, (13.23)
n1/2cp(n, k 
1/2) = 7rkr(k  1)1((2k  1)ak_1(n).
By these formulas the Fourier expansion (13.15) shows that E(z, s) has meromorphic continuation over the complex splane, and it satisfies the functional equation (13.24)
0(s)E(z, s) = 0(1  s)E(z,1  s).
In the halfplane Re s >, 1/2 the Eisenstein series E(z, s) is holomorphic except for a simple pole at s = 1 with (13.25)
3
ress=iE(z,s) = 3.
13. Convolution Lfunctions
238
For a general group r the Fourier coefficients c'ab(s, X), (Pab(n, s, X) are hard to compute, but once the meromorphic continuation is taken for granted one can derive the functional equation in an elegant fashion. The socalled scattering matrix (13.26)
4'(s, X) = (Wab(s, X))
(its rank equals to the number of inequivalent, singular cusps) plays a central role. In this connection it is convenient to arrange all the Eisenstein series into the columnvector (13.27)
E(z, s, X) = t(... , E0(z, s, x),...).
Selberg established (among many other things) the following theorem.
Theorem 13.2. The scattering matrix and the Eisenstein series have meromorphic continuation over the whole splane. In the halfplane Re s > 1/2 the scattering matrix has at most a finite number of simple poles in the segment 1/2 < s < 1. The poles of Ea(z, s, X) in Re s >, 1/2 are all simple, and they are at the poles of Soaa(s, X), so they are also in the segment 1/2 < s < 1. The point s = 1 is a pole of E0(z, s, X) only when X is trivial, in which case the residue is a constant function of z; more precisely, (13.28)
ress_iEa(z, s) = 1/vol (r\H).
The scattering matrix satisfies the functional equation (13.29)
and is unitary on the critical line, i.e. (13.30)
4(s, X)tfi(s, X) = I if Re s = 1/2.
The vector Eisenstein series satisfies the functional equation (13.31)
E(z, S, X) = `1'(s, X)E(z,1  s, X).
Proof of the functional equations. We have proved in previous sections the meromorphic continuation of the Eisenstein series for only congruence groups. However, assuming the first part of Theorem 13.2, we can establish the functional equations in full generality. There is no way to employ Poisson's summation as it could be, although not easily, in the case of a congruence group. Therefore, it is surprising that the functional equations are
13.3. Selberg's theory of Eisenstein series
239
a routine consequence of the apparently remote fact that A is a symmetric and nonnegative operator in the Hilbert space
L2 (r, x) = { f E A(r, x), (f, f) < oo} where (f, g) denotes the inner product
(f, g) = f \H f (z)9(z)dpz. Indeed, by Green's formula one shows that
y2(f+whence
(Af,g) = J
(A f, g) = (f, Ag) and (A f, f) > 0 provided f, g, Af, Ag are all in. L2 (r\Illl). By these properties it follows that the eigenvalues of A in
L2(r\Ill<) are real, nonnegative numbers. Based on this observation, we draw the following characterization of the Eisenstein series.
Lemma 13.3. Any f E A3(r, x) with Re s > 1, which at every cusp is bounded by a power function, is necessarily a linear combination of Eisenstein series. Proof. Since the Fourier coefficients of f (Qaz) are bounded by a polynomial while the Whittaker function decays exponentially, we get from the Fourier expansion
f (a z) = coy' + 0(1) as y  oo. Subtracting the Eisenstein series, we kill the leading terms toys and get a function
g(z) = f (z) 
c0E,(z, s, x)
bounded on H. Since g E A, (17, x) n L 2 (r, x), we must have A = s(1s) > 0, which contradicts Re s > 1 unless g = 0, i.e. (13.32)
f (z) = E caEa(z, s, X) a
Now we apply Lemma 13.3 for f (z) = Eb (z,1  s, x) with Re s > 1, getting Eb(z,1  s, x) _
P6a(1  s, x)E.(z, s, x)
by (13.32). This is just the functional equation (13.21) with s replaced by 1  s. Here the condition Re s < 0 can be relaxed by analytic continuation. Finally, (13.29) follows by repeated application of (13.31).
13. Convolution I,f actions
240
13.4. Statement of general results Since the analytically continued Eisenstein series Ea(z, s, X) for s not a pole is bounded by a power function of z at every cusp, we infer by the integral representation (13.6) and Theorem 13.2 the following important theorem.
Theorem 13.4. Let f E Sk(r,19), g E Mk(r, rl) and let
L(f ®9, s) _ t(... , La(f ®9, s), ... ) be the vector column of the convolution Lfunctions attached to the singular cusps a for r with respect to the multiplier system X = ft Then ,C(f ®g, s) has meromorphic continuation to the whole splane, and the vector function
A(f ®g, s) = (47r)6I'(s+k 1)L(f ®g, s) satisfies the following functional equation: (13.33)
A(f (9 g, s) = 4,(s, z9i)A(f ®g,1  s)
where 4 (s, X) is the scattering matrix associated with the Eisenstein series on A, (IF, X).
