Tata Lectures on Theta I

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Tata Lectures on Theta I

David Mumford With the collaboration of C. Musili, M. Nori, E. Previato, and M. Stillman

Reprint of the 1983 Edition Birkhauser Boston Basel Berlin

David Mumford Brown University Division of Applied Mathematics Providence, R I 0 2 9 1 2 U.S.A.

Originally published as Volume 28 in the series Progress in

Mathematics

Cover design by Alex Gerasev. Mathematics Subject Classification (2000): 01-02,01A60,11-02,14-02,14K25,30-02,32-02,33-02, 46-02 (primary); 11E45,11G10,14C30, 58F07 (secondary) Library of Congress Control Number: 2006936982 ISBN-10: 0-8176-4572-1 ISBN-13: 978-0-8176-4572-4

e-ISBN-10: 0-8176-4577-2 e-ISBN-13: 978-0-8176-4577-9

Printed on acid-free paper. ©2007 Birkhauser Boston BirkMuser 11$) All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhauser Boston, c/o Springer Science+Business Media LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 9 8 7 6 5 4 3 2 1 www. birkhauser. com

(IBT)

David Mumford With the collaboration of C. Musili, M. Nori, E. Previato, and M. Stillman

Tata Lectures on Theta I

Birkhauser Boston Basel Berlin

David Mumford Department of Mathematics Harvard University Cambridge, MA 02138

Library of Congress Cataloging-in-Publication Data Mumford, David. Tata lectures on theta I. (Progress in mathematics ; v. 28) Includes bibliographical references. Contents: 1. Introduction and motivation : theta functions in one variable ; Basic results on theta functions in several variables. 1. Functions, Theta. I. Title. II. Series: Progress in mathematics (Cambridge, Mass.); 28. QA345.M85 1982 515.9'84 82-22619 ISBN 0-8176-3109-7 (Boston) ISBN 3-7643-3109-7 (Basel) CIP- Kurztitelaufnahme der Deutchen Bibliothek Mumford, David: Tata lectures on theta / David Mumford. With the assistance of C. Musili ... - Boston; Basel; Berlin : Birkhauser. 1. Containing introduction and motivation: theta functions in one variable, basic results on theta functions in several variables. -1982. (Progress in mathematics ; Vol. 28) ISBN 3-7643-3109-7 NE:GT Printed on acid-free paper. © Birkhauser Boston, 1983 Third Printing 1994

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Graph of

Re -S^z, y^r) ,

- 0 . 5 £ Re z 0 0 = lim Z a exp n t->0 n n

(-TT

o n t)

n

= f(0) . Hence

£ ( x , it) c o n v e r g e s , a s a distribution, to the s u m of the delta

functions at a l l i n t e g r a l points x e 7L a s that it c o n v e r g e s v e r y n i c e l y , in fact.

t

>0.

We s h a l l s e e below

Thus £ ( x , i t ) may be s e e n a s the

fundamental solution to the heat equation when the s p a c e v a r i a b l e on a c i r c l e

x

lies

R/S.

§ 3. The H e i s e n b e r g group and theta functions with c h a r a c t e r i s t i c s . In addition to the standard theta functions d i s c u s s e d s o far, t h e r e a r e variants c a l l e d "theta functions with c h a r a c t e r i s t i c s 1 1 which play a v e r y important r o l e in understanding the functional equation and the i d e n t i t i e s

6

satisfied by 0 , as well as the application of £ to elliptic curves. are best understood group-theoretically.

These

To explain this, let us fix a T

and then rephrase the definition of the theta function £ (z, T) by introducing transformations as follows: For every holomorphic function f(z) and real numbers a and b, let (S f) (z) = f(z + b) b

o

(Ta f)(z) = exp (TT i a T+ 2TT iaz) f(z + at), Note then that S, (S. f) * S. + .. f and Taa (Ta f) = Ta + . f. ^l b2 i i a2 ^1^2 2 These are the so called "l -parameter groups".

However, they do not

commute ! We have: S. (T f)(z) = ( T O (z+b) b a a = exp(n ia 2 T + 2TTia(z+b)) f(z+b+a T) and T (Suf)(z) = exp (TT ia 2 T+ 2TTiaz)(Slf)(z +a T) a D b o = exp (TT i a T + 2n iaz) f(z+a T +b) and hence (*)

S o T = exp (2niab) T o S, . b a a b

The group of transformations generated by the T a 's and S 's is the 3 -dimensional group £

= o-#

«) V V b ) = *a. b+bl (ii) T

«1(Vb)

=ex

for

».'v b, i ZB

P(-2riaib)V*.b'

Ya a bc 2Z

* r x

11

(iii)

*a+p

b+q

= exp (2TTiaq) ^

fe

, Vp,qcZ;, a , b e j Z .

Hence (iii) shows that & . , upto a constant, depends only on

SL,b**/i2Z/2Z.

In view of Lemma 3.1 and the Fourier expansion just given for 0 . , it i s clear that a s a,b run through coset representatives of l / ^ S / 2 Z , we get a basis of V , . Note also that except for a trivial exponential factor £

is

just a translate of * . $ 4 . Projective emmbedding of (D/2Z+Z5T by means of theta functions. The theta functions £

defined above have a very important

a, b

geometric application. Take any l> 2. Let E T be the complex torus *o° ( u ) *oo< v >

+

* U | W * 0 l W *01 * O l W

^10w*10(y)*10

:

*oo (jt) *ooW*oo'«)*oo (v >

+

fl

and A -, we get further:

*01*01 W V « > *01 ( a r )

-*io (x) *io*io (u) *io (v )" *nW *n(y> *ii< u ) *n ( v ) = 2*01(x1)#01(y1)%1(u1)*01(v1). Substituting instead X+T for x in (R 2 ) and multiplying by exp(TTiT + 2TTix) so that * OQ (x) becomes * OQ (x) again while ^ ( x ) and * n ( x ) change signs, we get:

