Seminar on Complex Multiplication

Lecture Notes in Mathematics An informal series of special lectures, seminars and reports on mathematical topics Edited by A. Dold, Heidelberg and B. Eckmann, ZLirich

21 A. Borel. S. Chowla. C.S. Herz K. Iwasawa. J-P. Serre

Seminar on Complex Multiplication Seminar held at the Institute for Advanced Study, Princeton, N.J., 1957-58

1966

Springer-Verlag- Berlin-Heidelberg. New York

All rights, especially that of translation into foreign languages, reserved. It is also forbidden to reproduce this book, either whole or in part, by photomechanical means (photostat, microf'dm and/or microcard)or by other procedure without written permission from Springer Verlag. @ by Springer-Verlag Berlin 9 Heidelberg 1966. Library of Congress Catalog Card Numbe166--27636. Printed in Germany. Tide No. 7341

TABLE OF CONTENTS

I.

J-P. SERRE

:

Statement of results

II.

J-P. SERRE

:

Modular forms

III.

A. BOREL

:

Class Invariants I

IV.

A. BOP.EL

:

Class invariants II

K. IWASAWA

:

Class fields

S. CHOWLA

:

Remarks on class-lnvariants

VII.

C. S. H E R Z

:

Construction of class fields

VIII.

C. S. HERZ

:

Computation of singular j-invariants

V. VI.

and related topics

Seminar on complex multiplication

IAS, 1957-58 I-I

I

STATEMENT OF RESULTS

(J=P. Serre, Oct. 16, 1957)

w

The notion of complex multiplication. Let X be an elliptic curve.

As a complex Lie group, it is the

quotient of the complex plane ~ by a lattice ~, spanned by two periods ~'

~2' and since X is isomorphic to the curve defined by the periods

Z ~ l , z ~ 2 for any non zero z g C we may assume P t o

be spanned by 1 and

~ , where T has a positive imaginar# part. An endomorphism of X may be identified with an endomorphism of its universal covering ~mapping

~into itself; it is therefore the multi-

plication by a complex number z such that z, z~" g ~ .

The endomorphisms

of X form a ring A(X), which always contains the integers ~ , (the "trivial endomorphisms").

The other ones (if any) are given by complex numbers and

are called complex mult__~plications.

If A(X) # ~ , the curve X is said to

admit complex multiplications. "In general", X has no complex multiplication. that z defines a non trivial endomorphism of X. z =a

+bT

, z~

= c +d~

,

In fact, assume

Then (a,b,c,d integers, b # O) ,

whence a~

and T m u s t

+ b ~ 2 = c + dl"

belong to an imaginary quadratic field, say K; moreover z belongs

to the ring of integers ~(K) of K since it is in K and defines an endomorphism of a ~-module of finite rank, namely ~ .

Therefore, A(X) is an order

of K, (subring of ~(K) containing ~ and which has rank 2 as a ~-module);

1-2

one gets in this way all orders of all quadratic imaginary fields (if R is such an order, take for X a curve with lattice of periods R; since I ~ R, zR c R if and only if z ~ R, whence A(X) = R). Assume that A(X) = o(K), and that ~ c K.

Then ~ i s an ideal of

K, and conversely any ideal of K gives rise to a curve X such that A(X) = o(K).

Two such curves are isomorphic if and only if the corresponding

ideals are homothetic, i.e. belong to the s a m Let J be the modular function.

ideal class~

For the curve with normal equation

y2 = ~.x S _ g2 "x . g3 its value is '.',

,

Two elliptic curves are isomorphic over an algebraically closed field if and only if their modular invariants are equal. function on the ideal classes _~, ..., ~

By the above, j defines a

of K; the numbers J(k_i) are

"singular values" of j, and are called the class invariants of K; they are pairwise different, and have proved to be of fundamental importance in the study of the abelian extensions of K, to which w e now turn.

~2.

Unramified abelian extensions of an imaginary quadratic field. It is a classical result of Kronecker that every abelian extension

(i.e. normal extension with commutative Galois group) of the field ~ of rational numbers is contained in a field of roots of unity.

Thus, so to

say, certain values of the exponential function generate the maximal abelian extension of ~.

Such an "explicit" construction is also possible

for an imaginary quadratic field.

One has to use the class invariants

and also the values of a certain function, related to the Neierstrass

1-3 p-function (see ~). multiplication".

This theory is essentially what is called "complex

We shall first deal with unramified extensions.

The re-

sults pertaining to this case may be embodied in the following three theorems, where K is an imaginary quadratic field, ~ ,

(I < i < h), its

ideal classes. THEOREM I.

Th_~eclass invariants j (~) are al~ebraiq integers.

[Let us remark in passing that there is a sort of converse to Theorem I.

Namely, C. L. Siegel (Transcendental numbers, Annals of Math.

Studies 16, Princeton, 19~9, pp. 98-99) has deduced from certain results of Schneider that if z is an algebraic number in the upper half-plane not belonging to a quadratic imaginary field, then j(z) is transcendental.] THEOREM II.

K(J(~)) i_~sindependent o_~fi, (I <= i =< h), and is

the maximal unramified abelian extension of K. (Unramified means that every prime ideal of K decomposes in a product of distinct prime ideals with exponents i.) By class field theory, it is known that the maximal unramified abelian extension of K (Hilbert's "absolute class field") exists and that its Galois group G K is canonically isomorphic to the group CK of ideal classes; the next theorem describes how it operates on the j (ki)THEOREN III.

Let k g CK and let ~'k g GK be its image by the

iso_morPhism of class field theory.

Then -

9

m

For the proofs of the above theorems, we shall follow the method of the first part of Hasse's paper quoted in ~ , which presupposes classfield theory.

Nore specifically, Hasse establishes by function theoretical

arguments Theorem I and the congruence

1-4 -- J(k)N2

(i)

(~ prime ideal of K, ~ ideal class, N~ absolute norm of ~), for almost all (i.e. all but a finite number of) prime ideals of K having absolute degree one (i.e. N~ = p with p prime) with respect to ~.

The validity of (1) for

all ~ and Theorems II, III will then follow by class field theory. w

Al~ebraic interpretation. For any complex number j, there is an elliptic curve X defined

over ~(j), with invariant J; it may for instance be given by the equation y 2 - ~x 3 - h ( x + I )

,

(h = 27j . (j

26.33) -1) 9

If in particular j = j(k), with K, k as before, then K(j) is the smallest field of definition for all elements of A(X).

From this, it follows

easily that the numbers j (~) are algebraic and that L = K(j (~)) is independent from i. Moreover, given a prime ideal p of K, there is an algebraic procedure to obtain from X a curve, call it Xp, with invariant j (~-l.k) also defined over K(j). Deu~ing's method relies on the notion of reduction of X modulo a prime ideal ~ of L.

This means that we consider the curve with the

same equation as X, but the coefficients being reduced rood q.

It may be

shown that for almost all ~, this is possible and leads to an elliptic curve X which does not depend on the particular equation chosen to define X.

The curve X is then defined over a field of characteristic p, where p

is the prime number contained in q. To any algebraic variety V defined over a field of characteristic p # O, one can associate its "p-th power" V p.

Roughly speaking, V p is

I-5 obtained by raising to the p-th power all coordinates of the points in any affine model of V.

In particular we can consider xq for any q = pS

(s positive integer). With these notations, Deuring proves by a simple argument that the equality

holds for almost all ~ of degree I, whence the congruence (I). So far, it has not been possible to prove along these lines that the class invariants are in_te~al numbers. the analogue

~.

Deuring's proof for this fact is

with formal power series of the classical argument~

Ramified extensions. Let as before K - Q ( ~ ) ,

(d positive square free integer), be a

quadratic imaginary field, and X an elliptic curve with invariant j = j(k), where k is an ideal class of K.

Its group of automorphisms Aut X is the

group of units of A(X), i.e. of e(K). (resp. ~

This group is cyclic of order 2

resp. 6) when d # I, 3 (resp. d = I, resp. d = 3). The quotient X' of X by Aut X is an algebraic curve (of genus

zero), defined over K(j). Since X is a group variety, defined over K(j), its points of order n will be algebraic and so will be their images in X'.

We can now

state the main theorem of complex multiplication for ramified extensions : THEOREM IV.

Let K be an imaginary quadratic field, j a class

invariant of K, and X an elliptic curve with modular invariant j defined over L = K(j).

The maximal abelian extension of K can then be obtained

by adjoining t_~oL the coordinates i_2nX' = X/Aut X o_~fall points of order n of X (n = I, 2, ...).

I-6 In order to express Theorem IV analytically, we introduce the function T(u~ co l, cO 2) = (-l)e/2pe/2(u; cO1, do2)g(e)(cOl , ~ 2 ) where e = order Aut X, p(uj ODl, oD2) is the Weierstrass function,

g(2) = 27.35 g2.g3/A

g(3) 28.3 g /n .

g(6) , 29.36.g3/•

.

We have then: THEOREM IV'. Let K b_~ea_~nimaginary ~uadratic field, j a class invariant of K, Col' ~ 2 a basis for some ideal of K.

Then the maximal

abelian extension of K may be obtained by adjoining t_2 L = K(j) the a ~l+b co2 numbers ~ ( n ; Q)I' 032)' (a,b,n~Z=, n :> 0). The maximal abelian extension of K can also be obtained by adjoining to K the roots of unity, the values j(z) of the modular function for all z ~ K* having positive imaginary part, and square roots of elements in the field thus obtained.

For this and its relation to the

so-called Kronecker Jugendtraum, see Hasse's KlassenkSrper Bericht, Jahr. Ber. ~5.

D.M.V. 35,

1-55

(1926), w

Bibliographical notes. We content ourselves with some brief indications, without making

any attempt towards completeness.

As to the 19-th century literature,

we just quote: KRONECKER, Complete Works, Vol. IV, ~I, No. i~. The first two systematic and detailed accounts are to be found in.. H. WEBER, "Algebra", Band III, 1908,

I-7 R. FUETER, "Vorlesungen ~ber die singul~ren Moduls und die komplexe multiplication der elliptischen Funktionen"~ I (1922), II (1927), Teubner. Weber's book contains most of the essential results.

However,

although Weber already introduces and obtains several properties of the function ~ , both he and Fueter have to use other, more complicated, functions to generate the maximal abelian extension.

That this could be

performed by means of i~ only was first shown by Hasse: H. HASSE, "Neue Begr~ndung der komplexen Multiplikation", Teil I, Crelle Journal 157, 115-139 (1927), Teil II, ibid. 165, 64-88 (1931). Teil II and the above mentioned books follow the analytical method.

Hasse's Teil I combines analysis and class field theory. The purely algebraic approach was initiated and carried out

by M. Deuring.

See notably:

M. DEbrIEf, "Algebraische Begr~ndung tier komplexen Multiplikation", Abh. Math. Sere. Hamburg, 16, 32-~7 (1927). M. DEURING, "Die Struktur der elliptischen Funktionenk6rper und die ElassenkSrper der imagin~r-quadratischen ESrper", Math. Annalen 12~, 393-~26 (1952). It wss clear from the outset that a main obstacle to a generalization of Deuring's methods to higher dimensional abelian varieties was the lack of a good theory for reduction mod p in algebraic geometry.

This

was recently supplied by Shimura (Amer. Jour. Math. 7_~7,13~-176 (1955)), and was applied to higher dimensional extensions of complex multiplication by Shimura, Taniyama, Weil.

See:

G. SHIMURA, "On complex multiplications", Tokyo Symposium on algebraic number theory (1955), 23-30.

I-8 Y. TANIYAMA, "Jacobian varieties and number fields", ibid., 31-~5. A. WELL, "On the theory of complex multiplication", ibid., 9-22. The two-dimensional case had been considered long ago by Hecke (following a suggestion of Hilbert), using analytical methods. See: E. HECKE, "H~here Modulfunktionen und ihre Anwendung auf die Zahlentheorie, Math. Ann., 71, 1-37 (1912). E. HECEE, "Uber die Konstruktion relative Abelscher ZahlkSrper durch Modulfun~ionen yon zwei Variabeln", Math. Ann., 7.~, ~65-510 (1913). [(Added in 1965).

The Shimura-Taniyama-Weil theory has been

published: G. SHIMURA and Y. TANIYAMA, "Complex multiplication of abelian varieties and its applications to number theory", Publ. Math. Soc. Japan, 6, (1961). A systematic exposition of the analytic method is given in: M. DEURING, "Die KlassenkBrper der komplexen Multiplikation", Enz. Math. Wiss., Band I-2, Heft 10, Teil II. For further results, see: K. RAMACHANDRA, "Some applications of Kronecker's limit formulas", Annals of Maths., 80, 104-148 (1964).]

Seminar on complex multiplication

IAS, 1957-58 II-i

II (J-P.

w

MODULAR FORMS

Serre, Oct. 23 and 30, 1957)

The modular groupo Let E c C= be the upper half plane I(w) > O; the modular group

G is the group of automorphisms of E of the form: aw + b cw+d

w---~

(a,b,c,d 9 Z,

ad - bc = l)e

This group is the factor group of SL(2,Z) by its center ~ + l lo Let now r be a lattice in C=; we can choose two generators Wl, w 2 of ~

such that w I/%

of SL(2,Z)o

w 2 ~ O; they are determined up to a transformation

If we then put w = Wl/W2, we have w e E, and the orbit of w

under G does not depend on the choice of Wl, w2e

Two lattices ~

and

~ '

correspond to the same orbit if and only if they are homothetic, i.e. if the elliptic curves C ~

and ~ I

are isomorphic.

orbit Gw corresponds to some lattice (for instance, PROPOSITION I.

Conversely, every ~

= Z + wZ=). Hence:

The isomorphism classes of elliptic c1~rves are

in one-to-one correspondence with the orbits of G i_~nE. We denote by X the set E/G of the orbits of G in E.

Our first

task is to find a well-behaved set of representatives for X in E.

w

Fundamental domain for the modul~r Eroup. Let T be the translation w

symmetry" w

~

9w +

" wi__. ' one has TpS e G and S 2

i, and S the "inversion= 1.

I1-2

orbit of G meets D.

Moreover, two distinct points w and w' o_~fD are e~ui-

valent under G if and only if either w' = T'+lw, Rw' = ~ ~ o_~rw' = Sw, lwl = I. The set D is called a "fundamental domain" of the modular group G.

D

PROPOSITION 3.

The group G is generated by S and

PROPOSITION ~.

The stability group of p = e 2~i/3

(resp. of i) is of order 3 (resp. of order 2), and is generated by Q = ST (resp. by S).

Cnversely,

of E with a non trivial stability Eroup is p

under G to either p

any point

e~uivalent

or i.

These three propositions will be proved at the same Y

.#

I

time.

If w' - A w

aw+b = cw + d '~ a simple computation shows

that:

(i) I(w') = I(w)llcw + dl 2. For a fixed w 9

the set of all cw + d (c,d 9 ~) is discrete;

hence, the l(w') have no non-zero accumulation point.

Let then G' be the

subgroup of G generated by S and T, and let G'w be any orbit of G' in E. By the above 9 we can assume that l(w) is maximum on G'w; formula (i)9 applied with A = S, gives lwl _> I.

On the other hand, IR(TnW)I < ~ for a

suitable n 9 Z; since T does not change imaginary parts, I(TnW) = l(w), and therefore Tnw 9 D.

Thus, every orbit of G' meets D.

Let now w and w' = A w I(w') > I(w).

( A e G, A #

I) be points of D, with

Formula (I) then yields Icw + d I < I, and, since we can

assume c ~ O~ it follows that c = O, i. a = d = +- I, w t - w + b~ w h e n c e

If c = O~ then

b - ~ 1 a n d R w = ~ ~e

If c = I, t h e n w e

To

11-3

must have lw I - i, d = O, unless w is equal %0 p d = 0,I or d = 0,-i.

or p +

I, in which case

The formnla d = 0 gSves w t = a - i . Tasw.

Since

Sw e D, the first part of the discussion gives a = O, except when R(Sw) = -+ 9, i~

w =#

similarly for w = p + w'

=

a

-

i/(~

io

or p +

19 w h e n w

=p,

one can take a = -I, and,

Finally, the formulas w = p ,

+i) = a + p ,

d = I imply

whence a = O,i.

Propositions 2 and 4 follow readily from this discussion. Let now A 9 G,

and let choose a point w in the interior of D;

the orbit of Aw under G' meets D. BAw 9 D.

Hence, there exists B 9 G I such that

By propositions 2 and h, we have BA = i, hence A e Gw~ and pro-

position 3 is proved. Remarks. I)

It is possible to prove (for instance, by topological argu-

ments) that the relations between S and T are generated by (ST) 3 = I. Hence 9 the modular group G is isomorphic to the free product of a c[clic group of order 2 (correspond~n~ to S) and a cycl~c ~roup of order 3 (corresponding to ST). 2)

Let Q(xpy) - Ax 2 + Bxy + Cy 2 be a positive definite binary

quadratic form, with real coefficients.

Q(x,y) for a suitable choice of Wl, w2~

Such a form can be written:

-I=% * Y"r212, Applying proposition 2 to w = Wl/W2,

one then obtains the existence of a form Q' = A'x 2 + B'xy + C,y 2 which is equivalent to Q under the group SL(2,Z=), and verifies the inequalities:

A' >_c'>_. IB'I [This "reduction n process was already given in Gauss Is Disquisitiones arithmeticaeo ]

w176 Anal~tic structure and co~actific~t~on of X = E/Go We will first prove that the group G is "discontinuous" in E: PROPOSITION 5. (i) A 9G

Let K be a compact subset of E.

Then:

There exists a m~mber N s~cb that l(Aw) < ~ for e v e ~

and w e K. (ii) For ever~ compact subset K' of E, the set of A ~ G such

that AK ~

K' # ~ is f~niteo [ i~

3ira. AS( = | ].

Statement (i) follows from formula (i) and the fact that Inf. Icw + dl > O, when w runs through K, and c,d run through all pairs of relatively prime integers. Statement (ii) does in fact hold for every discrete subgroup of SL(2,R)/~ + I~ o

In the case of the modular group G, it can be checked in

the following way: Formula (I) shows that the number of pairs (c,d) associated with transformations A 9 G such that ~, ~

AK ~

K' # ~ is finite.

e G have the same (c,d) if and only if ~

Since T n ~ K tends to co with n, one has T n ~ K

~

But two elements

= TnA2 with some n e K' # ~ only for a finite

number of values of n, and (ii) follows immed~tel~. CC~OLLARY I.

The factor space X = E/G is Hausdorff (and hence

locall~ c o , act). This is a formal consequence of (ii): Let w and w t in E be inequivalent

under G.

Since Aw'

~ co,

there exists a compact neighbourhood U of w such that Aw' ~ U for any A q G.

Let V be a compact neighbourhood of w' and N c G be the set of all

A such that AV ~

U # ~;

by (ii), the set N is finite.

For any A e N,

let W A be ~ aneighbourhood of Aw I which does not meet U, and put U' = V N

A~ e ~ A'~A;

the set U' is a neighbourhood of w'.

If A e N~ one

II-5

has AU' c W A hence AU' N AU w ~

U = ~o

U = ~;

if A ~ N, one has AU' c AV hence again

This means that GU and GU' are disjoint saturated neighbour-

hoods of the orbits Gw and Qw~ 9 q.eodo COROLLARY 2.

The canonic~l pro~ect~on of the fundamental domain

D onto X is proper. AK

If K is a compact subset of E, one has to show that B = a ~ c is compact~

Property (ii) shows that

U

AK is locally finite, hence closed;

on the other hand, property (i) shows that l(w) is bounded for w 9 Bo

The

set B, being closed and bounded is therefore compact. COROLLARY 3.