One should be cautious when using Theorem 13.4 in general, because the scattering matrix is a meromorphic function of order up to 2. We don't have satisfactory estimates on (D(s, X) except for congruence groups, in which case t(s, X) can be expressed in terms of classical Dirichlet Lfunctions, and
consequently t(s, X) has order 1. By the integral representation (13.6) one sees that La(f (9 g, s) has at most simple poles at the poles of Ea(z, S, X). In the halfplane Re s 3 1/2 there can be at most a finite number of poles in the segment 1/2 < s 5 1; they are the poles of W.,, (s, X). There is no pole at s = 1 unless r) = 19 and f, g are not orthogonal. The residue is equal to (13.34)
ress_1 La(f ®g, s) =
47r
(f, g) L.
r(k) vol (L \>ii)
where
(f,g)k = J
\A[
y'f
(z)9(z)dpz.
13.5. The scattering matrix for ro(N) We shall give formulas as explicit as possible for the scattering matrix 4i(s, X) = X)) of the Eisenstein series in A3(ro(N), X), where X (mod N) is an even Dirichlet character. Recall that for a pair [a, b] of
13.5. The scattering matrix for ro(N)
241
singular cusps with respect to X the entry ,pnb(s, X) is given by the Dirichlet series (13.16) with coefficients (13.35)
Van C D a`);
Sab(0, 0; c) = (
(;
d)EB\aa'rab/B
here and hereafter, r = ro(N). To compute this sum we need an explicit description of the set aa , rab. Since cppb(s, X) depends only on the equivalence classes of cusps, we may stick to specific representatives. Recall that all cusps for ro(N) are represented by rational numbers
v with VIN, (u, v) = 1
(13.36)
and two such cusps are equivalent if they have equal denominator v and the numerators are congruent modulo (v, (see Proposition 2.6). However not v)(it is a property shared by the whole every cusp is singular with respect to X equivalence class).
Lemma 13.5. Suppose X (mod N) has conductor N* IN. Then the singular cusps for X of type (13.36) are exactly those with denominators vI N such
that (v, v) I N If X (mod N) is the trivial character, then every cusp is singular. If X (mod N) is primitive or N is squarefree, then the singular cusps are represented by
r = 1 with vw = N, (v,w) = 1,
(13.37)
v
so
of N. For such a cusp we take /a matrix (13.38)
Tp =
with a  6
(v
0 (mod w) and a  b
E SL2(Z)
1 (mod v). Then for a pair of such
cusps [a, a'] _ [1/v, 1/v'] we have Tn 1 rTn,
1(c
E SL2(Z) : (v,v')Ic,
(v,'w) d}
d) The cusp a = 1/v has width rn = to (see (2.29)), and a = TQ ( is its scaling matrix. Similarly for a'; therefore L
_
{(a/i?7i c w'io
b/
'1
dw/w' J
(w, v')a, (w, w')Ib (v, v') I c,
(v, w') Id
13. Convolution Lfunctions
242
Recall that N = vw = v'w'. Writing w = (w, v')(w, w') and w'= (w', w)(w', v), we show that the set oo troal consists of matrices of type P
_ A ww'/(w, w') B(w, w')/ ww'
(
D ww'/(w, w') )
C(v, v') ww'
where A, B, C, D are integers such that det p = 1, i.e. satisfying (w, V') (v, w) AD  (w, w') (v, v') BC = 1.
Now suppose X (mod N) is the trivial character. Then Saa, (0, 0; c) = cp((w, w') (V, V') C)
for c = (v, v') ww'C > 0, and by (3.16) we get 1 r(S  1/2)
Paa' (s)=r 2
r( s)
(ww/)9(v, v/)29
T\(w,w )(vfv )C)
(C,(w,v')(v,w'))=1
This series can be computed easily by factoring over primes. The following result was first established by D. Hejhal (by a slightly different computation; see [Hej]).
Proposition 13.6. Suppose X (mod N) is the trivial character and N is squarefree. Then for any pair of cusps [a, a'] = [1/v, 1/v'] with vw = v'w' _ N we have Paa' (s) = (P(S)paa' (s)
where W(s) is the scattering matrix for the modular group (see (13.19)(13.21)) and Paa'(s) = co((w,w)(v,v'))
fl (p2s PI(w,v')(v,w')
pIINN
D. Hejhal used the above results to arrange the scattering matrix for ro(N) as the tensor product
'(s) = w(s) ®PI N NP(s) where
Np(s) =
1
p1
psp1s
p2s1 p9pls
p1
).
This can be also written as NP(s) = .Mp(1  s)Mp(s)I, where
Mp(s) =
1
1S ) Cps
13.6. Functional equations for the convolution Lfunctions Hence
243
22s1 det 4)(s) = cp(s)h fl pp2s  1 PI N
where h = 2' is the rank of 4b(s), i.e. the number of inequivalent cusps. These formulas hold for N squarefree. If X (mod N) is primitive, all singular cusps are still of type (13.37) even if N is not necessarily squarefree. In this situation the former computations work but are somewhat longer, so we only formulate the results; they are taken from unpublished notes of N. Pitt (Rutgers, 1991).