(R

4>

:

' o o W *ooW *oo (u) *oo "*01 ( X ) V +

y )

*01(u> *01

* 10 (x) *10 *10 *10 " * n « * i i W *11 * n W

=2 V 3 ^ V y i> »io(ui> V ( VFinally replacing x by x+ T+l in (R ) and multiplying by exp (TTi T + 2 TTix), we get:

(R

5>

:

* o o W *ooW *oo » + £T)

" TTi T / 4 " T T i z ) ) *10
+

"

-

"

=2«*S1*UV1

+

»

-a^iiff^i^

. -

-.aWftf

- »

"

-^W'MV --itftfOtf

=2*oo1*cyi*101*ll1

21

We have listed these at such length to illustrate a key point in the theory of theta functions: the s y m m e t r y of the situation generates rapidly an overwhelming number of formulae, which do not hawever make a completely elementary pattern.

To obtain a c l e a r picture of the algebraic implications of these

formulae altogether i s then not usually e a s y . One important consequence of these formulae c o m e s from specialising the variables, setting x = y and u = v.

The important fact to r e m e m b e r i s

that £,,(0) =0 whereas £ ,£«., and £, n are not z e r o at 0 (cf. Lemma 4.1)1 11 oo 01 10 Then the right hand side of (Rg) i s 0; and (R & ), (R 2 ) + (R ) combine to give (noting that x

(A

1>

:

= x+u,y = x-u and u

*oo(x)\o(u)2+*ll*oo (0) *01 (0) ' *oo(x)*01<x>*00 s*o2i(x)*i2(/u)-*i2o(x)*oi(»)

*ll«* + ")*oo ( x H , ) *01 ( 0 , *10 ( 0 ) " *oo< x ) *U< x ) V u ) *10 ( u >

+

*10< x >*0lW*oo ( u ) *ll ( u )

*00*,1*o1 - *i0 ( x ) *01 ( x ) *oo ( , l ) *ll ( u )

* n (x+u)* 01 (x-u)* oo (0)* 10 (0)

= # 0 0 W# 1 0 (x)*oi,,oi(x)*io('l)*ii(u'+*io<x>i,ii(x)*oo(u)*oi(u)

IV. Equations for * (E

>oo<x> *oo

*01 W *2l< 0 ' + *l 2 0 ( x ) *10 ( 0 )

(E 2 ) : ^ ( x ) *o2Q(0)

= *02x(x) * 2 0 (0) - * 1 0 (x) ^ ( x )

1>

:

and

23

Specialising further by setting u = 0, w e find that a l l the above reduce to just 2 relations: <E1>

* o o ( x ) 2 *oo2 - * 0 l W 2 * 0 l ( ° ) 2

<E2) :

*

n

+

*io«2*102

W 2 * o o ( 0 ) 2 = # 0 1 W 2 * 1 0 ( 0 ) 2 - # 1 0 (x) 2

tf01(0)2.

Finally setting x = 0, we obtain Jacobi's identity between the ab '

M

theta constants"

namelv:

*oo(0)4^01(0)4

(M 2 )

+

^10(0)4-

We see now that the identities (E-) and (E 2 ) are equations satisfied by the 3 image qp (E-) in IP . We now appeal to some simple algebraic geometry 3 to conclude that cp (ET) is indeed the curve C in IP defined by the it

following 2 quadratic equations:

*oo 2 x i- v*>\ 2 3 By Bezout's theorem, it is clear that a hyperplane H in IP meets C in utmost 4 points. But the hyperplane Za.x. = 0 meets q>9(ET) in the points where a

o*oo(2x)

+ a

l*01< 2 x >

+ a

2*10(2x)

+ a

3*ll(2x)

=

°-

and there are 4 such points mod 2 A . So cpntE, ) must be equal to C! It is clear that the theta relations give us explicit formulae for everything that goes on in the curve C.

24

S 6. Doubly periodic meromorphic functions via d (z, T). By means of theta functions, there are 4 ways of defining meromorphic functions on the elliptic curve E T , and the above identities enable us to relate them: 3 Method I: By restriction of rational functions from IP : This gives the basic meromorphic functions

*oo< 2 z >

on E T . Method II. As quotients of products of translates of fl (z) itself: i . e . , if a

. . ,a, ,b , . . . ,b, c * 1 0 ( z ) *11 ( E ) is the formula (R 1 8 ) when x = y = u = v = z. To relate Methods II and HI, we can take the 2nd derivative with respect to u in the formula (A-Q) and set x = z and u = 0, We get: "(z) = 6 ^ ( z ) 2 + g 2 (T) and

^«"(z)-12^(z) .£'(*) Thus fp is a time independent solution of the (Kortweg-de Vries) KdV (non-linear wave) equation: u = u - 12uu , u t xxx x

= u(x, t)

28

§ 7. The functional equation of £(z, T) # So far we have concentrated on the behaviour of £ (z, T) as a function of z. Its behaviour as a function of T is also extremely beautiful, but rather deeper and more subtle.

Just as £ is periodic upto an elementary factor for

a group of transformations acting in z, so also it is periodic upto a factor for a group acting on z and T. To derive this so called "functional equation" of ^ in T; note that if we consider £(z,T ) for a fixed T, then although its definition involves the generators 1 and T of the lattice A quite unsymmetrically, still in its application to E

(describing the function theory and projective

embedding of E ) this asymmetry disappears.

In other words, if we had

picked any 2 other generators a T+ b, c T +d of AT (a,b, c, d e 7L, ad-be = 1 1 ) , we cou}d have constructed theta functions which were periodic with respect to z r-^- z + c T + d and periodic upto an exponential factor for z |—^ z+a T + b, and these theta functions would be equally useful for the study of E ^ . Clearly then however the new theta functions could not be too different from the original ones! If we t r y to make this connection precise, we are lead immediately to the functional equation of £(z, T) in T . To be precise, fix any (a c

\ ) c SL(2, 2L), i . e . a, b, c, d € 7Lt ad-be = +1 a

and assume that ab,cd are even. Multiplying by -1 if necessary, we assume that c > 0. Consider the function d((cT + d) y, T). Clearly, when y is replaced by y + l , the function is unchanged except for an exponential factor.