Let R be the e~uivalence relation ind~3ced on D by the

e~uivalence under G (tbe re.latJon R h~s been gi.v~n exp]icitly in prop~ 2)~ The canonical pro~ection D The map D/R (follows from cor. 2).

9X

induces an. homeomorphism of D/R onto X~

~ X is bijective (prop.2), continuous, and proper Since both D/R and X are locally compact, it is a

homeomorphism. After these preliminaries, we define an analytic structure on X in the following way: Let p : E on

: X denote the canonical projection.

an open set U of X is said to be holomorphic if

on p-l(u)~

A continuous function f fop

is holomorphic

One checks easily that the axioms of a complex analytic structure

are verified: if P i E is a point of order e (i.e. the stability greup Gp of P is of order e), we can find a local parameter Zp around P such that

%

operates on Zp by multiplication by e-th roots of unity, and ( ~ ) e is a

local parameter around p(P) . X [one can take for instance Zp = (w - P ) I ( w

- P)].

Let now X = X U

co

be the one point compactification of X l

A

we now want to extend to X the analytic structure of X.

Let ~

be the half

D

11-6

plane l(w) > i; it follows from formula (i) that the equivalence relation induced on E 1 by G is given by the translations T n. One has therefore p(~)

~

~ / T ; but the mapping w

) q = e 2~iw is an analytic isomorphism

of EI/T onto some open disk (minus the origin); putting q(oo ) = O, we then extend q to p ( ~ )

~J { o o ~

and take this function as a local parameter

A

around co on X.

This gives the desired extension.

A meromorphic function

A

on X may therefore be defined as a meromorphic function on E, inv~riant Imder G, admittin~ a power serles erpansion Fn> k anqn which converges aro~nd q = O. Such a function is called a modular function. PROPOSITION 6.

The space ~ is anal~tically ~somorphic to the

#%

sphere S 2 - C== By cor. 3 to prop. 3, X is homeomorphic to D/R, i.e. to a plane. The Riem~nn surface X is therefore homeomorphic to S=2, and, as is well known, this implies that it is analytically isomorphic to S=2. A

If ~

: X

~ C= is an analytic isomorphism, event modular function

is a rational function of ~

.

Such a ~

is determined up to an analytic

automorphism of ~, we normalize it by asking that it maps co and

p (~)

on-

to co and O respectively, and has residue equal to i at co (when expanded in a power series of q).

This particular choice of ~

will be referred to as

"the" modular function, and be denoted by j. Note.

Proposition 6 means that X is analytically isomorphic to C.

This can

also be proved by using the fact that the " h ~ q funddamen%al domain" J

D+ = { w

[ O
<~,

lwi > I ~

is ~nalytically isomorphic to a

and applying Schwarz's symmetry prlnciple.

half plane,

II-7

~4o

Modular functions of weight k. A (meromorphic) modular function of weight k (k 9 ~) is a meromor-

phic function h on E such that~ (2)

h(A.w) = (cw + d) 2k h(w)

and which admits a Laurent expansion in q = 9

(A a 0), 2~iw

with a finite number of

negative exponents~ h = Zn~ k anqn. m

A function of weight 0 is thus a modular function in the sense of 33~

A modular function of weight k which is holomorphic in E and at

is called a modular form~ Equation (2) is equivalent to the fact that the function of Wl, w 2 defined by:

h(Wl,W2) -

2k h(Wl/W2)

is invariant under the group SL(2,Z); note that f(wl,w2) is homogeneous of degree -2k. taw+b~ = dw/(cw+d) 2, one can also express (2) by saying that Since d ~cw~) the differential form of degree k

OJ = h dw k is invariant by G, and hence A

represents a differential form on X. E and at oo means that ~ PROPOSITION 7.

The condition that h be meromorphic on

is everywhere meromorphic on X. Let

oJ = h dw k be the differential form on X as-

sociated with the mod~11~r f~mction h of weight k~ let Wp(h) denote the order of h a_~tP and vp( ~ correspondin ~ point of x~

More precisely:

let w

(D

(h) and v "'

For any point P e Ej

) the order of cO at the

( CJ ) be defined simil~_rl~o O0

Then:

(3)

Woo(h ) =roe ((~J) + k

(~)

Wp(h) = e o v p ( ( D )

+ k(e

- I)

(e = order of P)e

II -8 Equation (3) follows immediately from the relation 2widw = dq . q For P e E, choose a local parameter Zp such that the isotropy group Gp operates on Zp by multiplication by e-th roots of unity~ and let t = (zp)e. We can then write: 60

= u tNdt k

(u holomorphic / 0~ N = V p ( 03 ))~

00

= u ek(zp) Ne + k(e-l) (dzp)k

which yields (h). On an algebraic curve of genus g, the sum of the orders of a differential form of degree k is equal to k(2g - 2)~ COROLLAHY I. (5)

Here g = O.

Hence-

If h is a modular f11nction of weight k~ one has:

woo (h) + 8 9

) +~w(h)

+ Z*Wp(h)

=~

where Z* means that P r~ms thro1~h a set of representatives of tbe equivalence classes of ordinary points (lee. points of order l)e [Formula

(5) can also be proved directly by integrating dh/h over

the boundary of D. ] Let ~

denote the vector space of modular forms of weight k~

If

h is a non-zero element of Mk, one has Wp(h) > 0 for all P e E and also Woo (h) > O. of ~

Formula (5) then shows that k > O.

For k = O, the only elements

are the constants; for k = I, formula (5) shows that ~

k = 2, one sees that every non-zero element h 9 ~

= O.

For

has a zero only on the

orbit of /o , and that these zeros are simple~ if h t @ M2, a suitable linear combination h' - ah has a zero outside this orbit, hence is identically zero;

this shows that dimeM 2 < i.

M 3 with i replacing p COROLLARY 2~

An analogous result holds for

. We have proved: a)

b) c)

~

= 0 for k ~ 0

and for k = Io

_ i 9rk.2,3. If h is a non-zero element of M 2 (resp. M3) P

it has zeros onl~ on tb 9 orbit of /9

(resp. of i), and these zeros are simple.

II -9

w

Eisenstein series. Let ~ = =ZwI + ~w2 be a lattice in ~, and let k be an integer,

with k ~ 2.

The Eisenstein series Gk( ~

) of degree k associated to f-~

is defined by the formula~ (1~rr~'l+r~2) -2k~ (m,n)/[O,O) The series G 2 and G3 are identical (up to a constant factor) with the g2 and g3 of Weierstrass's theoryz

g2 = 6~

(7)

g3 =z ~176

To the homogeneous function Gk(Wl,W2) = Gk( /-~ ) is associated a function Gk(W ) of w = Wl/W 2 by the formula: (8)

Gk(W) = (w2)2k Gk(Wl,W 2) = Zm,rj~, ' (m,n) # (0,0) (m+nz)-2ke

PROPOSITION 8e

For k ~ 2, the series (8) is abso]11tel~ convergent

in E and normall~ convergent ~n D.

Its m~m Gk(W ) is a mod~lar form of weight

k whose value at inf~n~t~ is eq~lal to 2 ~

(2k).

In D, we have IR(w) l < 8 9 and lw I > I, whence: Im + n z

j2

>m

2

- r~

which is a positive definite quadratdc form~

+ n

2,

This implies the normal con-

vergence of (8) in D, hence its absolute convergence in E.

The fact that

Gk(Wl,W2) is invariant by SL(2~Z) implies that Gk(W ) verifies (2)~ When w tends to infinity in D~ each term (m + nz) "2k, n / O, tends to zero~ and Gk(W ) tends to Zm#O m -2k = 2 ~ (2k); hence Gk(W ) is a modular form. Let now ~ form of weight 6; one has ~ hand:

= g3 . 27 g~~

since g 2 ( / O )

( ~ ) / O and ~

It is clear that Z~ is a modular

= 0 and g3(/O ) / O (COro 2 to prop. 7)9 is not identically zero~

On the other

II-i0

g3( hence A

) - 2o

( 60 ) = O.

- 280 & (6)

560

Formula (5) then shows t h a t w o o ( / k

) = land

that

/% is ever~wh~ / 00~Eo PROPOSITION 9~ and ~

: ~-6

Let

~ :~

9~

> h(~),

be the homomorph%sm h

~ M k be t.be.m~ltipS.icationbz/k , For k ~ 2 the sequence:

is exact~ Since 6

(Gk) = 2 ~

then function h / ~ (since Woo ( Z ~ )

(2k) # O,

6

is surjective.

is holomorphic in E (since / k

If now ~

(h) = O,

/ 0 on E), holomorphic at

= I), and of weight k-6, hence belongs to ~-6"

COROLLARY I.

For k > 2, the dimension of the space M k is Riven

dim, ~

=

i[

k/6] ~

k ~-I mo~.. 6

[k/6] + 1 if k H~[ I mod,6o

Proof by induction on k s using coro 2 to prop. 7o COROLLARY 2.

Ever~rmodular form is an isobaric polynomial in g2

and g3" Let k > 2.

There exist positive integers a,b such that

ba 2a + 3b = k 5 the form g2g 3 belongs to ~ , shows that every element h e ~ h=~

ba and ~ (g2g3) # O@

Prop. 9 then

may be written

b a3 + ~ h ' , g2g

with h' o ~ _ 6 ,

and our contention follows by induction on ko Remark.

Cor. I can also be obtained by the following argument s Let P

(resp. Q,R) denote the point at infinity (resp. the image of i, ~ Proposition 7 shows that ~

) in ~.

is canonically isomorphic with the vector space Jl k

of differential forms Gt~ of degree k on X whose divisors ( ~

) verify:

II -il

( ~ ) _>-~- [k/2]q- [2k/31R.

(9)

If K denotes the canonical class of ~, /'~ k is in turn isomorphic with L(Dk) , with D k = kK + kP + [k/2]q + [2k/3]R.

If k > 2, one has

deg(Dk) = -k + [k/2] + [k/3] >_ -I, and the Riemann-Roch formula (applied here to a curve of genus zero|) gives. dim.M k = dim.L(D k) = deg(D k) + 1 = 1 -

k + [k/2]

§ [2k/3],

an expression which is easily seen to be equivalent to cor=l. This interpretation of ~

(which applies as well to other dis-

continuous groups than the modular group) can also be used to derive explicit formulas for the Gk in terms of J.

(io)

~6o

One gets for instancet

g2~ 2 " "'3" 2 j(j.d2~33), 2

The, q-expan,sion,s of Gk, ~

13 d~ 3 g3~ "r~ 'i'2(j. 263~)

and J,

If n is an integer > i and s any n~mber, we put:

(~s(n) Zdln aS~ ='

PROPOSITION I0.

(ii)

%~-

The q-expansion of Ok, k _> 2, i_~s:

2 ~ (2k) 9 2.(2,,) 2k C - i ) " / ( 2 , ~ - i ) ,

z n=i |

O'2k-i(n)qn"

We start from the equaiity~ (m+w) -2 = . 2 / s i n 2 ~

Taking successive

(q = e2niw).

.= ( 2 ~ i ) 2 Zoo n=inq n

derivatives with respect to w, we get:

(i2) (s-i)J Zmez (re+w)-s - (-2~I)s Z=n-Ins'i

qn

On the other hand, we have=

Qk = 2 ~(2k) + 22~.i Z ~

(m,=)"2k. I

Combined with (12), this gives

%=25

(2k) 9 2.(-2~)2k/(2k-1)J Z ~ l d2k-1 qdn , n>l m

hence (11)o

11-12 PROPOSITION ii~ We have ~ =

(2,)12 q(l+~q+e.o) and

j = q-i + b ~ + blq * ... where the ai's and the biWs are rational integers~ Moreover, j is equal to 2633 g 3 / / ~ . ~ n Let us put U = Z~176 (3-3(n) qn and V = Zn=l O-5(n) q ~ Prop. IO then shows that:

g2 = 6 O G 2 = (2n)~ ~ ( 1

.24ou)

(13) g3

1~ 140 G3 = (2~)6-B-

(I - 5o4 v)

hence: (~)

A

= (2.) 12 ~ i

[(i + 24o u) 3 - (1 - 5o4 v)23.

2~3 ~ The fact that the aiWs are integers is therefore equivalent to the congruence (15)

(I + 240 U) 3 ~

(I - 504V) 2

mod. 2633 .

A little calculation shows that (15) is in turn equivalent to the congruence U ---"V

mod. 223, or

(~3(n) ~

congruence is a trivial consequence of d3 ~

d5

(~5(n) mod.223; this last mod. 223. This proves our

assertion on ~ . The function g ~ / ~

is a modular function which is holomorphic

on E and has a simple pole at infinity. a,b m ~~

Since g2( P

Hence g ~ / ~

) = J(#O ) = O, one has b ~ O.

= aJ + b, with One

the other hand

g2(~) = 4~4/3, and therefore the coefficient of q-i in the q-expansion of g ~ / ~

is equal to 2"63 "3.

Since the corresponding coefficient of J is

equal to I (by definition), this gives a = 2"63-3 , and j = 2633 g ~ / ~ . We then have:

j m (1 + 24o u) 3 q-1 (l+~q+...)-i and since the ai's and the coefficients of U are in ~, so are the coefficients of J, q.e.d.

11-13

PROPOSITION 12 ("q-expansion principle").

Let f = Zn>_N Xnq

n

be a modular function wh~.ch ~.s holomorpb.~c in E and has a pole of order N Then f ~s a pol~nom~a] in j of degree N:

at infinity.

f=

zn=N n n=OYn j 9

Moreover, the additive s1~b~ro1~ A _~f C genersted by the Xn'S is the same as the additive subgroup B ~ener~ted by the YntS. Proof by induction on N~ A = B = ~= o ~

If

~

N > O, put g - f - x

N = O, f is a constant Xo, and NJN.

The function g has a pole of

order _< N-I at infinity, and may then be written: g = Zn< N Yn jn. One has f = g + x_NJN , which shows that B(f) is the subgroup of C generated by B(g) and X N.

On the other hand, the fact that J has integral coefficients

shows that A(f) is also generated by A(g) = BCg) and X_N , q.eod~

w

Explicit formula for the q-expansion of ~ . This formula will not be needed to prove the main results on complex

multiplication outlined in Io

However, since it is of considerable interest

in itself (Ramanujan's functionJ), we give here a proof of it. essentially follow ~urwitz (Math.

We will

Werke~ Bd~ I, S. 1-67, 578-595).

PROPOSITION 13.

One has /~ = (2,)12 q - ~ K oo (l.qn)24 n=l ~ The product f = q -~T n=l oo (l_qn)24 is a holomorphic function on E, invariant by the translation T, and holomorphic at infinity.

It is

enough to prove it is of weight 6, since prop. 9 will then show that it is equal to ~

~

, and the factor ~

coefficient of q~

(16)

will be determined by looking at the

By prop.3, we then have to show that i f(-;) = w 12 fCw)

forweE.

II-lh

Since f does not vanish on E, one can pickup a determination of ~/24, for instance= = e~iW/12 U The ~

n=l ~ (i - e2~inw)~

function is holomorphic on E, and (16) is equivalent to

The rest of this w is devoted to the proof of (17).

For another

proof, see for instance CoLo Siegel, Mathematika~ I, p~ LF/4MA

i.

The series %(w) = Zn [Zm (m+nw) -2 ] converges absolutel~

and one has Gl(W ) dw = ~,~i d ~ /

~

o

(As usual, one removes the term corresponding to (n,m) = (0,0)). Using again the formula ~2/sin2 ~nw = Zm (m+nw) ~ 9 we get the absolute convergence in m (for fixed n)~ and the formula:

Gl(W) = 2 5 (2) + 2Zn~=l,2/sin2~nw = .213

8~2 z~n=l qnl(l~)2 *

whence the absolute convergence in n~ This can be written:

%(w)~

(,/6i) (I

24Zn~i zm=l ~

n qnm) aq/q.

On the other hand: d ~/.~

=

dq/2hq

-

Z n=l c~

n qn-ldq/(l-qn).

= (dq/24q) (I - 2hZ~. I Zm=l c~ n qnm), whence the lemmm~ LEF~A 2@

We have:

(18)

z

(19)

zn [~ (m-z + ~)-I (m+~)-l] = o,

[zn (m-1 + ~1-I (m+~)-1] = _ 2-iA,

where the series are absolutely- convergent,

if-15 (As before, we remove the terms (n,m) - (0,0), (0,I)). Let denote by H (resp. ~ )

the left side of (18) (resp, of (19)),

It is clear that the first summations are absolutely convergent. (m-I + nw) -1 (m+nw) -!

=

The formula :

(m-i + nw) -1 - (m+nw) -1

gives: lira Z ~

Zn (m-I +nw)'l (m+nw) -1 .

(m-I + nw) "I - (m+nw) -I

n --.,~ (D .

with the convention cota/w = 0 if a = O. We then have:

a series which is easily proved to be absolutely convergent.

Its sum is

given by: H = (w/w)

lim m--~

(./w)

lira

=

m

~

(cot~m-lw - cot~) (cot~m~l + cot~) = - 2~i/w,

---~(D

since c o t ~ tends to i when m tends to infinity. On the other hand, we have:

z (~-I + ~)-l (m+~)-I

m~ m

=

lim m

whence 5

((m-1 + nw)-1

.

(m+nw)-1)

- - ~ oo

((4-1 +nw)-i . (m+nw)-l)

- - ~ Q)

= O@

LEMMA 3,

1 We have GI(-~) = w 2 Gz(W) - 2,,iw,

Let us put

G(w) = ~. z m

n

(m+~) -2

We have: ~-GI=Z

n Zm

(m+nw) "2 (m-I + nw) -I,

which is absolutely convergent (as a double series). change the order of summations, and we get:

We can then inter-

=

o,

II-16

~-~-H-G. Using l e n a 2, we have then ~ ( w ) - G(w) = 2~i/w.

Using the obvious formulas

-~) w 2 Q(w),

%(-

=

we obtain lemma 3.

P ~ o f of (17). Lemma 3 may be written:

w

w

~

(w)<~ - 2 ~ ~lw.

Using lemma I~ this gives:

cd ~ / ~

> c- ~-:~.. , , ~ / ~

which means that

c- ~ - o ~/~ ~ Cw~. Putting w = i gives C

~-i

=

I, and (17) is proved.

+ ,,w/~,

IAS, 1957-58

Seminar on complex multiplication

III-I III

CLASS INVARIANTS I

(A. Borel, Nov. 6, ~I.

1957)

Introduction o Let X = ~ / ~ be an elliptic curve.

If X has a non-trivial complex

multiplication, the ring A(x) of all complex multiplications of X forms an order of an imaginary quadratic field.

Let an imaginary quadratic field K be fixed.

Then there is a one-one correspondence between the set of ideal classes _~, o.., ~

of K and the set of X such that A(x) coincides with the ring of all

algebraic integers in K; the correspondence is given by assigning to ~ elliptic curve with the invariant j (~).

the

(Cf. I ~I.)

The main purpose of the four following lectures is to prove that the class invariants j (k_i) are all algebraic integers forming a full set of conjugates over K, and also over q, and that K(j(~)) extension of K.

is the maximal unramified abelian

As mentioned in I, we follow Hasse's first method, which uses

analysis and class field theory. 22.

Modular correspondences. Let E be the upper-half complex plane, G the modular group and S the

closed Riemann surface obtained by adjoining co to E/G. onto the complex sphere by the function j (cf. II ~3).

S is mapped isomorphically So S iS an algebraic curve

of genus 0 and an algebraic correspondence of S into itself is called a modular correspondence. A method to generate such a modular correspondence is as follows: Let ~ be the group of all automorphisms of the Riemann surface E.

It is the

quotient of the group GL(2, ~ of real 2 x 2 matrices by its center.