Proposition 13.7. Suppose X (mod N) is a primitive even character and a, b singular cusps of type (13.37). Then cpb(s, X) vanishes unless a = 1//v, b  1/u, with vw = N, (v, w) = 1, in which case (13.39)
Xv(w)X,u(v)Nsir1/2r(s  1/2) L(2s  1, XwXv)
Wab(s, X) =
r(s)
L(2s, XwXv)
where X = XwXv is the unique factorization into characters to moduli w, v respectively.
The above result shows that if we arrange the singular cusps in a suitable order, then the scattering matrix is skew diagonal. One can write (13.39) in an invariant form by applying the functional equation for the Dirichlet Lfunction (take (12.7) for 2s 1 in place of s and XwXv in place of X) and factoring the Gauss sum T(XwXv) = Xw(v)Xv(W)T(Xw)T(Xv)
One obtains T(Xw)T(Xv)N13sir2s1
,Pab(s, X) =
In particular, for the pair of cusp a = 7 (13.40)
W000 (s, X) =
r(1  s)L(2  2s, XUXw) r(s)L(2s, xvXw)
oo and h = i
T(X)Nl3s7r2s1
0 we get
r(1  s)L(2  2s, X) r(s)L(2s, X)
13.6. Functional equations for the convolution Lfunctions We begin by considering the situation in which X = t9rl is a primitive character. Since (D(s, X) is skew diagonal, the vector functional equation (13.31) for the Eisenstein series reduces to the individual equations E. (z, s, x) = Wab(s, x) Et, (z,1  s, x)
13. Convolution Lfunctions
244
where b is the complementary cusp to a, i.e. ON = 1. Accordingly the functional equation (13.33) for the RankinSelberg Lfunctions reduces to (47r)sr(s
+ k  1)La(f ®9, s) = (Pab(s, X)(47r)s'r(k  s)Lb(f ®g,1  s)
provided X = i9rl is primitive. In particular, for cusps a  oo, b  0 we obtain
Theorem 13.8. Let r = ro(N), and let i9, r7 be two characters to modulus N such that i9(1) = r7(1) = (1)k where k, N are positive integers. Suppose that X = t9r7 is primitive of conductor N. Let f c Sk(r,19) and g E .Mk(r, r7). Put
A. (f ® s) = (21r)2sN 2'r(s)r(s + k  1)L(2s, X)L,,,(f (9 9, s) Ao(f ®
s) = (27r)2sN 2'r(s)r(s + k  1)L(2s, X)Lo(f ®g, s)
where L,,(f ®g, s) and Lo(f ®g, s) are the RankinSelberg convolution Lfunctions attached to cusps oo and 0 respectively (see (13.2), and note that oo, 0 are always singular cusps). Then we have the functional equation (13.41)
A.(f ®g, s) = e,Ao(f ®g,1  s)
where e), = T(X)N112 is the normalized Gauss sum. To achieve further simplifications we assume that all 19, 77, i977 are prim
itive of conductor N and that f, g are newforms. Therefore f , g are eigenfunctions of the involution operator W by Theorem 6.28. More precisely, by Theorem 6.29 we have (13.42)
Wf=
T(z9)N1/2aN(f) f
21A
where aN(f) = = (4ir)k1 f,.(N) is the Nth Fourier coefficient of f at cusp oo (up to displayed normalization). Hence we can express any Fourier coefficient at cusp 0 by the corresponding one at cusp oo; namely, we have (13.43)
fo(n) =
T('9)N1/2aN(f)f.(n)
A similar relation holds true between the coefficients of g. Combining both relations, we deduce that (13.44)
Lo(f 09,s) =
®g, s).
Inserting (13.44) into (13.41), we get a functional equation which connects the RanldnSelberg Lfunctions for the same cusp at oo (in this case we drop the subscript oo), namely (13.45)
A(f 0 g, s) =eA(f 0 g,1 s)
13.6. Functional equations for the convolution Lfunctions
245
where
e=T(t9n)T('t9)T(1))N3/2aN(f)aN(g).
(13.46)
Another interesting situation is for two newforms of distinct level, say f E Sk(ro(N1), X1) and g E Sk(Fo(N2), X2), where X1, X2 are primitive char
acters to conductors N1, N2 respectively such that X1(1) = X2(1) = (1)k. In addition suppose that (Ni, N2) = 1, so X = XIX2 is primitive to conductor N = N1N2 and the formula (13.41) applies. However. (13.43) reads
1/2aN,
fo(n) = T(Xi)N1 (f)f.(n) Consequently (13.45) holds true but with a different root number, namely s = T(XIX2)T(X1)T(X2)NlaN1 (f )aN2 (g). By the multiplication formula for Gauss sums the root number simplifies to (13.47)
E = X1(N2)X2(Nl)aN,(f)aN2(g)
The third situation we wish to analyze is for one cusp form f = g of weight k = 0 (mod 2) with respect to the modular group IF = SL2(Z). In this case we write a(n)nti21e(nz)
(13.48)
f(z)
L(f ®f, s) _
(13.49)
Ia(n)I2ns
so the integral representation (13.6) becomes (47r)1k
r(s + k  1)L(f ®f, s) = f\H ykI f(z)I2E(z, s)dpz.