It is not

29

o hard to rig up an exponential factor of the type exp (Ay ) which c o r r e c t s 0 ( ( c f + d ) y, T) to a p e r i o d i c function for y l—> y + l .

In fact, let

Y(y, f ) = e x p (Trie ( c T + d)y 2 ) * ( ( c T +d)y, T). Then a s i m p l e calculation s h o w s that Y(y+1, T) » Y(y, t ) (N.B:

a factor exp (TTicd) a p p e a r s , s o w e u s e

H o w e v e r , the p e r i o d i c behaviour of 0

for

cd even in the v e r i f i c a t i o n ) .

z |—> z+ T g i v e s a 2nd q u a s i -

p e r i o d for Y, n a m e l y ,

cT+d

c T +a

We give s o m e of the c a l c u l a t i o n s this t i m e : f o r m a l l y writing we have by definition:

, T)

T(

2 2 + 2TTicy(a T+b) +TTic exp[TTic(cT+d)y r J

^((cT+d)y+aT+b,T)

(a T

* b] cT + d

But *((cT+d)y+aT+b,T) Y(y, T)

a

exp[-TTia 2 T - 2 f f i a y ( c T + d ) ] * ( ( c T+d)y, T) exp(TTic ( c T + d ) y 2 ) i>((cT+d) y, T)

2 2 = e x p ( - n i a T-2TTiay(c T+d) -TTic(cT+d)y ). So multiplying t h e s e two equations and u s i n g a d - b e

Y(y +

* T) 2

3

1, w e get:

2

* e x p (-2TTiy(ad-bc)+TTic (.a T + * }

t ( y , T)

-TTia2T)

(ct+d) 2

= exp[-2TTiy-iIl-:-(a T(cT+d) - c(aT+b)2)] c T+d = e x p [-TTiy - - H i - ( a 2 T d - 2abc T - b 2 c ) ] c T+d

] .

30

But a 2 Td - 2 a b c T - b 2 c « a(ad-bc)T -ab(cT+d) + b(ad-bc) = (aT +b) - ab(cT +d). Now using ab is even, we get what we want. If we now recall the characterisation of *(y, T') as a function of y as in $ 1, namely, *(y, T ') i s the unique function (upto scalars) invariant under A^ where f'=(aT+b)/(cT+d)), we find Y(y, T) is one such. Hence we get: *(y, T) s q)(T)^(y,(aT+b)/(cT + d)) for some function qp(T). In other words, if y = z/(cT+d), then *(z,f) «5p(T) exp(-TTicz2/(cT+d)) * (z/(cT+d), (a T+b)/(cT+d)). To evaluate qp(T ); note that £ (z, T) is normalised by the property that the th 0 term in its Fourier expansion is just 1, i . e . , 1 j* *(y, T)dy = 1. 0 Hence cp(T)=

\ T ( y , T ) d y = } exp (nic (c T+d) y ) * ((c T+d) y, T) dy . 0 0 This integral is fortunately not too hard to calculate. First note that cp (T) = d( s +1) if c = 0 and so we can assume c > 0. Now substituting the defining series for & and rearranging terms, we get: cp(T) = C Z exp[TTi(cy+n) 2 (T+d/c) - TTi n 2 d / c ] d y 0 nc2Z '

Z exp (-TTin2d/c) £ exp( TT i(cy+n) (T+d/c) dy. ncZ 0

31

But (using again cd even) we have exp (-TTid(n+c)2/c) = exp (-TTin2d/c) and hence we get =Vz'T) 01 #1Q(

oo "

V '

'

)

=exp(Tri/4)#o(")

*n( " )

=exp(ni/4)* (")

) -

*10< " >

V ">

*10
d ) .

39

The condition that makes this definition work is that the factors (c T+d)

introduced in the functional equation (a) satisfy the

M

l-cocycle n

condition, i . e . , if we write e y (T) = (cT+d) k , where Y a £ then for all Y^Y^*^ * w e

nav

|j),

©

V T ) = VV , e Y 2 ( T ) This same condition, together with the fact that T

is normal in SL(2,ZZ),

(N) gives an action of SL(2, 2Z)/I*N on the vector space Mod, : if f is a modular form and v = ( ' c

,) , define a

f Y (f) =e Y (T)" 1 f( Y T). It is immediate that this is also a modular form of the same kind as y f : in fact, (a) for f implies (a) for f and (b) for f at p/q implies (b) for fY at Y(p/q). Finally note that if feMod^

and gcMod* ', then the

(N) product fge Mod ' . Thus K

Mr

Mod

«

© Mod, k k«ZZ+

is a graded ring, called the ring of modular forms of level N. Now we have the following: 2 Proposition 9.2. £

2 (0, T), *

2 (0, T) and *

(0, T) are modular forms of

weight 1 & level 4. Proof. To start with, condition (a) for 0

2

(0, T) amounts to saying that C ,

QO

the 8th root of 1, in the functional equation ( F ^ is 1 1 when (* j j ) « r

™

is immediate from the description of C (in fact, we only need c even and d = 1 (mod 4)). We can also verify immediately the bound (b)(ii) at co for

S

40

2

£ (0, T). In fact, the F o u r i e r expansion oo * shows that, a s Im T

(0, T) = I exp(TTin 2 T) n*7Z >oo, we have

*oo ( 0 '

T ) = 1 + 0 ex

( P(-nlm

T

»

2 hence £

(0, T) is every close to 1 when Im T> > 0 . Before verifying 2 (b)(ii) at the finite cusps p/q c Q , consider how S L ( 2 , Z ) acts on * (0, T).