Take a

111-2

subset H of G containing G such that GH = H, [H : G] = m ~ co and H "I - H (hence also HG = H).

For any M in H, let jM(z) denote the function j(M(z)).

depends only upon the class GM of M in H/G.

Then

Let Jl(Z), ...j jm(Z) be the func-

tions obtained in this way from all m classes of H/G.

Then it can easily be

proved that each element of H may be represented by a matrix with integral rational coefficients and that each class GM contains a triangular matrix. It follows then (see the proof of Theorem la below) that the s~,,etric functions of Jl' "''' Jm are integral nodular functions, i.e. polynomials in j.

Hence,

m

F(t, j) = ~ (t - Js(Z)) is a polynomial of t over C(J)~ and a mapping which s-i maps a point of S with "coordinate" j(z) to the points with coordinates Jl(Z), ..., Jm(Z) defines a modular correspondence. Taking any modular function g(z) instead of j(z), we can define gl(z), ..., gin(z) just as above, and symmetric functions of gl' "''' gm then give us nodular functions. of weight k.

More generally, let h be any modular form

For M in H, put hM(Z) =h(M(z))( dM(z) dz )k . Then hM(Z) again

depends only upon the class GM and we get m such functions hl(Z), ..., hm(z) from m classes of H/G.

A s~,2trlo function of weight # in hl, ..., h m is

then a modular form of weight ~ k. ~3.

The correspondences F n. We now consider a special kind of correspondences.

positive integer.

Let n be any

We denote by H* the set of all automorphisms of E given by n

az + b

M : z ---~ cz + d

,

ad-bc

- n , a, b, c~ d ~ z ~

and denote by H n the subset of Hn* consisting of those M satisfying

111-3

(a, b, c, d) = i.

Clearly, GHn* - H'n,

i

-I . H* and GHn = Hn,

= Hn.

Furthermore, a set of representatives for classes GM of Hn*/G is obtained by the automorphisms with matrices

(i)

,

0 Hence, [

a > O,

d > O, d >

b >

O, ad

= n

o

d

9 G] = 0-1(n )

d~J"

Imposing a further condition (a, b, d) = i

on matrices in (I), we get a set of representatives for Hn/G and it is not difficult to see that [Hn 9 G] = ~ ( n )

= n ~ F (i + pl_). pln is as follows: Let ~ be a two-dimensional

Another way of defining H n

lattice on the complex plane. primitive if ~ / P '

A sublattice ~ ' of ~ w i t h index n is called

is a cyclic group of order n.

Let ~ ,

and 00~, tO4 a basis of such a primitive sublattice ~'. cO~ - a ~ l

+ b~O2, 004 - co~ I + d c r 2.

Then M : z

~)2 be a basis of Put

~ aZcz ++ db is in Hn.

Con-

versely, every M in H n can be obtained in this way from some Y , ~ ' and their suitable bases OJl, 032 and ~ ,

cO4.

Thus if X is a curve with lattice of periods, ~ the correspondence F

n

defined by H n associates to X the curves having a lattice of periods primitive of of index n in ~ .

Analogously, H* associates to X all curves which have a n

period lattice of index n in ~ . Given such ~' primitive of index n in P, we can always find, by a theorem on abelian groups, a basis CO1, oj2 of ~such that n~91, 09 2 form a basis of ~'. H n = GMoG.

Hence, if M o denotes the automorphism z

~ nz, then

Therefore, right multiplication of G on H pe~,~.,tes the classes

GM of H/G transitivel~.

In general, this is not true for H~.

III-~ ~4o

The modular equations. Now, let ~ ,

.o., ~ ( N

= VJ(n)) be the representatives for Hn/G given

by the matrices (i) with (a, b, d) = I.

is(z)

= j(Ms(Z))

Put

,

s

= I,

.o.,

N

.

These are the functions associated with N classes of Hn/G and, as HnG = Hn, js(Z) ~

Js(T(z)) is a permutation of Jl' "''' JN for any T in Go Hence,

a symmetric function of Jl' "''' JN is invariant under z ~ T(z), T ~ G. N THEOREM 1. (a) Fn(t , j) - ~ - (t - js(Z)) is a polynomial of t and s =I j = j(z) ~rith integral rational coefficients, (b) i_~fn is not a square, the highest coefficient of j in Fn(J, j) is + l, (c) Fn(t , J) is an irreducible polynomial of t over the field C=(j), (d) Fn(t , j) = Fn(J, t), n > 1. Proof.

(a) An elementary symmetric function

0 ~ (jl(z), ..., jN(z))

of Jl' "''' JN is, as noticed above, invariant under G and is obviously a holomorphic function of z in E. q = e 2~iz.

Then, by II w

To see the behavior of ~

at co, put

j(z) = q'l(l + A(q)) where A(q) is a power series

of q with integral rational coefficients and A(O) = O.

Let a

s

Ms ~

as

b

0

d

'

e2~iMs(Z)

=~d

bs

d qS

2~i

,

s

Then,

a

a

-b s

js(z) -- J(Ms(Z)) - - S d s

= e s

S

(2)

d

---

q- ds

S

bd

(1 + A ( ~ d s

5) ) .

d

s

Iii-5 Therefore, ~

can only have the singularity of a pole in q at q = O, and, hence,

is a polynomial in j over C. ~

Furthermore, the coefficients ~fithe q-expansion of

are all algebraic integers in the field q ( ~ n ), ~ n

Galois automorphism of Q(Sn)/Q.

= e n

o Let ~ b e any

Then t~(~n ) = 5 nf for some f with (n, f) = l,

and we can write b

fb s

b

whe re

f

at' bt I 0 ,d t

,

a t = as, d t = d s, b t-- fbs rood. d s

is another matrix in (I) uniquely determined by M s. permutes the functions Jl' ~176 JN among themselves.

Hence, K(js) = Jt and Thus~ 15( ~y ) -o-~ for

any ~" in the Galois group of Q(~ n)/~ , and the coefficients of the q~expansion of ~

must be rational integers.

By the q-expansion principle (II ~6), ~- is

then a polynomial in j with integral rational coefficients. a

(b) Since n is not a square, ~

in (2) cannot be 1.

Hence, the leading

S

coefficient of the q-expansion of j - Js is a root of unity, and so is the leading N

coefficient of the q-expansion of the product Fn(J, j) = TU (j - Js )-

However,

s =I

this coefficient is equal to the highest coefficient of j in Fn(J, j) and, as it is rational, it must be +I. m

(c) This follows immediately from the fact that Jl' ~ transitively among themselves under is(Z) ~

Js(T(z)), T ~ G.

JN are permuted (Cf. the last

lines of w (d) Since z ~

nz and z --'~nz are both contained in Hn,

Fn(J(nz) , j(z)) = 0 and Fn(J(Z), j(z)) = O, identically in z ~ E.

Replacing z

III-6 by nz in the second equality, we get Fn(J(z) , J(nz)) = O.

Hence, as polynomials

in ~(J)[t], Fn(t , j) and Fn(J, t) have a common zero t = J(nz).

Since Fn(t , J)

is irreducible by c) and since the highest coefficient of t in Fn(t , j) is I, we obtain

Fn(J, t) = P(t, J)Fn(t , J) , with a pol~momial P(t, j) in ~[t, J].

It then follows that

Fn(t, j) -P(j, t)Fn(J, t) = P(J, t)P(t, J)Fn(t, j) , and hence, that

P(t, j) = P(j, t) - +i. en

If P = -I, we would have Fn(J, t) = - Fn(t , j) and Fn(J, j) = 0 so that Fn(t, j) is divisible by t - J, contradicting c).

Hence, P - i and Fn(t, j) = Fn(j, t).

Fn(t , j) = 0 is the modular equation for the degree n.

(For a given

J(z), which is the invariant of an elliptic curve X, the roots of Fn(t , j) = 0 are the modular invariants of the curves X i (i ~ i ~= N) which the correspondence F n associates to X.

Theorem Id means that then X is among the images of X.l;

this can also be seen geometrically by noticing that if P' is primitive of index n ~n ~ , then n ~ is primitive of index n in ~', and that the curves with periods C and n ~ ~5.

are isomorphic.

Class invariants. We next show that an elliptic curve X has non-trivial complex multi-

plications if and only if the invariant J(X) satisfies Fn(J(X), j(X)) = 0 for

III-7 some n > i.

Let X = ~ / ~

and let O01, CO 2 be a basis of ~ s o that J(X) = j ( ~ ) ,

~l Supposej first, that Fn(J(~) , j ( ~ ) ) = O, n > 1. Then j s ( ~ ) = j ( ~ ) ~= ~2 " for some s and M(oo) = ~ for some M in H . Hence, there exists a w in C such that n

=

b1

w~ I = a~ I + bO 2 , M~--~

d

woj 2 - cod I + d~D 2 , Then, w cannot be a rational integer and X has a non-trivial complex multiplication. Conversely, suppose X has non-trivial complex multiplications.

Then the ring of

complex multiplications A(X) is an order of an imaginary quadratic field K and we can find a w in A(X) such that NK/Q(w) = n > 1 and that w is not in any mA(X), m > 1.

Reversing the above argument, we see that j(X) satisfies

F n(j (x), j (x)) - o.

(of. below.)

We now fix an imaginary quadratic field K and consider the class invariants j ( ~ ) ,

i = l, ..., h, of K, i.e. the invariants J(X) of elliptic

curves X of which A(X) coincide with the ring of all algebraic integers in K. Let w be an algebraic integer in K such that n = NK/Q(w) is a square-free integer > 1.

Such a w always exists; if K = Q ( J - 1 ) ,

K = Q(J--m), m > 1 and square-free, take w -J~-m.

take w = I + J ~

and if

Let COl, oJ 2 be a basis of

an ideal in an ideal class k of K and put m

w~ I = ac01 + b~ 2 , w ~ 2 = c ~ I + d~D 2 ,

a, b, c, d ~ Z

Then ad - bc = NK/Q(w) = n and, as n is square-free, M : z ~

az + b cz + d is in H n.

.

u

(a, b, c, d) -- i.

Hence,

Since j(M(~O)) = J(cO), the invariant j(k) = j ( ~ )

III-8

immediately the following:

quadratic field a~e ~

integers.

Seminar on complex multiplication

IAS,

1957-58

IV-1 IV

CLASS INVARIANTS II

(A. Borel, Nov. 13, 1957) w

Introduction. The main purpose of this lecture is to obtain Theorem 3.

For

this we shall first establish some properties of certain functions formed by means of the discriminant A (see ~2~ 3). We follow the usual conventions of algebraic number theory in which often no distinction is made between an ideal a in an algebraic number field K and the ideal o(L).a it generates in a finite extension L of K.

In particular let al, a 2 be

ideals in ~ , ~ and L be a finite extension of ~ that aI divides ~

and ~ .

Then we say

(resp. that a I and a 2 are prime to each other) if

o(L).aI divides o(L).~ (resp. o(L).aI and o(L).a2 are prime to each other).

w

This is then true in any algebraic extension of ~

.The functions

and ~ .

~M"

In dealing with A , it will be convenient to use the homogeneous formulation.

We denote in the same way the automorphism of the upper half

plane E given by z

(az + b)(cz + d) "I

and the homogeneous linear transformation (Wl, w2)

~ (aWl+bW2 , CWl+dW2).

We recall that if h(z) is a modular form of weight k, then

h(wl, w2) = w22k h(Wl/W2) is a homogeneous function of degree -2k in Wl, w 2 and is invariant under the "homogeneous" modular group G.

In particular, the modular form A

IV-2 gives rise to a homogeneous function of degree -12, invariant under G:

& (wI, w 2) - w~ 12 a ~ ) and we have (I)

A(wI' w2) =

(2~)12 ~2

2.i w2 q(l + B(q)) ,

q " e

,

where B(q) is a power series of q with integral rational coefficients and B(o) = 0 (see If, ~5). For any M in Hn, put

12 ~(M(wl' w2)) (4i; ~2)

~,(w l, w 2) = n ThenCE

depends only upon the class GM of Hn/G and, from the N = ~ ( n )

classes of Hn/G, we obtain N homogeneous functions of degree O in Wl, w2:

~i(Wl,

w 2) - ( # H i ( w l ,

These functions are regular for Im G )

w2) ,

1 <. i <. N .

> 0 and are permuted among themselves

"2

under the transformations in G. Now, it follows immediately from (i) that ai (2)

~i(Wl , w 2) " ail2~di

qdi -

-

-

211i

I(i + B(~d i

.

o-a-),

and we see, by a similar argument as in III, ~ , that N

~n(t, J) " ~F (t- ~i(Wl, w2)) i-i

is a polynomial in t and J with integral rational coefficients. N

LEMMA 1. equal to (-1)P'lp12.

I_~fn = p i_~sa_prlme, ~ i ( W l ,

w2) is a constant and is

IV-3

Proof.

In this case, N - L~(p) = p+l and the matrices M i are

given by

Mi

-

Hence,

1

+i "

o

1

9

1

~i(Wl' w2 ) " ~ q P (3)

,(1 <. i < p);

o p

(I + B(~qP))(I + B(q))-I (i ~ i ~p),

~p+l(Wl, w 2) = pl2qp'l(l + B(qP))(I + B(q))-I,

(i " p + I),

N

and the q-expansion of ~.~i(WlJ w2) starts with the constant term l=s (-l)P-lp 12.- However, as this product is also a polynomial in J, it must be e~lal to the constant (-l)P-lp12. ~3.

Properties of the singular va!ues of the ~M. We now fix an imaginary quadratic field K. LEMMA 2.

Let a be an ideal of K and (al, ~2 ) a basi____~s of a such

> O.

Then, for any M i_.nnHn, ~M(~l, ~2) is a__nnalgebraic integer

that I ~ / ~

and, if n = p is a prime, the principa1 ideal (~M(~l, ~2)) is a divisor of (p12).

Proof.

By the definition Of~n ,

~n(t, j(a)

N =

"IT

i-1

(t

-

Ti(%.

9

But, as J(~) is an algebraic integer (III, Theorem 2), the left hand side is a unitary polynomial in t with integral algebraic coefficients.

Since

~M(~I, ~2),(= ~i(al, m2 ) for some i),is a root of ~n(t, j(a)) = O, it is an algebraic integer.

The second part is then an immediate consequence

o f LeIm~a 1.

Let a be an integral ideal of K.

Then, for any (fractional or

integral) ideal ~ of K, ~.~ is an additive subgroup of ~, of index equal to N~, as follows from the fact that N(~.~) = N(~).N(~).

Thus, if

(PI' ~2 ) is a base of ~, and (aPl+bP2 , CPl+d~2 ) a base of ~.~, where a,b,c,d~Z=, we must have ad-bc = N~; moreover (a,b,c,d) = I if ~ is not divisible by a rational integer > I, for instance if a is prime. A prime ideal ~ of K is said to be of first degree if every integer of K is congruent mod ~ to a rational number or equivalently if p = N~ is a prime number; ~ is unramified for K/~ if its square does not divide a prime rational number or equivalently if N~does not divide the discriminant of K/~. Under those two conditions we have ~ # 2 (" denoting complex conjugation) and (p) = ~.~. THEOREM i.

Let

(~

r ) b_~e~basis of an ideal a of K, ~ b_~e~

prime, idea____~lo_~fthe first de~ree inK, unramified fo__~rK/~, and p =N~.

Let

P ~ p (resp. P~p) be such that P(~I' ~2 ) (resp. P(%, ~2 )) is a basis of 2"~ (resp. ~.~).

Then

and if M~Hp, M~GP, GP, then ~M(~, #2) is ~ .un%t. Proof.

Let f be a positive integer such that ~f is a principal

ideal in K. We have t h e n ~ f integer of K.

=

(~), and ~

=

NK/~(~)

=

pf, where ~ is an

By the discussion preceding Theorem I, the elements P,P

are in H , and we can find PI = P' P2' ""' Pf in H such that P P PiPi_l ... PI(~I, ~2 ) is a basis of ~i.a (I ~ i <. f) and, in particular,

Pf-

Pl(% ,

=

(=i, %)12

Xi = ~ Pi (Pi-l'" "PI (%' ~2 ) ) = p

(Pie''Pl( l' 2)) (Pi-1-

,(l

i

f)

~-5 Then,

7Tx. f = pl2f a(Pf'"Pz(%, =2 )) . pl2f A(~5, =~2) = pl2f=-12 = ~12 i.1 9 a (~, ~2) a (%, ~2) " As every k.l is an algebraic integer by Lemma 2, the ideal (ki) divides -12 . (~12) = ~12f. But, by the same lemma, (ki) also divides (p12) = P 12 "~ f Hence, ()~i) is a divisor of ~12 = (~12, pl2). Since IK (ki_) = (~12)_ = p-12f, i=l it follows that (Xi) = ~12 for every i, and, in particular, (~p(~, e2)) = (h) = ~12. Similarly, (~ P(~I' ~2 )) = 12.

As p ~ ~, this shows that

GP # GP, and it follows from N

12 -12

i=IVF(~i (~, a2)) = (p12) = ~ that ?M(~, ~2) is a unit if GM # GP, G~. ~.

A formal c0ngruence. Let ~ and ~ be power series (in particular polynomials) of certain

variables ~rith integral algebraic coefficients and a an ideal of integral numbers.

As usual we write ~

~ nod a

if every coefficient of ~ - ~ is in a. We consider 2(p+l) q-series ~i(q), ~i(q), (1 < i < p+l), with the following properties (i) Their coefficients are integral numbers of ~(~p); ~ p, ~ p+l ~p'

~p+l have rational integral coefficients. The ~ i's (resp. ~i's)

(1 <_ i < p-l) are permuted by the automorphismq of the Galois group of

IV-6

(ii) They represent holomorphic functions in the upper half plane which are permuted among themselves by G. By (h) the ~i's satisfy these assumptions.

The same is true for

the Ji's because we have

si(q) " ~piq-1/P(1 § A(~ql/P)~ ,

(~)

Jp+l(q) = q'P(l + A(qP)) where A(q) has rational integral coefficients.

In fact, these will be the

only functions satisfying (i)(ii) to be considered in the applications. With indeterminates t, u we then put k=p+l

(5)

Gp(t, u, ~i' ~s ) " zJ~+l~(t'j-1 ~ ~')~k-l,k~s(u"~k)~ "

t,u,j with rational integral coefficients. (b) Gp(t, u, ~i(q),

~j(q)) =_ (%-~p+l(qll'(u p - ~p(qlP) sod p.

(a) is proved exactly as Theorem la of III. Since in ~(~ p), the ideal (p) is equal to (I-5p) p-I and since both sides of (b) have rational integral coefficients it is enough to show that both sides of (b) are congruent rood (l-~p).

By (i) above, we have ~ i

for (I <= j, i <= p).

Therefore

Gp i (t- ~p+l)(U-

~p)p

m ~j and ~ i =- ~j rood (l-~p)

+ p(u- ~p+l)(t- ~p) (u- ~p)p-1mod (l- ~p)

,

and our contention follows from the fact that rood p, we have (u-_~p)P--uP - ~ pP.

~5.

Applications. The modular e q u a t i o n f o r the degree p i s F ( t , j ) P

= O where

IV-7 Fp(t,J) = ~(t-J i) is a polynomial in t and J with rational integral coefficients (III Theorem la).

The following property of F

P

is a

classical result due to H. Weber (Acta Mathematica 6, 1885, p. 390). THEOREM 2.

We have Fp(t,j) --~ (t-J p) (tP-j) rood p, the integer p

bein~ prime. From (h) it follows by Fermat's first theorem that Jp+l(q ) m J(q)P

mod p

Jp(q)P ---j(q)

mod p

(6)

In order to apply Lemma ~, we put t = u, ~ i Then G

P

= ~ i = Ji (i < i <. p+l).