The Eisenstein series has the Fourier expansion
9(s)E(z,s) = 9(s)ys + 0(1 
s)yl9
+ 4y1/2 (> (a/d)9112)Ky1/2(2irInIy) cos(27rnx) n._1 ad=,
where 0(s) = 7r3I(s)((2s) (see (13.15)(13.22)). Hence the complete Lfunction (13.50)
A(f x 1, s) = (27r)2sI (s)I (s + k  1)((2s)L(f 0 7, s)
13. Convolution Lfunctions
246
is holomorphic in s E C except for a simple pole at s = 1, and it satisfies the functional equation (13.51)
A(f x f,s)=A(f x f,1s).
The residue of L(f ®f , s) at s = 1 is (13.52)
res9=1L(f ®f , s) = 3 (r() llf 112.
The convolution Lfunction L(f 0I, s) is useful for estimating the Fourier coefficients of f. Applying standard complex integration methods, Rankin and Selberg derived the asymptotic formula (13.53)
E la(n) 12 = CX + O(X3/5) n<X
where c is the constant given by (13.52). Hence it follows that a(n) << n3/10 Compare this bound with (5.19).
13.7. Metaplectic Eisenstein series Next we shall be interested in the convolution Lfunctions for a cusp form f of weight k on the modular group (its Fourier expansion is (13.48)) against the standard theta function
0(z) =
e(n2z) E ro(4).
Such convolution yields the following zetafunction: 00
Z f(s) _
a(n)n9, 1
which has the integral representation (apply Proposition 13.1 for l = 1/2 and + a in place of s) ; (13.54)
(47r)"Sr' (s+ 2
1)
Zf(s) =
J
`H11(z)E(z, 2 + 4)ctµx
where h(z) = 2y^+3 f(z)B(z) and E(z,s) is the Eisenstein series for the group r = ro(4) at cusp oo given by (13.9). This is called the metaplectic Eisenstein series. The general theory of Selberg applies here, giving analytic
continuation of E(z, s) to the whole splane. In the halfplane Re s >, 1/2
13.7. Nletaplectic Eisenstein series
247
there is only a simple pole at s = 3/4 (whose residue is some kind of a theta function); therefore, by the integral representation (13.54) it follows that Z1(s) is holomorphic in Re s > 1/2 (including the line Re s = 1/2) except
possibly for a simple pole at s = 1. Actually we shall see later that s = 1 is a removable singularity (see the next section concerning the symmetric square Lfunction).
One can also deduce the above analytic properties of the metaplectic Eisenstein series directly from the Fourier expansion. We shall only indicate a few vital steps to observe the pole. First write E (z, s) = y s
where , =
2s l E 'Ed Cdl + EC 41c>0 (d,c)=1 c
z+ d/c K z + d/c
y
s
Iz + d/c12

1. Then split the summation into residue classes d (mod c), 2 class apply Poisson's formula: and for each
E
z + m/c
(z+n/c)K(
JZ+m/cl2)
m_d (mod c)
=
s
y
(n
o00
(IZ+Yt12)S
cd) f o (z+t)K
n n
e(nt)dt.
For n = 0 the integral is equal to /3(s)y18, where = / 7 /t+i)K (t2 + 1)sdt = X41sI'(2s  1)/I'(s  r)I'(s + n), s a() J (\t  Z'
and for n # 0 it is equal to Vs(nz)lnls1, where ()"
V$(z) _
L
zt
(izt,2Ye(t)dt..
Note that V3(z) is holomorphic in Re s > 1/2, and it decays exponentially for z = x + iy with jyj + oo. More precisely, V3(z) << elay as y > 00 (to see this bound, move the contour of integration up to Im t = 21ry). We have V3(z) = e(x)V8(iy) and Vs(iy) = i2'7rsr(s + K.)1Wn,s1/2(47ry)
if y > 0, where W, ,,(y) is the Whittaker function. Next, summing over the residue classes d (mod c), we obtain Gauss suns 9. (C)
> Ed (dc) e
d (mod c)
(nd c
13. Convolution Lfunctions
248
and summing these over c we form the Dirichlet series gn(cc2s
ln(S) _ 41c, c>0
Collecting the above results, we arrive at the Fourier expansion E(z, s) = y8 + (13.55) + E Inj8lln(s)Vs(nz) v(s)y'8
ni40
where v(s) = Q(s)lo(s). Further explicit computations are technically complicated. According to the general theory the poles of E(z, s) in Re s > 1/2 are inherited from those of v(s). Here it is easy to compute lo(s); we have go(c) = i2 cp(c) if c = b2, or else it vanishes, and hence
E
+ i
10(s) = 1
,p(62)b4s
=
(1 + i)((4s  2)
1)((4s
21b, b>0
Therefore v(s) = /3(s)lo(s) is holomorphic in Re s >, 1/2, except for a simple
pole at s = 3/4 with residue (1 + i)/'r(I  s)r(4 + k). The other Gauss sums gn(c) can be computed in terms of the Jacobi symbol Xn(c) = (n) if (c, n) = 1, and consequently ln(s) can be expressed in terms of the Dirichlet Lfunction L(2s  1/2, Xn), among other things. For n squarefree Xn is the trivial character, so the Fourier coefficient ln(s) has
a pole at s = 3/4.