Let a * (1 l) & 3 = (° - 1 ) be the generators of SL(2, 2Z). Using Table V above, we check the following: C

*o> ^

=

^ 1 ( 0 ' T ) < [ * o > < T ^ = -**j> T)

[ ^ ( 0 , T)] a = * Q 2 o (0,T), C ^ t O . T ) ] C^ 2 0 (0,T)f = i* 2 0 (0,T),

=-i^ 2 Q (0,T)

[ ^ ( 0 , 1)]3 = - 1 * ^ ( 0 , T) .

So these three give an SL(2, 2Z) -invariant sub space of Mod-(4) . But then to check the bound at the finite cusps, it suffices to check it for all 3 functions at co because a suitable y cSL(2, 7L) c a r r i e s any cusp to oo. As for * (0, T), by the F o u r i e r expansions, we have: oo *

(0,T) = l+0(exp(-TTlm T))a s Im T

>co

* J O , T) - CKexp (-TTlm T/4)) T/4)) J This completes the proof of the proposition.

(In fact, a similar reasoning

shows that the analytic functions

TT

* ai bi(°. k i T ) - a i ' b i ' k i « Q - k i > 0

41

are modular forms of weight i and suitable level.

We will prove this

in a more general context below). The above proof also allows us to point out the following: 2 2 2 Remark 9. 3. The modular forms * (0, T ) , * (0, T) and * (0, T) look like oo oi 10 C /(c T+d) + (Error term) o

when T

> -d/c, where C= 0 or C = 1. Here T should approach -d/c

in horocircles of decreasing radii touching the real axis at -d/c, and the error term goes to 0 exponentially with the radius of the horocircle: more precisely, 2 (Error term) = 0(exp [-constant.Im T/|cT+d| ]). y (This is seen by estimating f ( T) as T as

T—>co).

> -d/c in terms of f(T )

42

Another simple fact which follows from the discussion above and which we need later is the following: (N) Remark 9.4. Let f e Mod, # Then k

f(T) = 0((lm T )" k )

as Im T

>0

To see this: let us recall the classical fundamental domain F for the action of SL(2,IZ)on H, namely, F = { T e H / | T | > 1 and | Re T| < i 3 (cf. Fig. 3, S 10, below). So we have H = U yF,

Y eSL(2,2Z).

Let

F' = { t e H / l m T * 3 * / 2 ) . Since FcF» , we have H = U Y F ' , v€SL(2, 7L). Take any TeH.

Then

Y

3-Y«SL(2;2Z) such that Im (v T) > 3*/2.

d' either Moreover if Y = (*c \),

Im T ^ 3 2 / 2 or c f 0. Now we make the following: Claim: 3a constant C 0 >0 such that |f(«r)l< C | c T + d|

whenever

Y = (* ^) e SL(2, 7L) is such that Im (y T ) > 3^/2. Observe that this claim proves the remark because c f 0 implies | c T + d | > \ c . Im T| = | c | . Im T > Im T , i . e . , [f(T)l < C (Im T)

for Im T < 3*/2, as asserted.

To prove the claim:

(N) since f e Mod, , we have: k fY(T) = (cT+d)' k f(YT),f Y €Mod[ N) and fW* f Y ,VY'er N . In particular, there are only finitely many modular forms of the type fY,

YeSL(2,Z).

Hence (by Def. 9.1, b(i))3a constant CQ >0 such that

V Y e S L ( 2 , Z ) , we have:

43

|fY(T)UCQ

(*)

if

ImT>3i/2

On the other hand, we have (by definition of fY) : (**)

fY"

(YT) = e

, f c f f V r ) = e ( T ) « T ) - (cT+d) k f( T ).

Clearly then (*) and (**) imply the claim.

44

§ 10. The geometry of modular forms Just as a set of theta functions with characteristics enabled us to embed the punctured disc = {w

/0IP : in fact, at the cusp

at oo, w = exp ( | n i T) i s the local coordinate on H / r , and we have: o

*„J°> T>

=

E exp ( n i n 2 T) = 1 + 2 £ w 2 n ncZ n eIN

*n1(0,T) = I exp(TTin 2 T+TTin) = 1 + 2 I ( - l ) n w 2 n U1 ne2Z ncK * l n ( 0 , T ) = £ exp(TTi(ri+i) 2 f) = 2 w* I w 10 nc2Z n«Z+

2(n 2 +n)

(*) In thinking about t h e s e diagrams in the non-Euclidean plane, it i s good to bear in mind a comment of Thurston: these diagrams make it look like the space gets very crowded and hot near the boundary; in reality, however, the space i s increasingly empty and quite cold near the boundary,

51

[ h e n c e * 2 (0, T ) - 4 w( 10

Thus T it

2 2 1 w2(n "^ ) ] . n«2Z +

is holomorphic in w, carrying the cusp w = 0 to ( l , l , 0 ) c I P 2 #

then it follows, by SL(2,Z£) - equivariance that f

But

i s holomorphic at the other

cusps too. Finally, we have the simple: Theorem 10# 1, The naturally extended holomorphic map *o &

: H

>C conic A:x 2 = x 2 + x 2 ]

/h 4

o

l

«

i s an isomorphism, Proof. In fact, both H/I\ and A are compact Riemann surfaces, and * 2 is a non-constant holomorphic map. Therefore, ramified covering of A. To see that T

T^ makes H/ r a (possibly)

is an isomorphism, we need only check

that its degree is 1. But if its degree is d, then over each point of A, there are d points (counted with multiplicities where T0 is ramified). Now consider the 2

6 points (1, t 1, 0), (1,0, + 1 ) and ( 0 , 1 , + i). Only cusps can be mapped to these and there are 6 cusps. Thus only the cusp T s i co is mapped to (1,1,0). But by the formulae above, (o.have: t)/*2(o,f)) 4L(«?Awe dw 10 oo r=0 to which means that T 2 is unramified at i co. Hence degree of T2 is 1, i. e . , f is an isomorphism.