= F (t,J), and, in view of (6), the right hand side in Lemma ~b P

is congruent to (t-jP)(tP-j) mod p.

Lemma ~ implies therefore that the

difference of the two sides in the formula of Theorem 2, when viewed as a power series in t and q, has all its coefficients in pZ=. By the q-expansion principle (II w

the same is then true when this difference is written

as a polynomial in t and j. THEOREM 3.

Let K ~___an imaginary quadratic field, p a prime ideal

of K of degree l, unramified for K/~, and k

the ideal class containing . p.

Then for any ideal k of K, we have

(7)

j

9 j (_k)

Let a be an ideal of k.

2 9

Since ~.~ = (N~), the ideal class k

contains ~ and our contention is equivalent to

(8)

j(~.a) ~j(a) p mod ~ (p = Np) .

Here p is prime, and p # ~. in Theorem 1.

Let ~l' ~2 be a base of a and P , ~ H p be as _

Furthermore, let c,d < p+l be the indices such that

-1

Iv-8

P~Mc, PGGH d. Then c # d since p # ~ and (9)

s(~.~_) - j~(a_) - jd(~_) - Jd(%/~ 2) . We now apply Lemma 4 to the case where ~ i " Ji' ~i " ~ i (I <= i ~ p+l)

and t - JP. This gi~s Gp(J(q)P,u,Ji(q)

~k(q)) m (JP-Jp+l) 9 u(

p-

P ~p)

rood p

which, together with (6), shows that Gp(J(q) p, u, Ji(q), ~k(q))=-0 rood p , G

being considered as a power series in u and q. P principle (II ~6), we have then

(lo)

By the q-expansion

N (jP,u,j) 9o rood p P

where (cf. Lemma 4), G

is G considered as a polynomial in j and u. P P Let now qo " exp'(2"i~I/m2)" The numbers J(qo), ji(qo), ~i(qo)

are algebraic integers.

Since Gp(Jq,u,Ji, ~j) may be ~rritten as a polynomial

in t and j with coefficients in pZ, its value for q = qo is a polynomial in u, whose coefficients are algebraic integers divisible by p.

(n)

Thus

ap(j(a) p, u, Ji(a), ~k(%, ~2)) TO rood p.

Let us now put u = ~ ( ~ ,

=2 ) = ~ d ( ~ ,

=2 ). Then by (ii) and the definition

of % ~ we get

(12)

(0(a)p - Sd(a_)) 7T (@d(~, ~2) - ~i(%, ~2)) -= 0 ~d p.

Iv-9 Since ~divides

(13)

(p), we have afortiori

(j(a)p - jd(z)) i7r ~ (~d (5, =2 ) " ~ i ( 5 ' ~2)) ----0 ~ d ~"

By Theorem I,

, a2) mod 2 and the right hand side generates the ideal ~12, which is prime to p. Therefore, (13) implies that

J(a) p - Jd(a) --=0 rood p and this, in view of (9), is the congruence (8). Remarks.

(I) We have Jp+l(Z) = J(p.z), so that the first con-

gruence in (6) is a congruence between the q-developments of j(z) and j(pz), that is between J(q) and J(qP). analogue to Theorem 3. numbers p ~ ,

(I~)

It may be viewed as a formal

In fact, for a suitable base ~ ,

a 2 of a the

a 2 form a base of ~.a so that (8) may be written

J(a_)p ----Jp+l(a_) _ rood ~ .

However, one cannot of course simply derive (lh) out of (6) by putting q = qo = exp(2~i~/m2)' because both sides are infinite series in q (and as a matter of fact, this would lead to a congruence rood p, not only mod p, which is not true in general.)

However, a formal congruence be-

tween q-series with coefficients in an additive group H of algebraic integers leads to a correct congruence for q - qo if both sides represent integral modular functions, because they can then be considered as poly-

IV-IO nomials in J with coefficients in H by the q-expansion principle, and j (qo) is an algebraic integer.

It is this fact which has allowed us to

derive (II) from Lepta ~. (2) M. Eichler, Math. Zeitshrift 6~ (1956), 229~2 has given a proof of the main theorems of complex multiplication which uses only j, and not A.

However, he does not obtain Theorem 3 in full, but shows

instead that one has either Theorem 3 or j (~.k~ ~ j (k)p msd p.

Seminar on complex multiplication

IAS, 1957-58

V-I V

CLASS FIELDS

(K. lwasawa, Nov. 20 and 27, 1957)

w

Introduction. As was proved in IV, w 5, the clsss invariants j(k) of an imaginary

quadratic field K satisfy certain fundamental congruences.

In the following,

we shall first give an outline of class field theory in its classical form and then show, using that theory, how we can deduce from those congruences arithmetic properties of the numbers J ~ ) and of the extension K(J~)) / Ke For the det~Ss of classical class field theory, cf. Hasse's KlassenkSrper Bericht quoted in I, w

w

Ideal groups. Let K be a finite algebraic number field.

Let m be a divisor of

K, i.e. a formal product of a finite number of prime divisors of K : m = ~ i Put m = m ~ mc0, where m ~ and mc0 are, respectively, products of non-archimee

dean and archimedean prime divisors in m I ~ i i ; m ~ may be then identified with the corresponding ideal of K.

In the following, we shall always

consider such a divisor m that m ~ is an integral ideal of K and that moo is a product of a number of distinct real archimedean prime divisors of Ke For a number ~

in K, we then write i I

if and only if ~)-Pi(~ -

mode m

i) => ei for every P-i dividing m ~ anddrp_j(~ ) > O

for every p_j dividing moo ;

here ~ i

denotes the normalized exponential

.

V-2

valuation of K belonging to ~i and<~Ej denotes the isomorphism of K into ~he real field corresponding to ~j. Now~ let I - I(K) be the multiplicative group of all non-zero ideals of K.

For any divisor m as considered above, we denote by Im = Im(K ) the s~b-

group of I consisting of all ideals of K which are prime to m ~ -and by Sm = Sm(K ) the group of all principal ideals ( J ) w i t h ~ = then a subgroup of Im and Im / Sm is a finite group@

I modo m@

Sm is

Any group H such that

Sm = H c Im will be called an ideal group of K defined mod~ m@

w176 The densit7 of a set of prime ideals. Let P = P(K) be the set of all prime ideals of K~

For any subset

M of P, we put ~,

(s; M) : ~

ilK/Q (p.)-s

The right hand side is absolutely convergent for R(s) > I and ~(s i M) is well defined in that domain.

Now, if the limit s~l s>l

exists, we call ~ ) of the set M in P@

~(s;

the densit~ (Dirichlet density or Kronecker density) Clearly,

~(P) - i@

Using the fact that the zeta-function

~K(S) = ~ p

(i - NK/Q(I~)-s)-I

has a simple pole at s = I, we can see by a simple computation that

~(s; P ) ~ where N

i log ~ K ( s ) ~ log s--l-,

indicates that the difference of the both sides of ~

bounded at s = I~

Hence

l~ s~l

I ~(s; P) / log F-1 = l,

is a function

v-3 and the density ~ (M) can be defined also by 1 ~(s; M) / log ~T-m"

~(M) - 1 ~ s~l Example,

Let

= absolute degree of E~

NK/Q(p) = pd (p . rational prime), d = deg p Let pT = p1(K ) be the set of all E with d = 1

and let P" = P"(K) be the complement of P' in P.

Since NK/Q(p) =

pd

> p

2

i Hence, l~_m ~(s; P") / log s-1 = O,

for p in P", ~(s; P") is bounded at s = 1. and

~(P,,)

= o,

~(P,)

-

~(P)

- 1.

More generally, if we put M' = M6~ P', M" - M g ~ P " for any subset M of P, then

~(M") = O,

l(M')

= &(M),

whenever one of ~(M') and ~ ( M ) exists. Now, let H be an ideal group of K defined mod. m and let k be any class (coset) of I m p .

For the set k o

P~ we have then

1 ~(_k ~ P) - ~(_k n P,) ~ ~,

(i) where h = [Im .- HI.

This is the theorem of arithmetic progression for the

field K, which generalizes the well-known theorem of Dirichlet for K = ~. An outline of the proof of E. Hecke for (I) is as follows (of. GSttin~er Nachrichten, 1917) :

Let

L(s;X) ~ . ' ( I - x ( ~ )

H

be the L-function of K defined for a character ~ It is first proved that L(s; ~ )

(~)-s)-l, R(s)>l, of Im/H (~' : over ~ in Ira).

can be analytically continued to a meromor-

phic function of s on the entire s-plane which has a unique pole of order 1 at s = 1 if ~

= ~o

and is holomorphic everywhere if ~

denotes the principal character of Im/H. easily from the definition that

$ ~ o ; here ~ o

On the other hand, it follows

v-4 h

%.

L(s; 96) = ~ ' ( i - NK/Q(~)-fs)~

R(s) > I ,

where ~ is taken over all characters of Im/H and f = f(~) is the order of the class of ~ in Im/H.

Hence, CO

L(s; ~ ) ~ ~ l ~-~' V~

% with rational integers a ~ ~65 ~o"

~ O.

R(s) > l,

Suppose now that L(1; ~ )

= O for some

Then the left hand side of the above would be holomorphic on the

entire s-plane and, by a theorem of Landau on the convergence of Dirichlet series with positive coefficients, the right hand side of the above must converge for all s.

Since a R

are integers, it would then follow that a y =

except for a finite number -- a contradiction.

0

Hence:

L(I; ~ ) + o

~~o"

Now, by a simple computation, we have

oo i NK/Q(p.)-V log L(s; ~ ) ~ z, Z=l C ~ (2)v s ,-, z, ~ (~) NK/Q(p)'s

S >I.

By the orthogonality of the characters 9~, it then follows that

~ ~(k -I) log LCs; ~ ) N for any class k of Im/~o

h "~k NKIQ(p)-S -- h - ~(s~ k ~ P),

Since L ( I ; ~

log L(s; %0 ) ~ l o g

) ~ 0 for~ 1 's'-'l;

960, we have

log L(S; ~ ) r - ~ O ,

I and the left hand side of the above is ~-~ log s--~-" Hence we get

(k n P) = l ~

~(s; _kO P) I !o~ ~ I = I

q~

s~l Remark~

By Theorems II, IV of ~6 below (which can be proved purely

algebraically), there exists an abelian extension E of degree h over K such that the zeta-function

~ E(S) of E is the orodllct of ~ L(s; ~ )

a finite number of factors of the form (i - q-S)-l, q > io has a simple pole at s = l, this immediately implies for ~

%o"

and

Since ~ E(S)

L(1; % )

~ 0

For K -- Q, such an E is given by a s~bfield of a cyclotomic field.

v-5 w

The inequality h =< n. Let L be a Galois extension of degree n over K.

A prime ideal 2 of K

is said to be completely decomposed in L if s is decomposed into the product of n distinct prime ideals of L, or equivalently, if ~ is unramified in L and ~ = ~/K([) for some prime ideal [ of L.

Let W denote the set of all

such ~ of K which are completely decomposed in L, and let P"
Then

[----~ ~ = NL/K([) defines an n to I correspondence

between the sets P"
([) = NK/Q(~) , we get

~(s~ P"'(L)) = n~(s; w,). However, the difference of P'(L) and P'"(L) is a finite set and ~(P"'(L)) = ~ (P'(L)) = ~(P(L)) = 1.

(2)

Hence, it follows from the above that

~(w) = ~ (w,) = !. n

Now, let m be a divisor of K as considered in w N

Define a group

= N (L/K) by m

m

Nm = Sm(K ) NL/K(Im(L))Clearly, Sm r Nm r Im and we put h

=

(L/K)

h

By the definition of W, Wm = W • also in N . m

=

[Im : ~ ]

"

Im is contained in NL/K (I(L)) and, hence,

As the difference of W and W

m

is a:ain a finite set, we obtain

from (I) and (2) that

n

=

h ' m

namely that

(3)

~

<.~n,

or

[~m " ~(LIK)] <__ [L 9 El.

This is called the second fundamental in e~l~]ity of class field theory and it

is

valid

for

a W m and for

any

finite

Galois

extension

L o f K.

V-6 w

Definition of class fields. Now,

a finite

Galois

extension

L of a finite a l g e b r a i c

number

field K

is called a class field over K if, in the above, hm = n holds for some divisor m of K.

If L/K is such a class field, there exists a unique divisor

f of K such that h

m

= n holds if and only if m is a multiple of f.

The

divisor f = ftL/K) is then called the conductor of L/K. Criterion.

If L/K is a class field, there exists a divisor m of

K and an ideal group H of E defined rood. m such that

i) ii)

c H, H f) P is contained in W except for a s~bset of density O.

Conversely, if there exist svch m and H, then

hm=n, Proof. that h = n. m

is a class field and

H =Nm.

If L/K is a class field, put H = Nm for a divisor m such

Then i) holds obviovsly and ii) follows from that the density

of the difference of N

m

H exist.

L/K

f) P and W

By i), we have Sm c

Nm c

m

is O. H c Im.

the theorem of arithmetic progression,

Suppose, conversely, such m and Put a = [H : Nm].

Then, by

2(H • P) = h-'" a However, by ii), m

( H N P) < ~(W) = l.n Hence

~m <= I

or p an <= h.m

It then follows from

(3) that a = i, hm -- n,

q.e.do

As an immediate application we notice the following: ~ / K and L2/K be class fields and let m be divisible by

Let both

fCLI/K ) and f(L2/K ).

It is then easy to see that m and H = N m ( ~ / K ) 6~ Nm(L2/K) satisfy the above conditions i) and ii) for the extension ~ a class field and hm( ~

L2/K.

L2/K ) = [ ~ L 2 : K], Nm( ~

Hence ~

L2/K is also

L2/K) = Nm(~/K)6~ Nm(L2/K).

Now, the condition N m ( ~ / K ) c Nm(L2/K ) is obviously equivalent with the condition Nm( ~

L2/K ) = Nm(~/K ) and, hence~also with

[ ~ L 2 : K] = hm(L I L2/K ) = hm(Ll/K ) = [LI : K], or, L I L 2 = ~ . fore, ~

~ L 2 if and only if N m ( ~ / K ) c Nm(L2/K ).

There-

V-7

w

Fundamental theorems. ii

,

i

i

|

We are now going to state the fundamental theorems of class field theory. Io

A finite extension L of K is a class field over K if and onlz

if L/K is an abelian extension (i.e. a Ga]ois extension with an abelian Galois ~oup). The most essential step of the proof is to prove n
m

for some m, when L/K is a cyclic extension.

This is called the first funda-

mental inequality of class field theory. II.

Given ar~ ideal grouu H of K defined rood.m, there exists a

unique class field L over K such that [Im 9 Nm(L/K )] = [L 9K] III. A_ prime divisor ~ of K is rsmJfied

and H = NmCL/K ).

in a class field L over K

if any only if p divides the conductor f(L/K) of L/K. Before going to state the next theorem, Artin's reciprocity law, we first make some preparations. of

Let L/K be an arbitrary Galois extension

degree n with the Galois gro%o G.

Let ~ be a prime ideal of K unramified

in L, and [ a prime ideal of L dividing ~.

Then there exists a unique element

CT in G such that (4)

~(~)

~ = NK/Q(~)

mod. P I

for any algebraic integer ~ in L.

This element ~ris called the Frobenius

substitution of P_ for the extension L/K and is denoted by O-p;

it generates

I

the so-called decomposition group of the prime ideal P, aad if f is the order of ~ p

and n = fg, then I

-PI = P ' where Pi are distinct prime ideals of L.

V-8 Now, a s s ~ e that L/K is a class field, i.e. an abelian extension. Then the Frobenius substitutions of ~l ~ ... --g P coincide with each other and we may denote these O - p

(i = l, ... g) simply b y C r

of K such that hm = n and let ~ = of an ideal ~ i n as above.

Im.

r

e

i~l~i i

9

Let m be a divisor

be the prime ideal decomposition

By III, everY~i is unramified in L and O - i

is defined

So, we may define O--~ by 9

Clearly, ~

~ (~

is a homomorphism of Im into G.

But

we have the fol-

lowing theorem: IV.

Th e homomorphism ~

~ O-~ind1~ces an isomorphis m

namely, the homomorph~sm ~_~ss11r~ective and the kernel is Nm. An immediate consequence of IV is the following: Prime ideals of K, which are contained in the same class of Im / Nm, are decomposed into the product of the same number of prime factors in L.

In particular, every

prime ideal in Nm is completely decomposed in L without exception. Finally, let L/K be a~ain a Galois extension of degree n and let O be an arbitrary element of the Galois group G of L/K.

We denote by PCY

the set of all prime ideals ~ of K such that ~ is unramified in L and that O-=O-p

for some prime ideal ~ of L dividing~. V.

Then

If c i_~sth__~enumber of elements i_~nthe c on~u~ate class o_~f(y-

i_~nG, then n

Notice that if L/K is a class field, V is an immediate consequence of IV and the theorem of arithmetic progression.

V-9

w176 Hilbert's absolute class field. Let i denote the unit divisor of K.

Then II(K ) = I(K), and

Sl(K ) is the group of all principal ideals of K.

The factor group I1/S 1

is hence the ideal class group C K of K in the ordinary sense and its order h is nothing but the class n~mber of K. Now, by the existence theorem II, there exists a class field L o over K such that h

=

[II ~. S 1]

=

[Lo : K],

S1

=

N1

(LolK).

The conductor f(Lo/K ) is then 1 and, by III~ Lo/K is an unramified extensian. Take an arbitrary unramified abelian extension Z/K.

Again by III, the con-

ductor f(L/K) is l, and since S 1 = ~ ( L o / K ) is contained in NI(L/K), L is contained in L O (cf~ the last lines of w176

Thus we see that L O is the

unique maximal unramified abelian extension of K. bert's absolute class field over K.

L ~ is called the Hil-

By IV, the Galois group of Lo/K is

canonically isomorphic with the ideal class group O K = II/S 1 of K.

w176 Con jugacy of class iuwriants. We now assume that K is an imaginary quadratic field and denote by _~, o.., _~ the ideal classes of C K = ~ / S 1.

The class invariants J(ki) ,

i = l, ...~ h, are then distinct algebraic integers.

Take a finite Galois

~xtension L over K containing all such class invariants j (ki) and denote by ~ l' "''' ~ s

the distinct elements in the set ~Y'(J(ki) ) where C~- runs

over the Galois grouo G of L/K and 1 < i < h.

and denote by P

We then put

the set of all prime ideals p of K with deg p = 1 such

that p is unramified in L and is prime to ~ .

It is clear that

V-10

~(P*) = ~(P') = 1.

Hence, for any subset M of P(K) with ~ (M) > O,

(M f) P*) = ~ (M) > O and the intersection M ~ Let ~ be a prime ideal in P*, k

-2

~ a n d p = NK/Q(~).

By IN#, w

the ideal class of C K containing

, we have then

j(~-Ik) , j(~)p for any ideal class ~.

P* is non-empty.

mod. ~,

On the other hand, if [ is a prime ideal of L dividing

p and O-p is the Frobenius substitution of P for L/K, then by (4), I

O-p(j(_k)) =_ j(k)p

mod.P.

It follows that O-p(j(k)) -- j(~-Ik)

mod. P.