13.8. Symmetric power Lfunctions The idea of RankinSelberg convolution provided inspiration for building many other Lfunctions. We shall close this book by discussing the simplest case of symmetric power Lfunctions. Throughout, r is the modular group, k is an even positive integer and f is a cusp form for r of weight k which is an eigenfunction of all the Hecke operators T, ,f = A(n)f,
n = 1, 2, ... .
We normalize f so that it has the Fourier expansion 00
f(z)
The Fourier coefficients
a(n)nkzle(nz).
a(n)nik,
satisfy the multiplication rule
a(?n)a(n) = E a(mn/d2). dI(m,n)
13.8. Symmetric power Lfunctions
249
The 1Iecke Lfunction associated with f has the Euler product 00 a(n)ns
Lf(s) = E
=
fl(1  a(p)p`
+
p
2s)1
n
I
whereas the complete Afunction (obtained by including a local factor at
p=00) A f (s) = iT81 1 s+
k
2 1
J Lf (s)
is an entire function, and it satisfies the functional equation A f(s) = ikA f(1  s). All these properties have been established in Section 7.2. We factor the Hecke polynomials of f into (13.56)
1  a(p)p' + p 2s = (1  appe)(1  Qpp 8)
where ap + ,Op = a(p) and apf3p = I. Recall that the Ramanujan conjecture (proved by Deligne) asserts that lapj = INpl = 1, i.e. ap = f3p. By (13.56) we get am+1  /4m+1
(13.57)
a(p") = n
p
=
ap  Qp
ap 7Qp. o
Next we reconsider the RankinSelberg convolution Lfunction no
L(f (9 f, s) = > a(n)2n8 By (13.57) one derives that L(f 0 f, s) has the Euler product of degree four
L(f ® f, s) = fJ(1 + ps)(1  an p8)' (1  p8)1 (1  l3Np 4)1 p
Here the first factor is somewhat different from the other three. Multiplying by C(2s), we obtain a more natural Euler product (still of degree four):
L(f x f, s) = C(2s)L(f ® f, s)
= fJ(1 p

apps)1(1
 p ")2(1

/app.v)I .
13. Convolution Lfunctions
250
The corresponding Afunction is defined by
A(f x f, s) = (27r)2sr(s)r(s + k  1)L(f x f, s). We have shown in Section 13.6 that A(f x f, s) is holomorphic in the whole splane except for a simple pole at s = 1; hence also L(f x f, s) is holomorphic
in the splane except for a simple pole at s = 1 with positive residue given by (13.52). Observe that the original SelbergRankin convolution L(f of, s) may have infinitely many poles on the line Re s = 1/4 at the zeros of ((2s). We also have the functional equation
A(f x f, s) = A(f x f,1 s). Closely related to L(f of, s) and L(f x f, s) are the following series: 00
(13.58)
a(n2)ns
Z1(s) _
and (13.59)
L (2)(S) = S(2s) Z f (s)
00
a(2) (n)n' t
where a(2)(n)
= E a(m2). m12 =n
By a(n)2 = 1: a(n2/d2)
On
it follows that (13.60)
((s) Zf (s) = L(f ® f, s)
and (13.61)
((s)L f)(s) = L(f x f, s).
These equations are quite interesting. First they yield meromorphic continuation of Zf(s) and L( )(s) to the whole splane. In the halfplane Re s >, 1/2 we already established (in the previous section) that Zf(s) is holomorphic except for a simple pole at s = 1. By virtue of this fact the equations (13.60)
and (13.61) imply that L(f 0 f, s) and L(f x f, s) are divisible by ((s) (i.e. each one has the zeros of ((s)) in Re s > 1/2 at any rate. Furthermore the
13.8. Symmetric power Lfunctions
251
simple pole of L(f 0 f, s) at s = 1 cancels out with that of ((s), so Z f(s) is holomorphic at s = 1 and does not vanish at this point. Precisely, we have (see (13.52)) (r(k)ti
11f112 > 0.
Zf(1) = 3
The same observation refers to Lf )(s), and we have k
Lf) (1)= 2([' k)
Ilfl12>0.