(In fact, it can be checked with the formulae we have at hand

that the cusps co, 0, {, 1, 2 & 3 are respectively mapped to the points (1,1,0), ( 1 , 0 , 1 ) , ( 1 , - 1 , 0 ) , ( 0 , 1 , i), ( 1 , 0 , - 1 ) and ( 0 , 1 , - i ) .

52

An important consequence of this theorem is: (4) Corollary 1 0 . 2 . The ring Mod' of modular forms of level 4 is naturally isomorphic to CC

*dl ( 0 - T)» *01(0'T)'*12C/0. rtH**oo-4l- *10>

2 i . e . , it is generated by A. (0, T) and subject to only the relation (J j). Proof. Let f e Mod. \

—

—

Then f/A

K

(0, T) is a meromorphic function on H/r,

OO

*

with poles only where £ (0, T) = 0, i . e . , only at the 2 cusps 1 and 3, and there 2 poles of order at most k (recall that just as A (0, T) has a simple zero at T = i oo , so also £

2

(0, T) has a simple zero at 1 and 3). Therefore, it

corresponds to a meromorphic function g on the conic A with at most k-folH poles at the points (0,1, t i). But A is biholomorphically isomorphic to the projective line DP XQ.

where (t.U)

via the map: » t * + t j , xx»

> 2 t o t 1 and x 2

are homogeneous coordinates on IP . Here t

O X

= 1 and t 1 = "t i Q

X

correspond to the points (x , x , , x 2 ) = (0,1, + i). So g corresponds to a meromorphic function h on IP h i s a x rational function of t j / t

with k-fold poles at t

- 1, t- = i i.

and by partial fraction decomposition of

rational functions, one checks easily that it can be written as:

*-<Mvv/ ( t o + t 5 , k for some homogeneous polynomial Q of degree 2k.Thus g

P(x rf

Hence

xltx2)/xko

for some P homogeneous of degree k.

Thus

53

f/*2* (0, T) = P ( # 2 . A.2. , * , 2 0 ) / * 2 k (0, T). 2 2 2 Finally, there can be no further relations between & , f , f because the only polynomials that vanish on the conic x x

2 O

- x

2

= x- + x 2 are multiples of

2 - x . , as required. £i

1

% 11. £ as an automorphic form in 2 variables. So far we have concentrated on the behaviour of 0 ( z , T ) as a function of z for fixed T , and as a function of T for z = 0.

Let us now

put all this together and consider £ as a function of both variables.

First

of all, it is easy to see that the functional equations on £ , plus its limiting behaviour as Im T Proposition 1 1 . 1 .

>co characterise £ completely.

More precisely:

d(z, T ) is the unique holomorphic function f(z, T ) on

1 as Im T

On the other hand,

>+co. This means that g(t) is

bounded outside a horizontal strip and hence by SL(2, JZ)-invariance, it is bounded everywhere.

Thus |g(T)-l| , if not identically zero, takes a positive

maximum at some point of F which cannot happen. So g(T) s 1, as required. The 4 theta functions &.(z, T) moreover satisfy together a system of functional equations that we have given in Table 0, 5 5 and Table V, § 9.

To

understand the geometric implications of these, we consider the holomorphic map: «:BP 3 ,(z, T)«

>(* o o (2z,T),d 0 1 (2z - T),* 1 0 (2z f T),* 1 1 (2z,T)).

The semi-direct product (l2Z) 2 KSL(2, 2Z) 4 acts on (CxH by

55

, fii (m,n; C c

bx , .) : (z, T )xl» d

_,z+mT+n >( -—-— , c T+ d

aT+bv ~ ), c T+d

and we have s e e n (fry the tables referred to above) that the 4 generators ( j , 0 ; I ) , (0, j ; D . (0,0; (l

*)) and ( 0 , 0 ; (°

the 4 functions £..(2z, T) into t h e m s e l v e s .

^))

In other words, the map f i s

13

equivariant when the s a m e group a c t s on IP

3

via:

(J . 0s« : ( x o . x 1 . x 2 . x s ) i

>(x2, -Ix,.xo.

(0.1:1) : (

"

>

Xxj.x^x,.

(0.0;(^):(

"

) «

(O.Ojf0"1):*

"

\

of this group transform

-taj) -x2)

> ( x 1 , x < j , X x 3 , \ x 2 ) where X = e x p ( n i / 4 )

)

>(x

x,,x1#-taj. O

0

Z

1

3

Now we have the following: Proposition 1 1 . 2 .

Let r

*

1

2

c ( J 2 Z ) (<SL(2,ZZ) be defined by

T* = { ( m # n ; ( a *))/(* * ) c r , m = | ( m o d l ) & n s % (mod 1)} . c a c a 4 o ° i

4c

Then T

2

i s a normal subgroup of (^ Z )

3

X SL(2, 2Z), it a c t s trivially on IP

l ( z , T) * l ( z ' , T') < — * > ( z \ T » ) = Y ( z , T ) for s o m e y tT* Thus f c o l l a p s e s the action of T quartic surface F in IP

3

.

on j , J a r e homomorphisms forward verification using the fact that {* ~|) I

56

from T

>Z/22Z.