But, the both sides of the above congruence are numbers in the set ~I' "''' Hence, as p is prime to ~

, we get

(5)

" P*, P_ I 2" Now, let k and k' be arbitrary ideal classes of C E.

of arithmetic progression,

~(k k ''l O

P*) = ~ ( k k ' - ~

By the theorem

P) > O and the

ideal class k k t-I contains a prime ideal p in P* so that k' -- k -ik. It then follows from (5) that the class invariants j(k) and j(k') are conjugate in L over K. ~(P~- (] P*) =

On the other hand, given any (3" in G,

~ ( P ~ _ ) > O by V and there exists a prime ideal p in P*

such that (5- = O-'p for some P dividing 2" Hence, by (5), any conjugate m

O-(j(k)) of a class invar~ant j(k) is another class invariant J(k'). Theorem i.

Thus:

The class invariants J(~), i = I, o.., h, of an

imaginary quadratic fie]d K form a complete set of conj~ates over K.

w

The extension K(j(k)) / K. From now on, we put L = K(J(_~), ..., j(~)), for the latter is

V-If a

Galois extension of K by Theorem Lemma~

i.

A prime ideal ~ i_nP* i_~scomp]etel Z decomposed i nnL if and

only if ~ is contained in S I (i.eo a principal ideal)~ Proof~

Let ~ be a prime ideal of L dividing ~o

pletely decomposed in L if and only if ~ p

= lo

Then s is com-

Since L = K(j(~), e

~

j(_~)),

~

it follows from (5) that C~p " 1 if and only if j(k_n-%) = j(~) for every ~, namely, if and only if k

-2

= l, or, ~ e

S 1.

We now verify that the two conditions of the criterion given in w are satisfied for m = l, H = S 1 and for the extension L/K. arbitrary ideal of L.

an

By the theorem of arithmetic progression, there exists

a prime ideal ~ of L such that deg [ = I and ~ =

(~)~with a number a in L.

We may also assume that ~ = NL/K(P ) is prime to ~ ~ P* and is completely decomposed in L. rained in Sl, and so is the ideal verified.

Let ~ b e

As ~(Slt] P*) = ~ ( S I ~

Then p is contained in

By the above lemma, p is hence con-

NL/K(~ ) = NL/K (a) NL/K(P ). Thus, i)

is

P)~the condition ii) is also satisfied

by the above lemmao By the criterion, L/K is therefore a class field and

h-

[L : K],

S 1 --Nl(L/K).

As explained in ~7, L is hence the Hilbert's absolute class field over K, i.e. the maximal unramified abelian extension of K.

Since L/K is now abelian

and j(~i ), I ~ i $ h, are conjugate over K, we can obtain L by adjoining just one j(~i ) to K. Theorem 2. quadratic field Ko

Thus the following theorem is proved: Let j(~) b_~eany cl~ss invarJant of 8n imaginary Then K(j(~)) is the maximal unr~mified abe]ian extension

of K. AS mentioned in w

we have then a canonical isomorphism between

the ideal class group C K of K and the Galois group G of L/K:

V-12

CK

=

G.

Let O-k denote the element of G corresponding to an ideal class ~under this isomorphism.

If ~ is a prime ideal of P

contained in the class ~,

then O~k_ = O--by the definition of the isomorphism.

We get therefore

from (5) the following explicit formula for the automorphismCY-k of L/K:

(6)

j( -i _k'), Now, let T

_k,

be an automorphism of the field of all algebraic

numbers.

Clearly, T ( L )

Since

is a quadratic field over Q a n d S ( K )

K

(L) = L. Galois

e CK

is the m~imal unramified abelian extension o f ~ ( K ) . - K, it follows that

Therefore, L/Q is also a G~lois extension.

Let7

denote the

group of L/Q

and let /~ be the element of ~ w h i c h maps ~ in L J to ~ (conjugate complex of m). Clearly, ~ 2 = 1. But p induces a non-tri-

vial automorphism on the imaginary field K. ~is G.

Hence the order of ~O is 2 and

the semi-direct product of the subgroup ~l~/o3and the normal subgroup Let ~

Since ~ =

be an ideal in an idesl class k.

Then/O(j(k)) = ~

NK/Q(~) is a principal id@II, ~ i s

(?)

= ~

= j(~.

in the class k -1 and we have

p(j(k)> . j(k-1).

Combined with Theorem I, this gives us the following Theorem i'.

The class invariants J(~i), i = I, o.., h, of an

imaginary quadratic fie3d K form ~ complete se__~_to_~fconj1~atesover Q. Therefore, if we put f(t)

=

h _ i=~l (t - J(ki)) , f(t) is an irreducible

polynomial of t with integral raticnal coefficients. Another consequence of (7) is the following: Let (Y be an arbitrary element ~ G'~

G a n d let
in ~ P

-I is again contained in G, and ~t follows from (6) and (7) that

V-13

(~,r

(i,~r

- p (jC_k'~,-~)) - j(~,), o=~(j(_k,))~

for any class invariant J(k'). Hence

f~rp -i " o "i, This, together with the fact t h a t ~

o- ~

is the semi-direct product

and G, shows that the structure of the Galois group ~ mined when we know the structure of G ~ C~.

o.

o~.f~.,p~

is completely deter-

IAS, 1957~8

Seminar on complex multiplication

~-i

VI

REMARKS ON CLASS-INVARIANTS AND R~LATED TOPICS (S. Chowla, Dec. 3 and Dec. i0, 1957)

Let p > 3, be a prime.

Suppose that the class-number of the imaginary

quadratic field R(~-:~) is 1.

(I)

In this case we have seen that

2 is a rational integer@

Here |

3

[I + 240 Z ~-3(n)q2n~ j(~) = J-

(2)

.

and q = e W ~

.

.

.

It is a remarkable fact (see Weber, Lehrbuch derAlgebra,

Bd. 3, 457-462) that not only the number (• but also its cube-root is a rational integer.

Under the same restriction on the class-number it is

also true (Weber, p. 504) that ~j(w) - 1728

I~

+ ~

1

2

is a rational integer. We can now formulate the Theorem

If the class-number of R(~-~) i_ssi, where p is a prime > 3,

then the diophantine equation - py

2

= - 1728

has a solution in rational integers with x = ~e" ~ / 3 ~ where [u} denotes the integer nearest to u.

VI-2

It is believed that p = 7, ii, 19, 43, 67, 163 are the only primes ~ 3 such that the class-number of R ( ~ )

is i.

Heilbronn and Linfoot have proved that

there is at most one more prime p with the property in question (by D. H. Lehmer we must have p ~ 109 for such a prime).

Chowla and Selberg (Proc. Nat. Acad. Sci.

U.S.A. 1949) have proved that the existence of a prime p ~ 163 with class-number of R(~-p) equal to i, would imply the falseness of the "extended Riemann hypothesis". The formulation of the above theorem is not without interest for the following reason.

Siegel's method (Act~ Arithmetica, Bd. i, 1936) does not lead to a deter-

minable constant c beyond which the class-number of R ( ~ )

is greater than i.

Also

the Thue-Siegel-Roth theorem on the finiteness of the number of solutions of certain diophantine equations (including the form ay 2 = b ~

+ f of our theorem) does

not lead to determinable constants c such that the equations have no solution when the variables in the diophantine equation are absolutely greater than c.

Our

theorem would be useful if, for example, one could use the methods of Delaunay and Nagell to prove the existence of determinable constants

Cl(~ ~) and c 2 such that

_ py2 = _ 1728 has no solutions in integers for x ~ e C l ~ and p ~ c 2.

One

could then deduce fr~a our theorem that there is a determinable constant c3 such that the class-number of R ( ~ - ~ ) is greater than i for p ~ c3. be remarked here that a recent claim by Kurt Heegner, (M.Z. 1954

It should also ) to have

solved this problem, seems unjustified. 2.

If m is a positive integer such that the classes of reduced binary quadratic

forms of discriminant - m have a single class in each genus, one uses Kronecker's "Grenz Formel" to determine explicitly (L) the values of j (w) when ~ = i ~ .

Vl-3 We introduce the functions I ~(.j) =ql-~

f(~)

=

~

( i _ q2n)

e'~

(see Weber, p. 113) Kronecker's formula ("Grenz Formel") states that (ibid, p. 531) lira S--~I+O

Z v (Ax2 + 2 B x y + Cy )

2. C' (I) + ~ log A ~ 4m -

9 2.

log

where m = AC - B 2 and

1

=B+iym A

'

2

A

(The prime in F, means that x = y = 0 is emitted from the summation where x,y range over all integers.)

From this, without difficulty, one deduces, for

example, that

(6 + 3,4~") (2 + ~ ) 3

= (2 + ~ ) 2 (55 + 1 2 ~ ) Using the formula f2~ (~)

we can now deduce the value in algebraic form of J ( ~ )

~3.

(4)

9

We add a few remarks on RamanuJan, s function ~(n) defined by

oo Z "r(n) x n = x 1

~

n)2~ (I -

1

(Ixl ~.,)

v'z-~ [cf. the denominator of (2)]. Ramanujan conjectured and Mordell proved that the function "~(n) is multiplicative, i.e.

(5)

7(mn) = ~(m)

T(n) if

(re,n)= 1

Ramanujan also conjectured that foLlPrime p,

(6)

l (p) l

2p -

the best that is known in this direction is that •(p) = O(p 23/4) which follows from A. Weil's estimate on Kloosterman,s sums

p-I ~. x=l

(7)

~(x+~) e

Here ~ is defined by x ~

i (mod p).

Another unsolved problem (raised by

D. H. Lehmer) is whether ~(n) can ever be O. 4.

The calculation of class-invariants J(~) can also be made to depend on

the classical modular equations of Legendre and Jacobi. details of the connection.

For 0 < k ~ i write W

(8)

K : Io

~ /z-k~-~# -- ~ "

k' is defined by k2 + k '2 = I and

(9)

K' : I ~ cr o ~l-k, ~ " # III

Also write

q = exp(- rr K'/K) Then co

(lO)

:

s

-.CO

2

C

F( 89 ~, Z

l~ k2)

We recall the main

Let

W

o

~ Z ~

o

~-~,~'~

If L I_

nK !

I--K

where n is a positive integer there is an algebraic relation between k and called the "modular equation")

n - 3,

4~

n-7

~

+ ~%'~' = l +

~

=l

Both results are due to Legendre.

By setting k = ~, (and so k' = ~) we get an

K'

algebraic equation for k when -~- = ~ a n d

n is a positive integer.

Abel remarked

that the resulting equation for k is "solvable by radicals". We quote the following results

(11) If-~-K'= 2 ~ ,

then k

= (/~- 1)4(2 -/~)2(~_~)4 x(8 - 34T)2(V~ - 3)~(4 _ ~ ) 2 ( ~ _ V-~)2 ~(6 - V~) 2

~%amanujan in a letter to Hardy dated 27 Feb. 1913) (12)

I f ~K' -=~

(~)

, then ~

#

is the real root of

+ 2x~ + 2 ~ + ~

- 1 = o

(after multiplication by x - I the left-side becomes 6

+ i

- ~

- x 2 - x + I)

The following solution originated from G. P. Young (Am. J. of Math. lO, 1888, 99-130) and was simplified by Cayley (ibid, 13, 1891, 53-58).

(I3) is

The real root of

Vl-6

i0 where

A, D = 3900o + 182oo~ ~ (172o + 92OV~)/235 + 9~I'C" B, c = 39000 - 1 8 2 ~

(1720 - 9 2 o J ~ / 2 3 5

- 94~

(See also Journal of the Indian Math. Soc. 18~ 1929-30~ p. 273 of "Notes and questions"). 5.

Finally we quote the following results of Deuring ["Die Typen der

Multiplikatorenringe elliptischer FunktionenkBrper H ] (G. Herglotz zum 60. Geburtstag gewidmet) in Abh. aus der Math. Sem. der Hansischen Univ. 14, 1941, 197-272. Let

J(~) = (f24(~) _ 1613 where ~ = ~i/~2 is the quotient of two numbers ~i and ~

which form a basis

of any ideal in an imaginary quad. field Z then by Weber (ibld, 540-541)

f(~)l~ is a unit of an algebraic number field if q is "roll zerlegt" in Z, otherwise

f(~)

the product of - ~ -

by a certain power of 2 with positive exponent is a unit.

From this result of Weber (based on a simple argument using Kronecker's Grenz Formel), Deuring deduces: A singular invariant j of characteristic 0 is either not divisible by any prime factor of 2 or by every such factor according as 2 is "voll zerlegt" or not in Z.

vz-? By more difficult arguments Deuring proves analogous results when 2 is replaced by 3~ 5, 7 or 13.

We quote from p. 271 of his paper (also p. 257 for a

relevant table) I J ist dutch keinen Primfaktor yon 3 oder 5 teilbar oder dutch Jedenw Je nachdem 3(5) in F roll zerlegt oder nlcht. J - 26 9 33 I J + i ist durch keinen Primfaktor yon 7 teilbar oder dutch Jeden, Je nachdem 7 in ~ roll zerlegt oder nicht. J - 5 ist durch keinen Primfaktor yon 13 teilbar oder durch Jeden~ Je nachdem 13 in Z roll zer~gllt oder nicht.

Vl-8 ~j6. In this section we shall give a new proof of the fundamental fact that the function

~3

co

I + 2~0 j (,r') -

Z 1

e 2"i?~ "co ~ i

O-3(n)e2n"x' j_ (I - e 2n~iT )24

is invariant for the transformations T ~ is, of course, obvious). O-a(n )

T + I and T ~

- l_ (the first T

Here Ira(T) > 0 and

"

Z

da

.

dln The proof is based on the functional equation for the Riemann-Zeta-Function

(i) and on Mellin's Integral (s - ~ +

(2)

i F~Y

S

it)

r(s)y'Sds = e'Y .

(~.~o) Here Re (y) > 0 and the path of integration is the vertical line 0"- k where k ~ 0.

~e first prove the Theorem

Write

a+l

(3)

co

Ha~) - z

~a

Cnje_2nwy

--~-

+ (-I)

I

P (a+ I ~ (a+l) (2,)a+l

Then if a is an odd integer > 3, we have a§

(h)

(-1) 2 ya+l Ha(Y) . Ha(1) .

There is a slightly different formulation for the case a - I, which enables

VI-9 us to study the effect of the transformation T ~ of j(T).

- ~i on the denominator

For the numerator we use our theorem for the case a = 3.

bining these two results we deduce the property of J(~') in question.

CornWrite

O0

(5)

G(y)

= G(y) = X G-a(n)e'2n"y I

and

I C6)

FCs) - ~(~) ~ (s-~)

co

~,a (n)

- z 1

.,

for ~

a +I]

.

n

From (2) and (6)

(7)

G(y) - 2.i

r (s)F(s) (2~) "Sds . 6"~a+2

From (I)

(8)

2 r(s~) ~ (s-a) cos (s-a),. 2 = (2.)s'a ~(1-s§

Since a is odd it follows from (6), (i), and (8) that a-I (9)

2 ~(s) r(s-a)F(s)sin(s.) (-I) -~= (2.)2s'aF(a+l-s) .

Multiplying both sides by (I0)

F(a+l-s) and using N(x) r(l-x) = ./sin(.x) we get

a+l r(s)F(s) (-I)-~" = (2.)2s'a-I V (a+l-s)F(a+l-s) .

From (7) and (lO)

VI-lO

G(y) - i

a+l (2.)2s-a-l(2~y)-S(.l) 2 r(a+l-s)FCa+l-s)ds

f

=a+2 t~oo I

a+l (2~)a+3+2ti(2my)-a-2-ti(.i)'-~--~(-l-ti)F(-l-ti)d%

t=-oo ~and now transform by t ~ - t) t=+co a+l = 2~iI L (2")a+3-2ti(2"y)'a'2+ti('l) 2 ~(-Z~i}FC-Z~i)at t a+l

I = 2.f

J (2.)a+l-2s(2my)'a'l+S(-l)" ~ O"=-I

I"(s)F(s)ds

so that (n)

G(y)

=

a+l (-l) 2 ~)a+Z

(2-~)s F(s)FCs)ds . ~=-!

Next, from (7) ,

G( )

(~-~+2)

-2~i'

i

" 2,,t

(2-Z~)S P(~)~(~)ds

S )s F(s)F(s)(Is (~--z) (2-~--~

§ %s~ o~ residues o~ ~n~e~r,n~ a~ ~ e , Thus, using (Ii)

(z2)

a+l G(~) . (-l) 2 ya+iG(y) §

I r(~§ ~ c~§

a§

+ r(z) ~ (z-a)(21~.) + ~ co~ ~(-a~ 1

.

, - a'~,, - ~ , , - O }

.

VX-II Since a is odd and > 3 it follows from (I) that ~(l-a) = O, and so we have from the last relation: a+l G(}) = C-l) 2 ya+l GCy)

+

r'(a+Z) ~(a+l)(2-~)a+l

+

S-(o) [(-.).

Using (I) and ~Co) = - 89this gives for a+l (-I) 2 ya+l {G a(y)+

odd a ~ 3,

a+l

C.I)T

~a+l) ~(a+l)} (2.)a+l

=

a+l Ga(}) + C.I)T F(a+l)~,(a+l) (2.)a+l

and hence, from (3), we obtain C4) which is our theorem. our theorem slightly for the case a = i.

ol 9 " - ~ ~

In fact when a = i, (12) gives

.,.~ ~ - ~.

or

From C13) we shall deduce that

e-2"y * C]~)

(I - e'2n"y)24

I

. y-12

OO

e -2"/y TT I

(1 - e "2n"/y) 24

Ve have to modify

VI-12

1

which shows the effect of the transformation T of J(~' ).

(15)

To prove (I~) we write

f(q) - q ~

(z-qn) 2~ - ~('I")

1 where (16)

q = e2trlT

.

We have by "logarithmic differentiation", O0

(17)

q

f,(q)

= 1 - 24

.

Z ~ ( n ) e 2n"i 1

g,(T)a!~

. ~'(~) 2.i q So

(~8)

2,d g < T )

"

From (13), (].9)

T 2 I1 - 2~ ? I

o'-l(n)e 2n"iT} 2n.i oo

= ~I - 2~ Z

O-l(n)e

1

or, us~g (17), (18), (19), I g'(- ~)

(20)

Hence

2'~i

"

. + 6iT

I + 6iT, 11

on the denominator

VI-13

(21)

~=

~2 g'('~) g(- ~) " ,2 -~

(22)

I log g(~) -- log g(-~) - 12 log'1' + log k

where k is a constant independent of ~ .

~,(f) I g(- ~)

(23)

.

Thus

k TI2

Setting T = iy this becomes

e-2"y

"~" (i - e-2n"y) 24 i

e -2.'/y _"I'r176176 _ (i - e-2n1~/Y)2)4 = ~ i Set y = I here to get k = i; (14) is proved.

k "

From (23) [or (14)] and the case

a = 3 of our theorem [(3) and (2)] we see at once that J(T) = J(-TI--). ~7.

By combining the results of Deuring in w with our theorem in ~i we get the: Theorem

If p is ~ prime ~ 19 and if the class-number of R ( J ~ )

is i,

the equation x 3 _ py2 . .

8

has a solution in rational integers and indee___~dwith X - ~ ~Le ~

where lul

the integer nearest to u. Incidentally we observe that if p > 19 is a prime, and the class-number of R(~-~) is I, then

is a multiple of 6'

VI-14 ~8. Hecke [Uber die Kroneckersche Grenzformel fSrreele, quadratische KSrper und die Klassenzahl relativ Abelscher KBrper.

Verhandlungen der Naturforschenden

Gesellschaft in Basel, Bd. 282, 1917, S. 363; see also his paper "Bestimmung der Klassenzahl einer neuen Reihe yon algebraischen ZahlkSrpern. GStt. Nachr. 1921, S. 1.] has proved that for a real quadratic field of discriminant ~ > O, the Grenzformel takes the following form: let ~ > 1 be a unit of the field:

(1)

~

u§

(u,v~l) 2'

and if ~ runs through a complete system of non-associated (with respect to ~)

(2) ~

numbers of the ideal

- [5, =2 ], % ~

- ~2 ,, - N V E

> o, N - N(~)

then we have

(3)

N(~)s ~

+

m

4Z

I

. ~.