The fact that Zf(s) is regular at s = 1 implies, by virtue of (13.54), that the residue of the Eisenstein series E(z, s) at s = 3/4 is orthogonal to u2+d.f(z)6(z) for every cusp form f(z). Comparing the Euler products for S(s) and L(f x f, s), we deduce from (13.61) an Euler product for Lf ) (s), namely
Lj 1(s) = (1
(13.62)
2p.,) `1(1
 ap3pp`S)`1(1
(appS)1
P
This Euler product is called the symmetric square Lfunction. Combining the functional equations for c(s) and L(f x f, s), we deduce from (13.61) a functional equation for Lf 1(s). Before stating this equar tion, let us modify the relevant gamma factors by applying the duplication formula
r(s) = 7r1/22slr
\2/ r (S 2 1)
This yields
r(s + k  1) =
1122sF "' 2. (S + 2
1) r (s k) 2
.
Accordingly we associate with Lf 1(s) the complete Afunction 2
2
JJ
2
and deduce the following functional equation:
Ar(s)=Af)(1 S). (2)(S) Hence we conclude that Al is entire (holomorphic in s E C), and so is L()(s). These properties of the symmetric square Lfunction were first.
established by G. Shimura in 1973 [Sh1J.
13. Convolution Lfunctions
252
By obvious analogy to the preceding investigations one may get an idea how to construct Lfunctions of higher degree (associated with a newfor111 f). A somewhat simplified construction would be the series 00
Zf
n) (s)
_
a(n°n)ns
1
or the Euler product
P
7,( 1)(.4) = L; (s) and ?J2)(s) Zf(s). However, the right thing to consider is
for any m > 0. Note that Zf )(s) (13.63)
L() (s)
_
(]
apesQpps) 1
P 0!5j<»e
This product converges absolutely in Re s > 1; it is called the mth symmet
ric power Lfunction of f. In particular we have L f)(s) = ((s), Lt )(s) _ L f(s), and L(2) (s) as before. Note that either Zf)(s) or Pf'n)(s) approximate L(T) (s) reasonably well in the halfplane Re s > 1/2, in the sense that the ratio is given by an Euler product which converges absolutely in Re s > 1/2 (to see this, compare (13.63) with (13.57)). However, such a similarity is false in deeper ranges, for one can show that if m is large the series
Z( ) (s) does not admit analytic continuation over the line Re s = 0 (the abscissa of meromorphy) while L("n) (s) is expected to satisfy the following properties
Conjecture. The symmetric power Lfunctions are entire. In fact, the complete function defined by
Af )(s) = 7r(m+1)Z I
C3 + rj l L(n)(s)
ojItn
/J
where rj are suitable constants, is entire, and it satisfies the functional equation of type
Aje)(s)=CAf)(1s). This conjecture is part of a larger program of Langlands which is rooted deeply in various subjects of contemporary mathematics. Even before its completion this profound program of Langlands has had a significant impact
13.8. Symmetric power Lfunctions
253
on the direction of modern number theory (see the comments on Artin Lfunctions in Section 12.6).
The meromorphic continuation of L!) (s) over the whole splane and the predicted functional equations have been established for m <, 4. Almost nothing is known about the symmetric power Lfunctions of higher degree. There are fascinating applications of the available results, in particular those for the symmetric square Lfunction (cf. [DI] and [Iw3]).
Bibliography
[AL]
A. Atkin and J. Lehner, Hecke operators on Fo(m), Math. Ann. 185 (1970), 134160.
[Asa] [Boc]
T. Asai, On the Fourier coefficients of automorphic forms at various cusps and some applications to Rankin's convolution, J. Math. Soc. Japan 28 (1976), 4861. S. Bochner, Lectures on Fourier Integrals, Princeton University Press, Princeton,
NJ, 1959. (CH) R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 1, WileyInterscience, New York, 1961. [Dav) H. Davenport, Multiplicative Number Theory, Markham, Chicago, 1967.
W. Duke and H. Iwaniec, Estimates for coefficients of Lfunctions II, Proceedings of the Amalfi Conference on Analytic Number Theory in 1989, Universita di Salerno, 1992, pp. 7182. [Dul] W. Duke, Hyperbolic distribution problems and halfintegral weight Maass forms, Invent. Math. 92 (1988), 7390. [Du2] W. Duke, The dimension of the space of cusp forms of weight one, Internat. Math. Res. Notices 2 (1995), 99109. [GR] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, Academic Press, London, 1965. [Hell E. Hecke, Uber eine neue Art von Zetafunktionen, Math. Zeit. 6 (1920), 1151. (He2) E. Hecke, Analytische Funktionen and algebraische Zahlen, Abhandl. Math. Sem. Hamburg, vol. 3, 1924, pp. 213236 (see his Werke, p.381). [Hej] D. A. Hejhal, The Selberg Trace Formula for PSL(2,R), Lecture Notes in Math., [DI)
(HT]
(Iw1)
vol. 1001, Springer, 1983. G. H. Hardy and E. C. Titchmarsh, Solutions of some integral equations considered by Bateman, Kapteyn, Littlewood, and Milne, Proc. London Math. Soc. 23 (1924), 126. H. Iwaniec, Fourier coefficients of modular forms of half integral weight, Invent. Math. 87 (1987), 385 401.