(2) Note that I\ is the least normal subgroup of SL(2, 2Z) containing L j)

: in fact, if N is the least normal subgroup in question, then the

fact that N = r

can be seen in several ways, v i z . ,

(i) Topological way: look at the fundamental domain for H/r (cf. Fig. 4): let Y. *F , 1 < i < 6, be the transformations identifying in pairs of the edges of the diagram, namely, v Y

i

,1 -4v 0 lh

,-3 -4x ,1 Ox , 9 - 4 , 4 5 M 4 1}' (16 V '

(

(

(

,5 - 4 , 4 -3}

.,9 4

and(

-16. -7 K

It follows that r is generated as a group by these y ' s . On the other hand, 4 i 1 -4 ) and it is easy to see that the y , 2 < i < 6 , are conjugates of y = ( n hence N = r . 4 (ii) Abstract way: recall that SL(2,Z£) is generated by a = r *) and b =(

) and hence their residues mod N generate SL(2, Z ) / N , and

modulo N

we find that a 4 = b 4 = 1 and b 2 = (ab) 3 = (ba) 3 2

Clearly then b

2 is in the centre and, mod b we have a 4 = b 2 = (ab) 3 = 1.

But this is a well-known presentation of the Octahedral group of order 24 (cf. e . g . , Coxeter-Moser, Appendix Table I). Thus

(*) But r

is not in general the least normal subgroup N containing (

In fact, for n £ 6, N is not even of finite index I!

n

)!

57

#SL(2,Z)/N = 48 = # SL(2,2Z)/r 4 . But N c T

and hence N = T\ , as required.

Combining the facts (1) and (2), we get: * (3) . T

1 2

2

is the least normal subgroup of (- 2Z) KSL(2, 7L) containing 7L

and (0,i,(J })). (4) f collapses the action of T : this is immediate since the same is true for the action of 2Z2 and (0, \ ;(* *)), as seen from the Tables O and V. Hence ft factors through ( C x H ) / r * . (5). Suppose f (z, T) = I (z 1 , T') = P, say. Recall from § 5 that for T fixed, we have =

o=

! \ a2

, 2 2

2

*(xoX2-Xl x

X

^< ^2- o

x 2

2.

V 3>

=^(xo-xJ)

and there are no non-zero x.'s for which all expressions on the right of (**) and (**') are zero, hence the lemma follows. Coming back to the situation of step 5, we therefore get that Tf = Y (T)

for

some Y « ^

N

w lift Y

to

Ytr

and let y(z, T) = (z n , T ! ).

Then | ( Z \ T ' ) = l(z", T«), i . e . , E

TI

^ts

to tneir

universal

0*0

coverings <E , i ; e . , it is induced by a biholomorphic map f :

^** * s * n e isomorphism of the fundamental groups

and E T , induced by f#

Then the derivative f

is a doubly periodic

entire function of z, hence it is a constant, i . e . , F(z) = Lz + M for some L, M e E Or, equivalently, Set of c o m p l e x t o r i E T modulo i s o m o r p h i s m s p r e s e r v i n g \ H / r

Proof.

n "

) ( the pair of a u t o m o r p h i s m s

L e t f: E_ T

b e s u c h that T = Y ( T ' ) . foaT=aT'of n

n

T a

T and 0

a b b e a b i h o l o m o r pr h i c map and y S L d )x cSL(2,2Z) *" » ^c

->ET, ' ~T

With L and M a s a b o v e , we find that

L ( z +u i ) + M = ( L z + M ) + l + \ , ii

cT'+d-l



i



c#

m

T

. f A

T'

d - l 2 0 (mod n) .

Likewise, f

r

o p ! - p T o f L ( z + - £ ) + M = (Lz+MJ + ^ + n c A - , n n n « T ^ ( c T ' + d ) T-T* . "^ n «A T , ,(a.l)T'n «ATt

a - l , b i 0 (mod n)

63

Thus both occur if and only if yeT

, as required.

Let us look at the particular case n = 4: we constructed in the previous section a diagram: CxH

H

->(CXH)/I\

->F

cFclP

-> H/r

->A c A c I P ^ o

o

We can add to this diagram the auxiliary maps a & B, from C X H to 4

itself.

If we define a\ & 0' on IP 4 4

V

(x

o' X l'V X 3 )l "

* x l ' x o ' X 3 ' "x2'

)t"

34:(

4

by

">(x2'-ix3'Xo'-ixl)

then I i s equivariant, i . e . , we have a commutative diagram:

«ExH)/r

I

-> F .

K «D X H)/r

F

In other words, to each a cA , we can associate the fibre IT (a) c F plus 2 automorphisms res a' & res 3' , and we have shown that distinct points a c A

o

are associated to non-isomorphic triples (TT~ a , r e s a ' , r e s 3 1 ) . 4 4 ->A ; a' , ft.1) is a kind of universal family of complex tori, 0 4 *

Thus (F

with 2 automorphisms of order 4 which sets up the set-theoretic bijection: Set of complex tori E T plus automorphisms \ A

o

S aT, PA modulo isomorphisms

j

More details on this moduli interpretation can be found in Deligne-Rapopart, Les Schemas de modules de courbes elliptiques,

64

in Springer Lecture Notes No. 34 9# Similar constructions can be c a r r i e d through for all n, but the formulae a r e much more complicated. §13.

J a c o b u s derivative formula We now turn to quite a startling formula, which shows that theta

functions give r i s e to modular forms in more than one way. Upto now, we have considered £..(0, T) for (i,j) = (0,0), (0,1) and (1,0). Since & (0, T)=0, this gives nothing new; however, if we consider instead

*z

U

|z=0

2 which we abbreviate a s £' (0, T ) , we find, e . g . , that &*AQ» T) is a modular form (of weight 3 and level 4), e t c . In fact, this is an immediate consequence of (Prop. 9. 2 and) the following: Proposition 13.1 # (J 2 ) :

F o r all T e H, we have Jacobi's derivative formula, namely, ^ ( 0 . t ) = . ^ ( o . T) # o i ( o . T) # 1 0 (0, T )

Proof. By definition, the F o u r i e r expansion of £' (0, T) is given by *1 '1 (0, T) = ~ z ° = 2TTi

( E exp(Tri (n+i) 2 T+2TTi(n+i)(z+|)) n*2Z z=0 E (n+i) exp[TTi(n+l) 2 T+ni(n+i)l ncZ

= 2TT E ( - D n nc2Z

(n+i) exp ( ni(n+i)

T) .