2

1

~(~) + el(S-l) + e2(s-1)2 + ...

where

where E is Euler's constant and

(5)

~2ev - i ~

~2ev + i ~

%J

h J § i~

i~

Herglotz (Ber. 8. a. Verhandlungen. Akad. Wiss. Leipzig. Math. Phys. Elasse, 75, 1923, 3-14) obtains some curious results from Hecke's formula.

The

following is a sample"

A direct evaluation of this definite integral is probably difficult'

IAS~ 1957-58

Seminar on complex multiplication

VII-I Vll

CONSTRUCTION OF CLASS FIELDS

(Carl S. Herz~ Dec. 18~ 1957 revised Nov. 1965)

SO.

Introduction. The purpose of this lecture is to give some explicit constructions of

class fields of imaginary quadratic fields by arithmetic means. Let

K

be an algebraic number field and K

H

the group of ideal classes

of

K.

The absolute (Hilbert) class field of

is the class field corresponding

to

H~

it is the unique maximal unramified abelian extension

H

generalj it is not easy to determine the structure of the group explicit construction of the field this way.

Suppose

K/k

H

is very difficult.

is normal with Galois group

~;

of H

K.

and the

The problem is attacked then

H/k

and by means of class field theory one can describe the Galois group exactly in terms of

N~

H~

and the action of

and restrict our attention to to the semi-direct product Galois group of writing

H

H/~

~H.

and that

in the form

K/~

~

on

H.

cyclic; in this case

is normal~ ~

of

H/k

Here we shall take P

k =

is naturally isomorphic

The rest is algebraic number theory: H/K

In

knowing the

is unramified we can go a long way towards

H = ~(~l,...~6n)

where the

9i

are roots of explicitly

determined polynomials. When is given by

then

L/~

is a quadratic field then

k_~ = ~ - I

simple indeed. K;

K

~ r H

Suppose

L

and

T

n ~ ~2

the generator of

is a subfield of

H

and the action of ~.

~

This situation is very

which is cyclic of degree ~

is normal with a dihedral Galois group.

K~

over

It is easy to describe

dihedral extensions of the rationals; thus the problem is to find such fields which are unramified over

on

L

and this does not require complete knowledge of the

VII-2

class group

2.

The cases ~ =

2, 3, or 4

are simple enough so that explicit

calculations can be carried out. In summary, if the exponent of the class group number field

K

divides 12, the Hilbert class field

arithmetic means. H = K(j(~))

If

where

K

j(~)

H

H

of a quadratic

is easy to construct by

is an imaginary quadratic field we have (V, Theorem 2) is any class invariant.

Except for very small discriminants,

the calculation of the class invariant is intractable unless ~ ~4 X Z ~ - 2

H ~ Z~-I

and in these cases the arithmetic construction of

H

or is respectively

immediate, almost immediate. w

Preliminaries from class field theory. LEMMA I. Proof.

I_~f K/k Let

abelian extension of extension of

K,

Let

O

is normal then

K O = K.

Since

H

is normal.

~

on

H

F

F

Then

and hence K/k

Let

H

~O

H O = H.

(action on the right).

K

Th@n the Galois ~roup o_~f H/K

Then

K.

correspondin~ i_sscanonically

H.

is an extension of

depends on the particular element

H

H/k. by

u ~ H2(~,H)

~.

According to the Artin The precise determination of

u

class of

In order to avoid going into details

under a natrual projection.

here we restrict our attention to subfields of

K/k

k = ~

are ramified over

F

describing the extension in question.

Accordin~ to the Weil-Shafarevitch Theorem; K/k,

K.

is an unramified

be the class field of

denote the Galois group of

Reciprocity Law,

H/k.

2, the group of ideal classes of

o_~f ideal classes.

isomorphic as a h-module to

be the Hilbert class field of

is the unique maximal abelian unramified

H@ ~ H

ARTIN RECIPROCITY LAW.

Let

H/k

denote the Galois group of

there is a natural action of

to the group

H

be an isomorphism of

it follows that

~

Let

is the image of

u,

the canonical

(it suffices to assume that all abelian

k).

COROLLARY OF SHAFAREVITCH-WEIL THEOREM.

If

K/~

is normal with Galois

Vll-3

group

~

then the Galois Erou~ of

H/Q

is isomorphic to the semi-direct product

n_H. For

K

an imaginary quadratic field the appeal to the Shafarevitch-Weil

Theorem can be avoided. the class invariant

We have seen in V that

j~)

K.

Let

T

be the automorphism of

complex conjugation,

the restriction of

of

(j(~))~ = ](~) = j ( ~ ) .

Moreover,

~

to

can also be described as the automorphism of ~The elements

~ r H

> - Vq-,

H/~.

H/~

Thus we have

where

h = ~H

j (&l)

K

H/Q

given by

is the non-trivial automorphism

Let

kI

H/9

determined by

> j (&l)

be the unit class; then

T

.

are associated with automorphisms >

of

where

is a root of a polynomial with coefficients in

which is irreducible over

K/~.

H = K(j(~)) = ~(V~-,j(~))

2h

~

,

j(~l )

distinct automorphisms

= class number of

morphisms exhaust ~, -i (V~)T k__~ =I/d- and

> j(&-l)

K.

Since

the Galois group of

H/Q.

~I'''''~'

~kl'''"Tkh

[H : Q] = 2h,

of

these auto-

By direct calculation we have

J(kl ) ~ - I k T-- = J(--ik)kT_ = j(~-I)T = j(~-~) which proves that

w

K/k

class field of

F

is isomorphic to the semi-direct product

H/k

If

K. F

be abelian with Galois group The genus field of

~

must contain

a subgroup

~

of

LEMMA 2. the elements

K/k

is the Galois group of

the c~mLLutator subgroup of in

Hence

The genus field and the group of genera. Let

of

T-I~T = k__ T.

k I-T

~'. 2;

F. Thus

Since F'

H/k K/k

~,

and let

H

be the Hilbert

is the maximal abelian subfield then

G

is the fixed field of

G r'

is abelian, the canonical image of

is canonically isomorphic as a

n-module with

this subgroup is called the principal genus.

The principal genus where

~ e 2,

T c ~.

G

is the subgroup of

H

zenerated by

VII-4

Proof. by

~.

Let

We know that u

c F

F'

since

is canonically isomorphic to an extension of

be a representative of

identified with the products generate

F

H

u

9 C ~.

The elements of

u -I k- I u k = kl-7 . 7 9--

k and 7 --

F

are

These co~m~utators

is abelian.

The factor group

~

= H/~

is called the group of genera of

K/k.

is canonically isomorphic under the reciprocity map with the Galois group of

It B/G.

The essential fact used in the construction of the genus field is given by PROPOSITION I.

Let

K/k

unique maximal abelian extension Proof.

Let

the discriminant of

Ik K/k

be abelian. G/k

denote the group of ideals of and

PK

in the course of this proof. K/k

if

class field to

K.

NH/K PH = PK "

field of

H/k

K/k,

is the

K/k.

relatively prime to K~

again

an assumption made for all ideals

as subgroups of

By definition~

Theorem~

k

the group of principal ideals of

An abelian extension

NK/k PK = NL/k PL

K/k

with the same ramification as

relatively prime to the discriminant of

as

The genus field of

L/k

Ik.

NH/K IH = PK '

Since the genus field

G

has the same ramification

Now let

H

be the Hilbert

but by the Principal Ideal is the maximal abelian sub-

we have

NK/k PK = NK/k NH/K PH = NH/k PH = NG/k NH/G PH c NG/k PG NK/k PK = NK/k NH/K IH = NH/k IH = NG/k NH/G IH = NG/k IG and so K/k 3

NK/k PK = NG/k IG = NG/k PG " Thus and

G/k

any extension extension so

L

L/k

G/k

has the same ramification as

is the class field to the subgroup L/k

we have

NL/k PL c NL/k IL

with the same ramification as

is a subfield of

NK/k PK

of

Ik.

Since for

it follows that for any abelian K/k

we have

NG/k IG c NL/k IL ;

G.

Proposition I leads to a simple effective procedure for constructing genus fields.

To do this we examine

NK/k PK "

Given a prime

p

of

k

let

Up

VII-5

denote the group of elements

of

k

relatively prime to

p

and let

V

be the P

subgroup of

U

consisting

of those elements which are congruent

to norms from

K

P modulo arbitrarily that

high powers of

(~) r NK/k PK

p

E0

E0

the discriminant

where "congruent"

of

n

and the intersection

K/k,

including possibly real infinite primes as "has the same sign".

is the subgroup of

and U

K/k

is

consisting

of elements which are congruent

to

P

The situation is particularly the rationals

of prime degree

p ~ 0, I mod n *

In case

[Up : Vp] = n

n-th powers modulo arbitrarily high powers of

Z r ' P

is taken over primes

it follows that

U np c Vp Un P

condition

P

k

is to be understood

cyclic of prime degree

where

NV

9

is the group of units of

dividing

Then a necessary and sufficient

is ~r

where

p.

since

Un = U P P

n.

[Up : U~] = n

simple when

Ramification

unless

the group of units modulo

p.

nl~(p r)

r

p .

for

K

is a cyclic extension of

can only occur at primes where

Moreover,

=Z r P

~(pr)

is the order of

is cyclic for

p ~ 2

Hence

p ~ 0,I mod n

with the single exception

At oo there is no ramification when [U~

is odd, for then

n

U=== Uo~ ;

but

: U2] = 2. We conclude

then

n

V P

= Un P

occur when V 2 = [l,v]

for all

n = 2. 2 9U 2

that if

is cyclic of prime degree n

finite or infinite except for

There are three possibilities

where

It is always the ramified primes,

p

K/~

and

p = 2,

p

ramifies

which can only

in the exceptional

case:

v = -1,5, or 3. true that

PQ/NK/Q PK N N Up/E 0 A V = = -p

or, for that matter,

where

p

runs over

any finite set of primes containing all

VII'6

the ramified primes.

Using the Chinese Remainder Theorem, we obtain a direct

product decomposition best written in the form (a&l isomorphisms being canonical).

(n Up)/(E 0 N Vp) ~ r p

P~/NK/~ P K = Fp = U P /VP

where P0

if

E0 c VP

and

~p = ( U p 0 ~ U p ) / E 0 ( V p 0 ~ Vp)

being a fixed ramified prime such that

-i ~ Vp0

~(Bp)/~

cyclic of degree n and whose ramification is described by

~p ,

Q

corresponding to

~

-

class field of ~(Sp).

~

is

corresponding to

Let

K/~

G = K(813...,St)

arbitrary

Since the genus field

~p

,

G

is the

it must be the compositum of the

d

(excluding

P0 m -I mod 4

for

n

odd and

extension of degree for

(fii)

n = 2

for

n = 2

b_eecyclic of prime de~ree n.

where

if

p n

and and

pl,...,p t P0 =oo

K

is cyclic of de~ree n ove_._./r ~

(ii)

namely the

What we have proved is

the discriminant

(i)

which is

p

THEOREM I. K/~

.

E 0 ~ Vp,

The reason for writing

the product this way is that there is a unique extension

class field of

if

The genus field of

are the distinct prime divisors of

if

K

is imaginary quadratic and an

is real quadratic) where for each

Pi'

~(ei)

and specified this way:

given (so

p ~ 0, I mod n),

which ramifies only at p ----Irood 4,

~(8)

is the unique cyclic

p,

e = ~-,

p ~ - I mod 4,

@ =VCp

or

8 =~p0 p

according to the

imaginary or real case, (iv)

for

n = 2

8 =~'2

if

and

or

so

K = ~(V~),

the cases are

d = = 8 mod 32

8 =~/i-I or ~ e = ~-2

p = 2,

if

/~0

if

d z4

mod 8

according to

d ---8 mod 32

according to

It may not be obvious how we arrived at (iv). that if

2

ramifies then

K = ~(~0 )

d < 0

or

d < 0

d > 0 or

d > 0.

To see it one observes

where the discriminant is

4d 0.

Then

VII-7

since

I - dO ~ V 2 -i ~ V 2 are

and

F 2 = U2/V 2

d O m -2 mod 8

3 ~ V2 ;

and

-i -c V 2 ;

_~(2~0)

considers at ~

H+~

One puts

G+

or

where

pl,...,p s

Therefore either

isomorphic to

~ F . p d P

Let

q > 0

q ~ V

and of

7p'~ q

q

G+ = G

or

d > 0

Q(V~)

we can or

we must use

Q(~--2) Q(V~0)

K

G/~

K

one

which is unramified except H+/Q

and then

G + = G(~P0 )

according to whether

dividing the discriminant.

is

I~/r'

n

and~ as we have just seen~ this is

has been extended to

p = p'

if

n

G.

Then

acts non-trivially~

be a prime such that

/ = I.

q ~ d~

q ~ Vp, ~

n

acts on each

p = p"

if

q ~ Vp,,

n

acts

(assume

The interpretation of the reciprocity symbol gives the fact

remains prime in

~(0p,)/~

on the residue class extension at (G/~q)

in

r2

d < 0

in the quadratic cases). Then q determines a generator P0 The reciprocity law for the extension Q(Sp,)/~= _ says

~p,

q

When

If

The reciprocity law makes this isomorphism explicit.

we shall write

trivially.

In neither of these cases

are the distinct prime divisors of the

PO ~ -i mod 4

Assume that the action of ;

3 ~- V 2

3 m i - d o mod 8 so

P0 =----i mod 4.

The Galois group of

that

5 ~ V 2.

the maximal abelian subfield of

or not there is a prime

that

so

I~2 = Up0 ~ U2/E0(Vp0C'7 V2).

5 -c V 2

then

The remaining possibilities

the maximal abelian extension of

discriminant.

d0~- 2 mod 8

In the former case

5 m i - 4d 0 mod 8 hence

If

In the classical treatment of real quadratic fields

G + = _~(~l,...,V~s)

7p,

~(V~.

d o m -i mod 4.

instead where

REMARK.

~(Sp)

1 + d~0.

P0 = oo and the class field corresponding to

according to whether and

is the norm of corresponds to

in the latter

do we have take

i - do

= (~yp,)-i

its divisors in

and q.

yp,

This determines

On the other hand

G/~_ are fixed under

induces the Frobenius automorphisms

n.

yp,

uniquely.

q must remain prime in

Hence

(K/~q)-lq

It follows K/~

since

is the generator of

VII-8 which induces the Frobenius automorphism on the residue class extension at (K/~,q)

Clearly

Galois group of generator of

is the image of G/~

~.

p]d

q

such that

q

Let

Given an action of

~

K/~

_

such that

~

is a

remains prime in

K/~

and determine the

p'

K

and the

with the Galois group of

for

Fp

such that

= (p~d r p ) / ~

~(@i)/~_ _

9 = n P' 7p.

where

imaginary quadratic one takes

hence also on if

7p %

_

Fpi

G = ~(el,...~9 t)

p'

generates

denotes those

Pi

acts non-trivially o_nn @ . . 1

For

and 2

Id yp

has not been given in advance it

on the genus field

and the group of genera is

H3

G

'

~ =

be cyclic of prime degree n with Galois group

there is a choice of generators

on

on

Hence

q ~ V . P To summariz% we state

reciprocity law identification of

H

~

K/~.

such that

THEOREM 2. ~.

under the natural projection of the

onto the Galois group of

If the action of

suffices to choose as those

(G/~,q)

q.

d-

G.

4 rood 8

or

Then the

p'

T

are the

to be complex conjugation

pld

such that

d - - 8 mod 32.

The concrete isomorphism of

~!~ with

Each genus can be represented by a prime

~

( ~

pld

of

K~

rp)/ll is made this way. but since conjugates belong

to the same genus it suffices to specify the rational prime The groups in question are isomorphic to rational primes

p ~ -I mod 4

ql,...~qt_ I

Z t'l ; =n

q

which

~

divides.

so it remains only to give

representing generators of

%

in correspondence

with certain elements of

~p

N ~ 9 Given yp as an automorphism of G/~ ~ let pld P be the automorphism of K/~__ obtained from the natural projection of the Galois

group of

G/~

onto the Galois group of

trivial (this is no restriction at all). can find integers

el~...,et_ I

as elements of the groups primes

ql~...,qt_ I

~p

(qi > 0

K/~ .

Take

Pt

so that

is non-

Since

such that

~ is cyclic of prime degree we e. ~Pi ~Pt I = I. Now consider the 7p

defined by certain congruences in and

7p t

qi ~ Vp0

~

and choose

in the quadratic cases) such that

VII-9 e

Pi qi ~ YPi Vpi ~ ~ t THEOREM 3.

Vpt~j~i, tN

The canonical isomorphism

Vpj

(p~d ~p)/II

> ~

i__sscompletely

e.

specified by

YPi Yptl --->qi ~

i = l~...~t-l.

viewed a~s the Galois group o_~f G/K

and

~

Here

(p~d ~p)/n

is to be

as the firoup of ~enera of

K.

e.

Proof

The

l generate a subgroup A ?Pi YPt 6 = I, but ~ I where ~ generates

of

9 8 ~ A and of

we have A ~ Z t'l

-=n

= (p~d l'p)/~.

G/K = K(01,...,gt_l)/K.

K/~

and in each

follows that

~(Sj)/~

Ki

remains prime in

~.

Therefore we may identify

for

j ~ i,t

A

qi

. For

P A ~

qi

splits in

~(ei)/~.

splits completely in

K(61~...~i_l~gi+l~...~t_l)/K

K(ei)/K , where

is any prime of

Ki

which leaves

0j

~i

fixed for

splits in

K/~

K

dividing

it and

qi"

The

with the automorphism of j ~ i

and which induces the ~i

the residue-class extension at

is the same as the residue-class extension at

= [I},

with the Galois group

but does not split in

Frobenius automorphism in the residue-class extension at Since

F

Thus

According to the reciprocity law~

reciprocity law demands that we associate K(~I3...~0t_I)/K

~

p|d

qi

of

of Ri

K(0i)/K. of

~(9i)/~.

K(ei)/K

The auto-

e.

morphism

?i

of

Galois group of

~(6i)/~

corresponds to

Yi Yt I ~ A

when

A

is taken as the

K(el~...~et_l)/K.

The above three theorems give a complete description of the theory of genera for

K/~

cyclic of prime degree n.

This is equivalent to the theory of

the representation of integers by certain homogeneous forms of degree n in n rational variables with integral coefficients. tedious when

n > 2.

The explicit computations are

In the quadratic case the computations are nearly trivial~

especially since there is no choice of generators to worry about. For contrast~ we shall give on illustration of Theorems I, 2, and 3 in

the

cubic

case 9

VII- I0

Illustration. occurs at

Take

Pl = 3, P2 = 7.

G = Q(61,62)

K = ~(@)

q = 5

remains prime in

where

Y3

follows that

~3 "

(Also

H =%.)

~(62).

~_+4 mod 9

Ramification

2 > Y377

59 <

Y7

to elements

~ = Y377

m +--2mod 7.

According to Theorem 3 the non-

and

x,y,z ~

2 > Y3Y7 .

2 <

The theory states that

of a principal ideal, i.e. there exist

We see that

Hence, by Theorem 2,

and

~3 = 72 ' ~ 7 = 72 .

principal genera are given by ~=

3 62 - 3"762 - 7 = 0.

and

~(6), ~(61) , and

corresponds to elements

It necessarily

@ 3 - 216 - 28 = 0.