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11w2]
(lw3]
H. Iwaniec, Introduction to the Spectral Theory of Automorphic Forms, Bibl. Rev. Mat. Iber., Madrid, 1995. H. Iwaniec, The lowest eigenvalue for congruence groups, in Topics in Geometry (memorial volume of Joseph D'Atri) (S. Gindikin, ed.), Birkhi user, 1996, pp. 203212.
[1w4]
H. Iwaniec, Mean values for Fourier coefficients of cusp forms and sums of Kloosterman sums, Journme Arithmetiques (1980); London Math. Soc. Lecture Note Ser., vol. 56, Cambridge University Press, Cambridge, 1982, pp. 306321.
H. D. Kloosterman, On the representation of numbers in the form ax2 + bye + cz2 + dt2, Acta Math. 49 (1926), 407464. [Lan] S. Lang, Introduction to Modular Forms, SpringerVerlag, 1976. W. Li, Newforms and functional equations, Math. Ann. 212 (1975), 285315. [Li] [Lom] G. A. Lomadze, On the number of representations of positive integers as the sum of nine squares, Acta Arith. 68 (1994). 245253. (R.ussian) [Mall A. V. Malyshev, On formulas for the number of representations of integers by positive quadratic forms, in Current Problems of Analytic Number Theory, Nauka i Tech., Minsk, 1974, pp. 119136. (Russian) (Klo]
T. Miyake, Modular Forms, SpringerVerlag, 1989. H. Petersson, Zur analytischen Theorie der Crenzk eisgruppen. IV, Math. Ann. 115 (1938), 2367; 175204; 518572; 670709; Math. Z. 44 (1938), 127155. [Ran] R. A. Rankin, Modular Forms and Functions, Cambridge University Press, Cambridge, 1977. H. Salie, Uber die Kloostermanschen Summen S(u, v; q), Math. Z. 34 (1931), 91[Sall [Miy]
[Pet]
109. (Sar]
P. Sarnak, Some Applications of Modular Forms, Cambridge Univ. Press, Cam
[Sell
bridge, 1990. A. Selberg, On the estimation of Fourier coefficients of modular forms, Proc. Symp.
[Se2]
(Shl]
Pure Math., vol. 8, Amer. Math. Soc., Providence, RI, 1965, pp. 115. A. Selberg, Collected Papers, Vol. I, Springer, 1989. G. Shimura, On modular forms of half integral weight, Ann. Math. 97 (1973), 440481.
[Sh2] [Sie]
(Ser]
(Ter) [Wei]
G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton Univ. Press, Princeton, NJ, 1971. C. L. Siegel, Lectures on Quadratic Forms, Tata Institute, Bombay, 1967. JP. Serre, Modular forms of weight one and Cairns representations, in Algebraic Number Fields (A. Frohlich, ed.), Academic Press, 1977, pp. 193268. A. Terras, Harmonic Analysis on Symmetric Spaces and Applications, Vol. I, SpringerVerlag, New York, 1985. A. Weil, Uber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 168 (1967), 149156.
Index absolute inner product, 6.5 adjoins operator, 6.4 ambiguous class, 12.5 angle of a Kloosterman sum, 4.4 Artin conjecture. 12.6 Artin Lfunction, 12.6 automorphic form, 2.7 automorphs of a quadratic form, 11.2 Bernoulli numbers. 1.3 Bgroup of integral translations, 2.5 chart, 2.3
circle method of Kloosterman, 11.4 class group, 12.2 class group Lfunction, 12.4 complete Lfunction, 7.2 complex conjugation operator K, 6.7 congruence group, 2.4 congruent number, 8.2 congruent theta function, 10.1 conjugacy class, 2.2 consistency condition, 2.6 converse theorem, 7.4 convolution Lfunctions, 13.1 criterion for cusp forms, 7.4 cusp, 2.3 cusp form, 2.7
cusp forms for the modular group, 1.6 cuspidal zone, 2.3 decomposition group, 12.6 Dedekind sum, 2.8 Dedekind zetafunction, 12.2 deformation factor, 2.1 determinant of a quadratic form, 11.2 diagonal symbol, 5.5 different of a field, 12.2 differential operator of Euler, 9.2 Dirichlet class number formula, 11.2 discontinuous group, 2.3 discrete group, 2.3 discriminant function, 1.2 discriminant of a field, 12.2 discriminant of a quadratic form, 11.2 discriminant of an elliptic curve, 8.1 double cosets, 2.5
cigenvalue rl of W, 6.7 Eisenstein series E(z,Q), 11.3 Eisenstein series for the modular group, 1.3 Eisenstein series of weight zero, 13.2 Eisenstein series with a multiplier, 3.2
ellipsoid KA, 9.1 elliptic curve, 8.1 elliptic function, 1.2 elliptic. hvperboic. parabolic motions. 2.2 equidistribution, 11.6 equivalence class of forms, 11.2 equivalent quadratic forms, 11.2 etafunction, 2.8 Euler product for llecke Lfunction, 6.8 even quadratic form A[xl, 10.1