In t e r m s of the variable q - exp (TT i T), the local coordinate at the cusp ioo, we get: (0,t) = -2TrCql/4-3q9/4

# 11

So the formula (J 2 ) reads a s

+

5q25/4-

].

65

[ q ^ - S q 9 ^ BqM/4.7q4»/4+... 1

[! 2q+ 2q 4 + 2q V . . j ^

[l-2q + 2q 4 V + ...]Cq 1 / 4 + q 9 / 4 + q 2 5 / V..] which the reader may enjoy verifying for 3 or 4 terms (we have taken the expansions on the right form page 1.16, $ 5 above). We may prove this as follows: start with the Riemann theta formula (R 1Q ) above and expand it around the origin, getting:

C

+

" 2 +

*oo **oo*

+

--

, ti 2 +

#

-"*0l * 01

y

+

,f

_. -

] C

V H o

2 U

+

t --

] C

*11

nt v +

il

3 v

+

-'-l

W;1^XV..][#10H#;/+...]C#01^#;A-^I^+*CV8+---]

where x 1 = i(x+y+u+v), y1 = |(x+y-u-v),u 1 = i(x-y+u-v), v x = i (x-y-u+v). 3 Now comparing the coefficients of any cubic term, say x , on both sides (the result i s the same for all the cubic terms), we get? 6 *ll*10V>oo

=

ar*oo*01*10*Ll

+

» oo v 01 10 11

8 *oo*01*10*ll 8

oo 01 10 11

Or, equivalently, 0

. S L *il

too *oi *oo

But in view of the Heat equation

j

*01 *10 '

66

"fc2

-

A

*

the above is also equivalent to 0 . ^ [ t o g * ^ . l o g ^ - l o g # M - log # 1 0 ] , or

' i i / ' * L S . « is a constant function of T(on H). Letting T i i oo 01 10 we see that asymptotically

>ico.

*/i ~ ' 2TTexp (ni T / 4 ) and t> * * ~ 2 exp (-rri T / 4 ) , hence the 11 OO 01 10 constant is -TT. This proves the formula (J 2 ). As a consequence of this formula, we have: 2 Corollary 13,2. £» (0, T) (besides being a modular form of weight 3 and level 4) is a cusp form, i . e . , it vanishes at all the cusps (since at each 2 cusp one of the 3 modular forms £.. (0, T), (i, j) / (1,1), vanishes). We shall find later a large class of differential operators which applied to theta functions give modular forms.

However, only isolated

generalisations of Jacobi's formula (J2) have been found and it remains a tantalising and beautiful result but not at all wellunderstood! § 14. Product expansion of 0 and applications We shall devote the rest of this chapter to discussing some arithmetical applications of the theory of theta functions.

No one can doubt that a large

part of the interest in the theory of theta functions had always been derived from its use as a powerful tool for deriving arithmetic facts. We saw this already in § 7, when we evaluated Gauss Sums along the way in proving the functional equation.

67

We will divide the arithmetic applications into 3 groups constituting the contents of this and the subsequent sections. The first group consists in a set of startingly elegant evaluations of infinite formal products which go back to Euler and Jacobi. Their connection with theta functions comes from the idea of expanding £(z, T) in an infinite product. However, these product formulae are special to the one variable case. Since the zeros of *(z, T) break up into the doubly infinite set z = \ + £ f + n+m T; m , n c S , it is natural to expect that 0 will have a corresponding product decomposition. In fact, note that exp[TTi(2m+l)T- 2TTiz] = -1 2Triz-TT(2m+l) T (2n+l)TTi,n %7L
z = i ( 2 m + l ) T + l ( 2 n + l ) .

This suggests that £ ( z , T) should be of the form T T (1+ exp[TTi(2m+l)T-2triz]) me2Z upto some nowhere vanishing function as a factor.

To obtain convergence,

we separate the terms with 2m+l > 0 & 2m+l < 0 and consider the infinite product: p(z, T) = 1

rj( 1 + e x P^i(2m+l)T-2mz])(l+exp[TTi(2m+l) i+2TTiz]) )

To see that p(z, T) converges (absolutely and uniformly on compact sets), we have only to show that the 2 series E exp [TTi(2m+l)T t 2 n i z ] meZ£ + have the same property: in fact, if Im z d >0, then

68

fexp [TTi(2m+l)

T

t 2TTiz]|^ (exp 2nc)(exp -TTd)2m+1

e t c . , hence p(z, T) converges strongly.

Clearly p(z, T ) has the same

zeros as £(z, T). NOW we have the following: Proposition 1 4 . 1 .

An infinite product expansion for £(z, T) is given by

(J 3 ): *(z, T ) = Y 7 ( l - e x P T T i ( 2 m ) T ) y T

£(l+exp[TTi(2m+l)T-2TTiz] ). (l+exp[TTi(2m+l)T+ 2niz])} .

Proof. We write the right hand side as C(T)

p(z, T ) . Observe that the

convergence of the function c(T) = | | (1- expTTi(2m) T) meIN is immediate, and is noTwhere vanishing on H. On the other hand, p(z, T ) has the same periodic behaviour in z as £: in fact, we see that a) p(z+l, T ) = p(z, T ) (clear from definition of p(z, T ) ) , b) P ( Z , + T,T) = Y J d + e x p [TTi(2m+l) T - 2TTi(z+T)]Xl+exp[ni T-2rri(z+ T )]). meIN y~{" (l+exp[TTi(2m+l) T+2TTi(z+T)]) meZ+ = \ | (1+exp[Tfi(2m-l) T-2TTiz]) texp(-TTiT-2rriz) # meIN (1+ exp[TTiT+2TTiz] )} \ \ = exp (-TTiT- 2tTiz)# p(z, T ) . Therefore, we must have (*)

*(z, T) * c ' ( t ) p ( z , T)

(l+exp[ni(2m+3) i+2TTiz])

69

for some (nowhere zero) holomorphic function C'(T).