The genus field is, according to Theorem I,

@I3 " 361 " i = 0

where

where

m -c U 3 f~ U 7

We have

is the norm

such that

m = x 3 + 714y 3 + 16"7z 3 + 6x2z - 3xy 2 + 9.Txz 2 - 12.Tyz 2] , iff

m --=+I mod 9

a prime

and

q ~ 3,7

m =+_I mod 7.

The cubic reciprocity law states that for

the congruence

28 ~ w 3 - 2 1 w m o d has a solution iff

q

This is equivalent to

q a b Y3 Y7

corresponds to q

__a_b Y3 Y7 = I,

where

being the norm of an ideal.

of non-principal

ideals while

presentation of

T, Y3'

and

2"59 77

Thus

2

i.e. and

q --_+2k mod 63. 59 are norms

is the norm of a principal ideal.

The

as automorphisms of the appropriate fields is

too tedious. Let us return to the quadratic case. quadratic field (also assume in

K,

K = ~(~)

(P0 m)

= I).

and

d

be the discriminant of a

a positive integer relatively prime to

We have seen that

m

d

is the norm of a principal ideal

i.e. X

has solutions 21d.

m

Let

x,y ~ ~

s

2

-

dy2

(;) = I

---- 4 m

for each odd

There is one class in each genus, i.e.

x,y ~ ~

exist then solutions

quadratic residue

mod p ,

x,y ~ =Z

and if

m

exist.

~ =% Here

pld ,

and

m ~ l,v mod 8

if

iff whenever solutions (;) = +I

if

is a prime, the splitting of

m

is a m

in

K/~

Vll-ll

is determined by the classical quadratic reciprocity Example i. 7 = 727377

Take

Generators

d = -84.

for

%

Then

x 2 + 2 ly2 = m

H = G = Q(~,

~/~3-, V~TJ.

We have

are given by

19 ~----->7277 The equation

law.

5 <

> 7377

.

has i~tegral solutions whenever

it has rational

solutions. Example 2. and a generator

Take

for

~

d = -87.

is given by

Then

G = ~(~J~, Aft).

2 <

> 73Y29 .

to the principal genus, and, indeed, the equation solution

x = 5/2, y = 1/2

primes dividing other hand then

7

give non-principal

Therefore if

We shall see later that Take

3 P-7

d = -39.

H = Q(~-~,~-~E~5) Then

2.

P_~

is in the principal genus, but

if

2 <

G ( - ~ j %/~).

is given by

P_2 ~ P_2

since

> 73713

2

.

Let

for

~

Take

since

d = 40.

is given by

integral solutions whenever m > 0

Hence the

K

On the

dividing

is unramified.

where We have

In fact

g2 _ 5 6 7 = y3 K

P_~ ~ ( 2 ) =

x 2 + 39y 2 = 4" 22

I = 0.

and a

is not principal. Hence

7

dividing To see

P_~+T;

with

so

y ~ 0,

_H N_ Z 4.

Example 4. generator

has the

P--2 be a prime of

p2

P_.~ were principal there would be a solution to

y -c =Z" Here

corresponds

is in the principal class.

%

this, observe that

+ 87y 2 = 4-7

P--7 is a prime of

generator for Obviously

7

classes in the principal genus.

P--7 is in the principal genus and

Example 3.

2

Notice that

7 = 73

but it does not have integral solutions.

162 + 87" 12 = 73.

_H ~ =Z2 • Z 3 .

x

We have

3 <

Here

G = __Q(~/~.

> 7275.

The equation

We have x

2

T = 75 9

- 10y

2

= m

The

has

it has rational solutions and there is no need to take

32 - I0"12 = -I.

Example 5. has integral solutions

Take

d = 21.

for all

m

Here

G = K.

such that

x 2 - 21y 2 = _+4m has solutions whenever

The equation

(3) = i

(3) (7) = i.

and

x 2 - 21y2 = 4m

(7) = I.

Therefore

If we impose the restriction

Vll- 12

m > 0

there is a change:

is not a norm.

In this example

Example 6. = Y2 m m~l

or

Take

T = YI7'

mod 8~

The equation

w

2

- 34y

2

= -I.

Structure Let

H

k]

G = ~(V~, ~/17).

- 34y x

2

2

= m

- 34y

There is a choice of

has rational

2

= m

solutions

has rational

solutions

but not necessarily with the

H+N

~4

for all

+

then

sign.

there is no integral solution

while

= ~

= ~

~2

of the group of ideal classes. K/~

be an abelian extension of order n. of

K

~I-T

Then for any

corresponds

an ideal of

2

Whenever

In this example

generated by the

N = ET~] ~.

x

ideal but 17

H + = 9(qq-3, VrJT).

(5/3) 2 - 34(1/3) 2 = -I ;

the group of ideal classes of

but

Here

has integral solutions

The simplest example is x

H = K,

d = 136.

~ ~2 k mod 17.

x 2 - 34y2 = -Mn

to

the ideal (17) is the norm of a principal

and

T E ~

~ E H

G

Recall that

the principal

where

~

genus,

i.e.

is principal.

is a subgroup

of

~.

denotes

the subgroup

is the Galois group of

we have that

kZ -N E ~.

It follows that

Hn

Hence the group of genera

K/~.

Put

On the other hand,

to the ideal class of the norm of an ideal in

9,

H

~

and this, being

the group of n-th powers in ~

= H/~

is a subgroup of

H/Hs . In case the genus field here was

t

n G

is a prime we know is given by Theorem

one prime

factor

THEOREM 4. For

I.

as the Galois group of The fundamental

G/K

numerical

where

invariant

where

t = number of distinct prime (excluding

~

n = 2,

~

non-trivial

if

Let

~ H/H2

factors of the discriminant

P0 m -i mod 4 K/~_

for

if

K

~

K/~

is real quadratic).

_ b__eecyclic _of prime degree n.

n > 2,

d of

Then

N

Z t-I -- =n

~

i_~s~ quotient ~roup o_~f H/Hn

which is

H n ~ H.

COROLLARY.

Let

h

Proof of Theorem 4.

b_~e the class numbe___..~rof According

K.

Then

nih

iff

to Theorem i the Galois group of

t > i. G/Q

Vll-13

is the direct product of

t

When

Hence

n = 2,

= H2 on

N = i + T.

for

H

n = 2.

leaving

the n-group H/H_n ;

When

H_n

H/H_n.

H

n > 2

i.e.

that

G ~ H

k-c_~,

_~

however,

G = Hn

but I],

n

acts

acts on

fixed point of

~ in

further,

ideal classes,

denote

let k

ideal classes

the subgroup

which are invariant

The sequence

and

>H1-T>H--->!~_ _

~

>i

have the same order (observe

_kn = _kN = I.

One may argue directly

Z t-I " -- =n

.

the last statement

k_7 = k.

hence

that

there must be a non-trivial

H ~ H_n

1--->~_ is exact;

we cannot conclude

~

Therefore

If

this implies

I],

k T = k -I

since

Thus the cyclic group of order n,

composed of ambisuous

under

k I'T = k 2 ,

This shows that

invariant.

To elaborate of

cyclic groups of order n.

that if

Since H

n

is a prime,

that

G N

it follows

H/HI'I).

that then

contains an element of order n

For

H I] , ~ Hn

. the

group of fixed points, must be non-trivial.

REMARK. The old-fashioned treatment of genera of quadratic fields was based on the isomorphism For

of the genus

gives no information, G/K.

For it is just

field

however,

In the real quadratic

of the units ef K ,

_~.

n > 2

One proved directly

K.

case,

in a very straightforward

the analysis

n

that

on

_~ _N =nZt-l"

Hn is quite elementary and yields

about the representation

the action of

~T = k-l.

extension of

G

but this is ultimately

THEOREM 5.

LIQ

_~ with

imaginary quadratic f i e l d s the analysis of

the construction

of

of

of

of

_~

~

fashion.

It

as automorphisms

involves

consideration

irrelevant. H

may be complicated.

For

n = 2

Therefore we have immediately Let

Then

L

K/~

be ~ quadratic

is normal over

~.

L

field and Moreover

if

an unramified L/K

abelian

is cyclic then

is dihedral. A consequence

of Theorem 5 is that when

K

is quadratic,

H

is the

VII- 14

compositum of dihedral extensions of

Q.

These are, in principal~ manageable as

the splitting fields of polynomials with rational coefficients. Theorem 4 gives the number of 2-primary components of the group of ideal classes of a quadratic field. where

h

For ~

an odd prime~ the question of whether

is the class number is best handled by examining the possibility of the

existence of a cyclic unramified extension this topic in w

for the case

In case field

H

~/h

H~_ Z2 -I

~ = 3, then

_H = ~

L/K

of degree

~.

I shall treat

cf. Theorem 6. and so

H = G.

The absolute class

is then given by Theorem I; all that is involved is the simultaneous

adjunction of square roots of rational integers.

The table below shows that for

Idl

(The first exception for

is

small these trivial cases are quite common. d = 145,

_H ~_ Z4. )

fields with a given

H

d > 0

One knows that there are only finitely many quadratic as class group.

I do not know which finite co~-utative

groups are realizable as the group of ideal classes of an imaginary quadratic field.

VII-15

Let

K = ~(~)

structures with

be a quadratic

field of discriminant

of the groups of ideal classes

for all

d ~ -168

d o > -I00 when these are outside the list of

other than

d < 0.

d _> -168

Below we list the

and also

d = 4d 0

but give a group

Z~-I.

H

-d

(1)

3, 4, 7, 8, ii, 19, 43, 67, 163, ?

Z2

15, 20, 24, 35, 40, 51, 52, 88, 91, 123, 148, ... 84, 120, 132, 168, ... 420,

...

Z4

39, 55, 56, 68, III, 136, 155, ...

z8

95, 164, ...

z 2 x z4

260,

Z3

23, 31, 59, 107, 139,

z2 • z3

87, 104, 116, 152,

z4 x z 3

356, ...

Z5

47, 79, 103, 127, 131,

_z2 x =z5

119, 143, 159, ...

=Z7

71, ...

_z2 x z 7

151, . . .

Zll

167, . ..

...

...

...

...

VII-16

w

Unramified

cubic extensions 9

Suppose Galois group of

L/K L/K

w ~ K.

Then

L(W)/K

by putting

is cyclic of degree 3. and let

L(W)/K

= W.

Let

P

We extend

~ ~- L(W)

uniquely determined by the choice of the pair

o=

,

a, e~ a

It is easy to check that

l+p

=

and ~

O

where

Now suppose

K/Q

@

We suppose

to an automorphism of

e

such that

(O~)

~O-i

=

w

L(w)/K ;

is

~

up to multiplication

by

8 ~ K(WI

by

g2-

=

a~e ~ K~ 0 ~ L.

L = K(0)

for the

denote the unique automorphism of

There is an element

Define quantities

O

be a primitive cube root of unity.

is cyclic of degree 6. ~

which has order 2 9

W

Choose a generator

-~-

g2p- 1

Indeed

03 - 3a0 - ae = 0 . is quadratic and

the non-trivial automorphism of

K/~

L/Q

is dihedral 9

to an automorphism

T

We can extend

of

L(~)/~

with the

properties WT Put

= W

~ = ~T-I.

Hence

,

Then

8 ~ K(W),

by

2

(W-~2)~

T

2

= 1 ~

pT = Tp ~ 2

80 = ~TC-O

and we are free to replace if

8 = -I.

.

(W2~)T(-I~-i)

= (~T)(W-I

~

if

by

(i+8~

L/K

is unramified

it follows that for a prime

y E KI

~3 ~ KI ;

hence if

~ ~

then

where

of

L(w)/K(W)

K(W)

if

K = ~(~-d~

is unramified.

~]~ 3

is a prime of

is no loss in generality in assuming that

KI ~3

and

then

~3]~3.

~]~3

to

divides

~

of a ~ Z

~3

is

only if

~3(l+p) = a 3 p

factors in

then

is an integer of

divisible by the cube of any integer (other than a unit) in KI

8 ~ -I;

~-I) = ~T-I we replace ~ = ~T and

K I = ~(%/-/~-d).

a~e ~ ~. If

hand

2

With these replacements we may assume that

is determined uniquely up to a factor Moreover

OT = TO

Since

On the other 3 ~ I~ 3. KI

K I.

~3 ~ K(~)~

There

which is not

The norm from

The conclusion is that the rational prime KI

into non-principal

prime ideals whose

p

VII-17

cubes are principal. The above reasoning leads to THEOREM 6. number

h.

Put

Let

K = ~(~ Let

K1 =

mental unit of

KI~

be an imaginary quadratic

a

the norm of

hI g ,

field with class KI,

be the class number of e

and

g the fundaThen

the trace of

31h iff

at least one of the followin~ conditions holds

(iI)

a = +i,

(i2)

e =-. 0 mod 9

(i 3)

a = -I,

(if)

31h I .

e-- +2 rood 27

e--_+4 rood 9

K(@)/K

l_~f (i) holds then

K

i__ssa__nnunramified cyclic cubic extension of

where

@3 _ 3a6 - ae = 0. REMARK.

Cases (i2) and (i3) can only occur if If

Proof. L = K(@) ~3

suppose that

then

9 = ~ + ~P and

where

is a unit.

31h

If

~3

~3 = 6

K

has an unramified cyclic cubic extension

~3 ~ KI .

6 P.

The extension

can only ramify at primes dividing 3 in in

K/~ ,

~(@)/~ a = I

and hence

K(@)/K

3 ~ e).

cases condition Finally~

if

When

K.

If

e m +2 mod 27

31d

then

real quadratic field

KI

has no units other than

where

3 ~ d ~(6)/~

~ gk

@3 _ 3a8 - ae = 0

then

3

does not ramify

ramifies at 3.

(because when

that

we can

3 ~ d

K(e)/K

In turn we must have

but in all

be unramified.

has an unramified cyclic cubic extension, but a

~(V~I)

of the imaginary quadratic

by a factor

the analysis is more complicated,

(i) is necessary and sufficient

31h I

~

K(8)/K

is ramified iff

is unramified at 3 iff and

As we saw above, either (ii) holds or

is a unit then~ modifying or

31d.

can have such an extension only if the class number

field

__Q(~I )

is divisible by

3

since

Q(~I

•

Examples. (f I)

This is the case most frequently encountered~

e.g.

d = -23~ e = 25;

)

VII-18

d = -31, e = 29; case

d = -107

Also

h I = 3,

d = -116, e = 56; is more interesting.

g = 215 + 1 2 ~ .

e 3 - 3@ - 430 = o.

Note that

not yield an unramified dividing 3 in

d = -152~ e = 2050;

K I.

One has

Using

N(II+9~-~) 2

(il) we get

The absolute

but

3 mod [3

~6~I

class field of

= 133

K1

e = 79.

and

H = ~(V~l-~7,@)

N ( 1 7 + 2 3 ~ ) = (-2) 3

extension since

d = -1979,

h = 3

where

~3 = (17 + ~ / 2 where

is

[3

The

does

is the prime

H I = ~(V~'~@I)

where

3 81 - 3"1381 - II = 0. (i2)

d = -3.7.11

The absolute

.

Here

H Z ~

class field in

d = -3"29

.

class

field is

H = ~(-~/~ ~ ' ~

(ii)

d = -3.229

.

One has

H ~ ~2 X ~3 "

of the cases

H~

(i) apply.

h I = 3,

~2 X ~3 '

hI = 1

6 = (9 + ~ 1 7 ) / 2 .

6~).

hi = I,

N(26 + 3 ~ - 6 ~ )

The absolute

6 = (5 +~/-~)/2.

= 193 ,

K I = ~(2~)

On the other hand K

and

6Y%).

We have

The absolute class field to time, since

We have

H = ~(Nr/~, q ~ J - 7 , ~ ,

(i3)

in fact,

Here

X ~3 "

is

,

31h.

Also

~ = (15 + ~ - 2 ~ / 2

N(6 + ~) = 53

H = (~-~, ~/-~,

the absolute class

hence

field to

and

(6 + ~)Y3). KI

is

,

(6 +

21h

and,

so none 6 ) 2 ~ I mod 9.

At the same

H I = ~(~2-~-9-, el)

where

3 @ 1 - 3.19@ 1 - 52 = 0. w

Unramified Let

cyclic extensions K = ~(~a~

is an unramified L/~

of

2 ~ .

Then

G

where

d = d+d_

d < 0

field of discriminant

cyclic extension of degree 4.

the relations

where

(If

be a quadratic

is given by generators

satisfying

of de~ree 4.

F

S

and

2 ~4 = i = T ,

is an unramified

is the genus field. ;

T,

K

d+ > 0

and

Let

from Theorem

d_ < 0.)

F

extension of

the factors are distinguished

this means

is the fixed field of

OT = TO 3.

It follows

Suppose

L/K

We know that the Galois group of

where

quadratic

d.

o,

be the fixed field K;

hence

F c G

I that

by

Take

R

the fixed field of

T.

Vll-19

Then

R

is quadratic over Q( dV~++); so we can write R = Q ( ~ + , ~ where _~2 -c _~(~-~+) and ~ = -~-. On the other hand R/_~ is not normal; hence ~ _~ and

L = _~(V~_, ~r~.

the actions of both

O

One finds easily that

and

T.

It follows that

~14~

~14~

reverses sign under = m~_

where

m -c Q. I

The result is N~ = ~ I+O = m 2 d where

N

denotes the norm from

Q(~-~+)

to

Q=

'

There is obviously no loss in generality in assuming that square-free integer of square of a unit if

~(~+).

This fixes

~

~ is a

up to multiplication by the

~(~/~+) has all its ideals principal.

In general, what one

can say is given in LEMMA 3. ~(~dfa-+). Write i_n.n ~ ( ~ + )

Let

d

= do

In order that

F(~

in

k

Hence

~

be a rational prime;

o_/_r d_ = 4d 0 ,

i_~s (~) = m2d Proof. Let

F.

p

where

splits)

do

odd,

and

and

N~ = (do)

be a prime of

must be even unless

k

~12.

m

then

p

Thus

has n__ooprincipal ideal factors.

and suppose

Suppose k~

splits i__nn

The factorization of

~

k

exactly divides

it is necessary that

~I d_ .

pkflm2d - .

is odd.

and

F

pld

4 ~ d0 .

~(~+)

be unramified over

Taking norms, we have that (p

where

if

~Id0

k

and put

must be odd.

be a square pf = NIh.

Therefore

Everything is proved except for the case

2

in

F.

f = i

d_ = 4d 0 ,

This requires a detailed examination of the equation

locally at the primes dividing

x

2

- ~ = 0

The desired result is obtained, but we

omit the details. A simple description of the whole story is THEOREM 7.

Let

K = ~(~)

be ~ quadratic field of discriminant

necessary and sufficient condition that

K

d

is a norm from

_~(V~+) and

2

d.

have an unramified cyclic extension

o_~f de~ree 4 is that there exist an admissible factorization, splits i__nn Q ( ~ + )

if

d = d+d_

21d ..

~.

such that

Vll-20

Proof. says that filled.

2

If such an extension exists we have

splits in

Then

~(~f~+)

do = i + ~

must have the form

if

mld - .

for some

(~) = _dm I-~_ .

by modifying suppose

m

m).

is not principal.

Therefore

such that

(8) = ~

(812)% , 2 x - ~ = 0

i.e. we change

(la) =

m 2d

where ~

does not ramify over

has all the

it can be made so

~12.

Since

~

2,

is square-free,

and there is an integer 2.

with no loss.

8 ~ ~(~r~+)

We replace

m

by

The equation

at any prime not dividing F.

for

2

The ramification at

since 2

is handled

we have

and the calculation of

b > 0

J

b2d+

;

be fixed such that

can always choose the sign of at primes dividing I mod 4~ mod 4 II:

2.