factor system of weight k, 2.6 courier coefficient at a cusp, 2.7 courier expansion at a cusp, 2.7 courier transform, 1.1 fractional part, 1.1 Flicke involution W, 6.7 Flobenius aut.omorphism, 12.6 Frobenius conjugacy class, 12.6 Fuchsian group, 2.3 flichsian group of the first kind, 2.3 functional equation for Hecke Lfunction, 7.2 fundamental domain, 2.3 fundamental units, 12.2 gamma function, 7.1 Gauss sum, 4.3 Gauss sun
12.2
generating function of Poincare series, 3.1 genus characters, 12.5 genus group, 12.5 genus mass, 11.2 genus of quadratic forius, 11.2 geodesics, 2.1 Crossencharacter, 12.2
group of fractional ideals, 12.2
harmonic function, 9.2 harmonic polynomial, 9.2 HasseWeil Lfunction, 8.1 Hecke congruence group, 2.4 Hecke eigencuspform, 6.6 257
Index
258
Hecke eigenform, 6.4 Hecke 1function, 7.2 Hecke operators, 6.2 Hilbert space of cusp forms, 3.3
homogenuous space R', 9.3 hyperbolic isometrics, 2.1 hyperbolic measure, 2.1 hyperbolic plane, 2.1
inertia group, 12.6 inner product, 3.3 integral basis, 12.2 integral representation for Lfunction, 7.2 isometric circle, 2.1
isotropic vector, 9.1
Jacobi function 8(z, v), 10.1 Jacobi inversion formula, 10.2 Jacobi sum, 8.3
Jacobi symbol extended, 2.8 Jacobian of a diffeomorphism, 9.1 jfunction, 2.1 jinvariant function, 1.3 Kloostorman sum with a multiplier, 3.2 Kronecker symbol, 12.3
Laplace operator A, 13.2 Laplace operator AA, 9.1 lattice, 1.2 Legendre function P,m(z), 9.4 Legendre polynomial 9.4
level of a congruence group, 2.4 Lfunction with Grossencharacter, 12.2 linear fractional transformations, 1.3 local densities 69(n,Q),6o (n, Q), 11.2 metaplectic Eisenstein series, 13.7
modular function of weight k, 1.3 modular group, 1.3
multiplicityone principle, 6.6 multiplier system of weight k, 2.6 Neumann series, 5.5 newform, 6.6 normal operator, 6.4 normal polygon, 2.3 number field, 12.2
principal genus, 11.2, 12.5
principal ideals, 12.2
quadratic form of level one, 11.3 quadratic form of Liouville type, 11.3 quadratic form of Mordell type, 11.3 quadratic form of Weil type, 11.3 quadratic form of WeilEichler type, 11.3 quaternion group, 2.3 Ramamijan sum, 6.7 Ramamujan rfunction, 1.4 ramified primes, 12.2 RankinSelberg integrals, 13.1 ray class group, 12.2 real and complex embeddings, 12.2 regulator, 12.2 representation number r(n,Q), 11.2 residue class field, 12.6 Riemann surface, 2.3 Riemann zetafunction, 7.1 ring of integers, 12.2 root number, 8.1
Sali6 sum, 4.6 SatoTale measure, 4.4 scaling matrix, 2.3 scattering matrix, 13.2 ShimuraTaniyama conjecture, 8.1 Siegel mass formula, 11.2 sifting operator Sv, 6.7 signature, 2.3 singular cusp, 2.7
singular series, 11.5 slash operator, 2.6
space of modular forms of weight k, 1.6 special orthogonal group SO(n), 9.3 sphere 9.3 spherical function, 9.2 spherical function Y(O, sp), 9.4 spherical polar coordinates, 9.3 stabilty group, 2.3 standard fundamental polygon for the modular group, 1.5 standard polygon, 2.3 Stokes' theorem, 9.2
surface Laplacian 0', 9.3
orbit of a point, 2.3 orthogonal group O(n), 9.3
symmetric power Lfunction, 13.8 symmetric square Lfunction, 13.8
parabolic vertex, 2.3 periodic function, 1.1 Poincark differential, 2.1 Poincar6 series, 3.1 Poisson summation formula, 1.1 positive definite quadratic form, 9.1 primary integer, 8.3
Tchebyshev polynomials, 4.4
prime discriminant, 12.5 principal congruence group, 2.4
theta function 8(z), 10.1 theta function 8x(z), 10.5 theta multiplier, 2.8 thetafunction, 2.8 total mass, 11.2 trace of motion, 2.2 tridiagonal metrix, 10.4 twisted sutomorphic form, 7.3
Index
unimodular transformation, 1.3 unit group, 12.2 upper halfplane, 1.3
volume element, 9.1
Weierstrass equation, 1.2 Weierstrass equation for an elliptic curve, 8.1
Weierstrass pfunction, 1.2
Weil's bound for Kloosterman sums, 4.3 Whittaker function, 13.7 width of a cusp, 2.4 zonal spherical function, 9.4
259
ISBN 0821807773