(m 0 (~a,b)

a

both even or both odd

is odd). is a solution to the norm equation.

so that

x

2

- ~ = 0

does not ramify over

The proof requires an analysis of cases.

d_ m I mod 4 ; in

a,b ~ Z

N~ gives

2 2 4m0d 0 = a -

residue

where

(8/2)2k~ , F

~ = m0=

is relatively prime to

is relatively prime to to

= (a + b Y+)/2

d+~

k

m

~i+~ = (2)

R

then

is not square-free~

is the square of an ideal in

this way:

Let

~

We may also assume that

is exactly divisible by

The prime factorization of

m 0 = Nm_ ;

properties previously given (if this

and Lenmm 3

Now suppose the condition is ful-

~ ~ ~(~+). Put

d_ = N(m'l~)

~(Nf~+).

here the condition is that

We F

Case I is

be a quadratic

The remaining cases are

d+=--8 mod 32~

d_ m 1 mod 4

III:

d+ ~ 1 mod 8~

d_ m -4 mod 16

III':

d+ ~ I mod 8,

d_ m -8 mod 32.

The ramification conditions become complicated when

21d ,

and we omit the tedious

details of the case analyses. The unramified cyclic extension of degree 4 is of the form

L = ~(~_,~).

VII-21

Here are some examples~ including all known cases where

d

d+

d_

d < 0,

~

41d ,

8

-7

-1 + 2 V~-

~4

-184

8

-23

-3 + 4 V ~

{4

-248

8

-31

-568

8

-71

-68

17

-4

4 +V~

-64

41

-4

32 + 5 ~

-260

65

-4

8 +~65

-292

73

-4

1068 + 1 2 5 ~

-356

89

-4

500 + 5 3 ~ - 9

-772

193

-4

1764132 + 126985~193

~4

-136

17

-8

(3 +~fl~)/2

-328

41

-8

(19 + 3 " ~ ' 1 ) / 2

~4 ~4

-55

5

-ii

3 + 2 ~r

z4

-95

5

-19

-I + 2 ~ -

Z8

-155

5

-31

-7 + 4 ~

z4

-39

13

-3

(-I +~/~)/2

{4

-III

37

-3

(-5 +'~-7)/2

{4

+145

29

5

II + 2 V ~

{4

1 + 4~f2--

remains prime in is fixed under

9(~I-4-5) and T ;

cf. w

{8

-1 + 6 , ~

~(V~

d+

from

~

_H

-56

(In the real case one distinguishes

and

~4

{4

{8

{2 x {4

d_

{4 {4

by testing:

but splits in

~(V~);

here

7

hence %/2-9

~4

Seminar on complex multiplication

IAS, 1957-58 VIII-I

Vlll

COMPUTATION OF SINGULAR J-INVARIANTS (C. Herz, Jan. 15, 1958)

Let C be an elliptic curve.

By a "model" of C we shall mean here

a non-singular embedding in the projective plane.

The first fact to be

noted is that the class of non-singular plane cubics and the class of models of elliptic curves are identical. available.

However there is a much deeper statement

The natural equivalence relation for algebraic curves is bi-

rational equivalence, i.e. isomorphism of the associated function fields; the natural equivalence relation for projective models is projective equivalence, i.e. two plane curves are projectively equivalent if and only if one may be transformed into the other by a collineation. equivalence is a priori narrower than birational equivalence.

Projective The second

main fact is that for elliptic curves the two notions coincide provided that the ground field is algebraically closed of characteristic # 2 or 3. This is substantially the same as saying that one can choose a standard cubic model for each elliptic curve which, as it turns out, depends on a single parameter.

This parameter is essentially the modulus for the curve.

The general homogeneous cubic polynomial in three variables, H(x o, x l, x2), has nine coefficients and so the cubic curves H = 0 form an 8-dimensional projective space.

If we subject the coordinates to a linear

transformation ~-, a non-singular 3 X 3 matrix, the coefficients undergo a transformation by the symmetrized kronecker cube of the transposed Hit(x) = H ( V x ) of ~ .

The group of these transformations, obtair~d by

taking all non-singular

~ ' s with coefficients in the constant field, acts

VIII-2

on the projective 8-space.

The homogeneous space obtained by factorization

is the variety V of all plane cubic curves.

It is easy to see that H may

be put in the form

"(-o,

§

c x o.

The remaining transformations consist of the diagonal vgtrices TU. is the projective 3-space

(a, b, c2, c3) modulo the action of the group

induced by the diagonal matrices computations.

Thus V

~.

This reduction step simplifies the

What we are looldng for are invariants of cubic curves, --

the first things needed are the ray-invariants,

i.e. the homogeneous poly-

nomials,g, in the coefficients of the general cubic which tlnder the action of a matrix ~ - t r a n s f o r m b y

being multiplied by a power of the determinant

of ]7-. Any ray invariant,g, is of course a ray invariant in the reduced case; the converse is true by the "unitary trick" which is applicable because the general linear group modulo the diagonal subgroup is compact. Now if T[" a c t i n g on I~

)

is ?.,iven by

x~

i s given by the m a t r i x

0

~ "if

the a c t i o n

0

I a2~,

p3

o

o ,slr 2 y3

1

Since this is diagonal, a basis for the ray invariants of degree m is formed v x y by the monomials g ~ a o c2c 3 where u + v + x + y = m. gl-T = (det ~ ) m g

Since we demand

we have the equations 2u = m, 3v + x = m, u + 2x + 3y = m~

The general solution is v = u - z, x = - u + 3z, y = u - 2z where z is another integer and 3z > u > 2z.

Taking z = 1 there are two possibilities:

VIII-3

u - 2 corresponding to the monomial g2 = a2bc2 and u = 3 corresponding to g3 = a3b2c3"

It follows that all other ray invariants are polynomials in

g2 and g3" A=

~2 - 27g~ = a6b3 (c32- 27bc~) is another ray invariant.

It

has the virtue that /~ - 0 is a necessary and sufficient condition for the cubic curve to be singular, i.e. either have a linear factor or a double point,

j = 26-33 ~2 ~ - l is defined for all non-singular cubics.

If V ~

is the sub-variety of V defined by ~ = 0 then it is easy to see that V - V ur~formizing o is an affine line and j is a~parameter. V - V ~ is the variety of moduli of elliptic curves. A has another virtue. curve, C.

Suppose H(Xo, Xl, x2) = 0 is an elliptic

Then k du=

0

x

0

dx

0

2

x2 Z2 k ~ H / ~ x j 3"0 j is a differential of the first kind on C, and is independent of the choice of the point (~2' ~ '

ko ) in the projective plane.

of a change of coordinates on du. is H*(x*) = H(~x*)

= HTK(x*).

du

Let us examine the effect

Suppose x = ~ x * .

In the new system

"

O

x*O

O

k2

x2

dx 2

2

Then the new equation

VIII-~ Of course du* = ~du where ~ is a constant multiplier since differentials of the first kind differ only by a constant factor. arbitrary we can put k = ~ k * Hence ~ t d e t ~

and 1 2

However since k is

and it becomes evident that du = (det TU)du*.

= •*/A

, -- for this reason expressions of the

form ~*/Z~

are occasionally called multipliers.

~ du 12 is an invariant of

the curve.

Could we choose A 1/12 invariantly, du C = Al/12du would be a

differential of the first kind canonically attached to the curve. The group operation on an elliptic curve has a simple geometric interpretation.

TakB a model and let (OL) and (~r) be two points on it.

These points determine a line which intersects the cubic in exactly one other point which we write as ( - ~ inflection (~).

~ ).

The curve has a unique point of

It is easy to check that the operations defined give rise

to an additive abelian group with identity element ((7). Since the group operation is defined by lines it is clearly invariant under collineations. There are exactly three points on C, other than (~), whose tangents pass through (~).

If (~) is such a point, (2~) = ( ~ ) , according

to the description of the addition operation, and the converse is true.

Thus

if (~i) and (~2) are two such points, (7~3) = (-~l - ~2 ) is a third. These three points are collinear; call the line they determine L C.

Projection

from (~) gives a two-fold covering of C onto LC; two points on C go into the same point of L C if and only if they are inverses of one another.

(The pro-

jection is not defined at (~') so we extend by continuity, -- (~) is projected along the inflection tangent.)

Let ~ o

be the projection of ~ o n

L C.

The cross

ratio ( ~ o ~ l ~ 2 ~ 3 ) = k 2 is invariant under projective transformations; however the three points (~l),

(~2),

(~3) may be permuted at will so

VIII-5

that k 2 is an invariant of C only up to the action of the s~,.~,etric group on three letters.

If we

write k '2 = 1 - k 2, the full invariant is

j = 2 8 (1 - k2kr2) 3 Suppose C* is an elliptic curve which forms an m-fold covering of C.

Since the covering map must be a homomorphism of the group structures

with suitable base points, inverses are mapped into inverses.

Thus if we

have a model of C* with inflection point (~*) mapping onto a model of C with inflection point ( ~ ) ,

there is a well-defined map of LC. onto L C which

is a m-fold covering of one projective line by another. to take the models in the projective plane with ( ~ ) inflection tangent x ~ = O, and the line L C as x 2 = O.

It will be convenient

at x ~ = x I = O, x 2 = l, The covering map of LC.

onto L C then has the form

x~

where Q is a homogeneo~

Q(x*, xl)x ~ ,

xI

polynomial of degree m - I and P is one of degree m.

The covering map of C* onto C may then be described at all but a finite * *

number of points by the additional transformation x 2 = R( is a homogeneo~

r a t i o ~ l ~ n c t i o n of degree n - 1.

explicit form of R. * *2

and XoX 2

*

The equations of C a M

*

- T (Xo,

We shall now find ~ e

C* are XoX 22 - T(x o, x I) = O

)= O respoctively where T and ~

with the coefficient of x~ different from ~.

are homogeneous c ~ i c s

The differentials of the first

kind are x du

x*o dx* o

dx o

o

"

and XoX 2

, Xl)X 2 where R

du xo x2

VIII-6

Since Xo, x I are homogeneous polynomials of degree m in Xo, Xl,

Ix~ dx~ i

Hence du - --

.!

xo* dx* o

.

9(x , ~ ) I d *

~ _ ~ l 9

o

I au~_. but we must also have du

•

a constant multiplier.

~ ~

~.

. m < ~

+ Q-ax

( ~x I

9(x o , x l)

This tells us that R =

I ~(~, P) h~ I ~-7--~I

~

~

Q

= ~du where ~ is o

i

----=----r-~I

~(~, ~);

~ 9

-•

*

Substituting in the e~uat~on for the

o I ~(Xo,Xl) l)

curves we firg that

Q( ~P )2 * ~ + 2x o

m

~ (~tP) ,

~

+

Q-lx*2

* * ~(Xo,xl)

~(q,P)

~(Xo* ' Xl* )

o

9 T (Xo, x.~.)

- T(x*~, P). There can be no denominator arising from the Q-I term on the left. the only possible linear factors of Q are x

O and

Hence

the factors of T*; all

other factors of ~ must be at least quadratic. What we have just described is the Jacobi transformation principle. It leads to the algebraic computation of the singular class invariants j. Some explicit computations are performed in Weber: Vol. III, sections 8-10.

Lehrbuch der Algebra,

The results, of course, coincide with the in-

variant eq~lations for j obtained by analytic means.

These equations will

hold in any of the fields we have considered, i.e. the analytic method gives results valid over rather general ground fields.

The difficulty in making

computations from the class equations is due to the fact that the coefficients in the q-expansion of j are intractably large: j . q-I + 7 ~

+ 196, 88~q + 21,~93,760q 2 + ....

It is in fact easier to

VIII-?

compute ~sing k 2 .

However we

shall now proceed to the analytic case and

examine the class equation by investigating the subgroups of finite index in the modular group.

Let P denote the full modular group and H a (not necessarily normal) subgroup of index I~.

We shall write ~ = Za~=lHVao

Let • be the standard

fundamental domain for the modular group; ~ has a simplicial decomposition with 2 faces, 3 edges, and 3 vertices, namely ~ stand for the transformations S" ~

ico, i, and ~ .

~ ~ + l, T: ~

S and T

~ - 1/~.

ioo is a

fixed point of S and all its powers, i is a fixed point of T (T2 = I), is a fixed point of S - ~

((S-~) 3 = I).

P is generated by S and T and

the only relations are consequences of those given above. A fundamental domain, 3 , for the subgroup H is defined as follows. ~.

At each interior point 1 ~ & ~ w e

assign ~points V l ~ , ... V~pZ" in

These points are inequivalent modulo H for if V a ~

a fixed point of ValUV b. ValUV b - I.

This can occur for ~ ,

= UVb~

then K is

t~# ioo, i, ~ only if

Hence if U 8 H, V a = UV b contrary to the assumption that for

a # b, V a and V b represent different cosets.

Thus ~ has 2 ~ faces and 3

edges. The vertices require special treah~ent.

First we take ~ -

ioo.

Let us define an equivalence relation V a ~ V b if and only if VaSkVbl ~ H for some integer k.

This provides a grouping of the cosets of

rood H into equivalence classes ~ , each class K

c

come together.

... Kd with Ko/H of order n c.

there is one vertex of ~ above ico in ~ a t

which n

O

For

sheets

VIII-8

A similar but simpler analysis holds at %" " i and Z" " f 9 V a and VaT represent two different sheets of ~ unless VaTV-Ia ~ H; let ~ i be the number of solutions of this equation.

Then there are 2 ( ~ -

~i )

vertices of ~above i in ~ at which 2 sheets come together and ~i unbranched vertices.

Likewise Va, V a(S'IT), and Va(S-~)2 represent three

distinct sheets of ~ unless Va S-ITv-I ~ H in which case all three coincide~ a let ~ 89

be the number of solutions of the last equation. - ~)

and ~

There are

vertices of ~above f in ~ at which 3 sheets come together

unbranched vertices. The total number of vertices of ~ i s d + 8 9

+ 8i ) + ~ ( ~ +2 ~

Hence, in particular, ~ i s a sphere if and only if 2 ~ - 3 ~ + ~(V+ 2~

) = 2, i.e. ~ -

+ d + 89

).

8 i)

6d + 3 ~ i + ~ 8 ~ - 12.

The most important subgroups for our purposes are the transformation groups H arise as follows.

and the congruence groups ~m" Let M be the matrix (~ 0).

The transformation groups U & H m if U ~ Cand

MUM-I 8 ~which is equivalent to saying that U has the matrix representation U = (F

with ~, ~, ~ , ~ integers, ~

- ~

- i and ~ -~ 0 rood m.

Writing ~ = Za~l(m)HmVa the matrices MV a can be chosen to run through the set (0 ~) where m~_ i, 0 <_ ~< ~, ~ write V ,~,~ for Va.

= m and (s, ~, ~) - I. We shall

The invariant polynonial is J(t, j) " ~

(t-j~[~a'~' ~ ). M jM defined by jM(~) = J(mU) is invariant un:~er the action of H . Thu3 j

is a meromorphic function on the Riemann surface ~m corresponding to Hm. The Galois group of the invariant equation J(t, j) = 0 over the field ~(j) is ~ / ~ m ~ere

~ m is the largest subgroup of Hm which is normal in ~.

~m is represented by the matrices (~

) ~ (o

I) mod m with ~, ~, Y ,

viii-9

integers and m ~ - 6 ~

= I.

If m is a prime ~ / P m

is isomorphic to

SL2(Zm) modulo its center; this group is simple for m ~ 5. The computations involving singular invariants J are easy to carry out when the Riemann surface ~ m

corresponding to Hm is a sphere.

I = m ~ (I + ~). We have to calculate d, ~i' and ~ f . plm The equivalence relation at ico, V 6,~ ~ V~, 6,, ~, may be formulated

Here,=

as ~O (O ~

~(m)

i )(0

k ) = U(~' i

~' ~l) where U ~ ~.

The matrix on the left is

6~ k~) and the equivalence holds only if ~' = ~, ~' = ~ and 6' ~--6 + k~

mod ~ .

For simplicity assume m is square-free.

Then (m, ~ ) = 1 and ~'

can run through all the values O, I, ... ~ - I for any given ~.

Hence for

each ~I m we have an equivalence class K~ containing ~ cosets.

The number

d is dCm), the number of divisors of m. The equation Vm,6, ~ T V ~ 6 ~ 62 + l m O

mod m.

8 H

is equivalent to ~ - m and

The analogous equation at the vertices above ~ =

equivalent to ~ = m and 62 - 6 + l ~ O from now on that m is a prime > 3.

mod m.

~ is

For simplicity, suppose

Then ~ ( m )

=m+l, d = 2, 8 i = 2 or O

according to whether -I is a quadratic residue mod m or not,

8~ = 2 or 0

according to whether -3 is a quadratic residue mod m or not.

The condition

for ~ m

to be a sphere is m+l - 3 8 i + ~ .

This holds for m = 5, 7, 13

and no other primes > 3. We shall now carry out an extended analysis for the case m = 5. On ~ 5 there is one vertex above ~ =

ioo at which there is only one sheet

and one vertex at which 5 sheets come together.

Above ~ - i there are two

vertices at which 2 sheets come together and two unbranched vertices. Above ~ =

~ there are two vertices at which three sheets come together.

VIII-IO A( I We know in advance that ~l'

( %_.,

it is invariant under Hm.

~2 ) )

is a meromorphic function on ~

m

since

The only possible zeros and poles occur at the

vertices lying o v e r t = ioo. Let q be the local uniformizing parameter 2 mi e at the unbranched vertex o v e r t ' - ioo. Here

A ( ~ O I , O~2) = 512 q~ ~ Z1 (~i, ~2)

(l-qn)-2~

5~n

The function has a four-fold zero; since it has only one pole, the pole is also ~-fold and this function is the fourth power of a function f with only one single pole. parameter.

Since ~ 5

Now consider the function fj.

branched vertex above ~ = ioo. it has a three-fold zero. on 3 5 .

is a sphere,,f is a global uniformizing f has a six-fold pole at the

At each of the two vertices above ~" =

Hence fj = ~

where v is a meromorphic function

Since v has a double pole at the pole of f, v is in fact a

quadratic polynomial in f.

Let

-2~ ~2 ql = e be the local uniformizing parameter at the branched vertex above ~'= ico. At this point j = q,-5 + E where E is a power series in non-negative powers of q'. f is defined by

SO that f = q,-I + ..., f2 = q,-2 - 12q,-I + ...~ and v " q,-2 _ 2q,-i + .... Hence v - f2 . 10f has no poles; it is thus a constant.

At the other vertex

j . q-I + ... ,f = 53q + ... so v = 5 + ... and the constant term is 5.

We

VIII-II

have computed v - f2 + lOf + 5, J =

(f2 +lOf +5) 3 f

9

We now make use of a very important property of the transformation groups:

namely (TM)Hm(TM)-l = H m.

~ m if f is.

Thus f(TM) is a meromorphic function on

In the case m = 5 let us write ~ = f T M

i.e. 7 ( ~ )

= f(- ~ ) .

The q-expansion of ~ is the same as the q'-expansion of f, i.e. ~ = q-1 + Since f - f, f~ has only one pole, and we conclude that ~ = 53f-I. ~= ~/5,

- ~ i-

=r

so f = f and hence f ( ~ / 5 )

choice of sign f2 + lOf + 5 ~ 0 ;

= ~ 5 3/2 . Nith either

hence the sign is the same as that of

j(~/5),

but j is positive on the imaginary axis.

f(8/5)

= + 5 3/2 9

(5p ~ )

For

Accordingly

is an ideal basis in the field ~ ( ~ )

representing the principal class.

From the expression of j in terms of f

one has j = 23.5.(25 + 1 3 ~ ) 3 for the singular invariant of the principal class.