TO show that

cf(T) = C(T), we will use Jacobi's derivative formula (J ) from the previous section* In fact, substituting z + | , z+£ T , z + | + \ T for z in (*), we get: *

01

(z, T)

C'(T) TTt(l-exp[TTi(2m+l) T-2TTiz])(l-exp[TTi(2m+l)T+2TTiz])} meZL

* 1 0 (z, T) = C'(T) (exp TTi T/4) [exp n i z + exp ( - n i z ) ] . " f T {(l+exp[ni2mT-2iTiz])(l+exp[TTi 2m T+2TTiz])} mclN (z, T) = ic'(f) (exp rri T/4) [exp TTiz - e x p ( - n i z ) ] .

0

TT C(l-exp[ni2mT - 2TTiz])(l -exp[rri 2m T+2rriz])} . mclN Thus we get: *oo ( 0 '

T T t1 + ex PTTi (2m+l) T ) 2 me2Z + , 2 * A 1 (0, T) = C'(T) T T (1 - exp TTi (2m+l) T) 01 + m eZ£ *

10

T ) = C (T)

'

(0, f) = 2C«(T) (exp TTi T/4) T"[ (1+exp TTi(2m) T ) 2 mclN

£'(0,

T) - -2TTC'(T) (exprriT/4)

11

-r-r 2 \ ] (1 -exp TT i(2m) T ) .

melN

(The last one is obtained by writing *ll^ z * T* = f e x P n i z " e x P (-TTiz)]f(z) and noting that simply ^ ' ( 0 , T ) = 2TTif(0)). Now substituting into Jacobi's formula (Jo)> w e £ e t :

70

-2TTC(T)(exPTTi T/4) "["""[ (1 - exp TTi(2m) T) mflN S

- 2 C « ( T ) 3 J T (l+exPTTi(2m)T)2 ]~T (l-expni(4m+2)T) mcIN mcZT*

I ( (l-expTTi(2m) T) mcIN

c'(tr

J T (l+exPTTi(2m)T) T T x+( l - e x p TTi(4m+2) T) mcIN me2Z

Now cancelling 2nd part of the denominator against the terms in the numerator corresponding to m = 1, 3, 5 , . . . , we get:

C'(T)

J T (l-expni(4m) T) mcIN

2

| I (1+ expTTi(2m) T meIN Writing l-exptTi(4m) T= (l+expTTi2m t ) ( l - e x p Tri2m t) and cancelling gives 2

c'(*) 2 »

T T d - exPTTi(2m) T) mcIN

= C(T)

.

But since lim C(T) = 1 , this shows that Im T >co c(T)

f T ( l - exPTTi(2m) t ) , meIN

as required. This proves the formula (Jo). Some applications: In terms of the variables q = exp n i t and w = exp TTiz, the formula (J«) reads: ,_ x (P ) :

2m T~T/* *mx T T f/ix 2m+l 2m+l 22WWt, A, 2m+l 2m+l -2 m ^m - zX. . L q w = \ |(l-q ) [ | {(1+q w )(l+q w )J . mtffi mcIN m«2 r

An elementary proof of this striking identity can be found in Hardy and Wright, p. 280. Setting w = 1 and w = i respectively give equally striking

71

special cases:

(p 9 ): 2

2m+l v 2

qm - T T W m > T T

z

mcS

«*q2m+1)

nic2Z+

m e IN

z (-i)mqm2 - JT(i-q 2 m )TT d-q 2m+1 ) 2 .

(P 3 ):

me2Z

me2Z +

m e IN

However, the most striking variant of all arises when we look at *!

i (0*3 f): we have

*" » 2

6 *,

( 0 , 3 T ) = (ex P TTi/6)(ex P TTiT/l2) ^

1

6'

1 2

(1 + £ T , 3 T )

OO

= (expTTi/6)(expTTiT/l2) T | (1- expTTi(2m) 3 T). meIN ~J I mc2Z

{(1-exp [TTi(2m+l) 3T t Tri T ] ) )

« (expTTi/6)(expTTiT/l2) J " [ (1-exp TTi(2k) T )keIN On the other hand, we have *-

\.\

(0,3T)=

S exp[TTi(m+l) 2 3 T +2TTi(m+I)iD

meZ

6

6

D

= (exp Tti/6)(exp n i T / 1 2 )

m 2 Z (-1) exp TTi ( 3 m + m ) T . mf2Z

Thus we get in terms of q:

(P4):

Z (-i) m q 3m2+ni = TTd-q 2 m ) me2Z

m e IN

which was first proved by Euler. A final identity of the same genre is found by returning to the formula for £

and substituting c( T), we find:

72

*'(0,

T ) = -2TT(exPTTiT/4) 7~J (l-ex P TTi(2m) T) . mClN

But we have (from § 13) I V J O . T ) = -n(expTTi T /4) YL (-l) m (2m+l)(exp iri (m2+m) T ) . 11 mc2Z Thus we get in terms of q :

Z (-D m (2m + 1) q m 2 + m = 2 T T (1 -J™)3.

(P5) :

m«2Z mcIN Combining (P.) and (Pj.), we deduce that t*l

x

(0, 3 T ) f = 2 ^ - *»

(0.

T)

6'? hence that £, £

(0, 3 T ) has value zero at all the cusps. It is the simplest

, with this property. Among higher powers, a, D [*

is the famous

^ O ^ T ) ] 2 4 =exp(2TTiT)yj (1-expTTi(2k) T ) 24

A-function of Jacobi.

The reader can check easily

from (Pr), (J 2 ) and Table V that it is a modular form of level 1. It is the simplest modular form of level 1 vanishing at all the cusps! (P 4 ) and ( P J are in fact the first two of an infinite sequence of evaluations of the coefficients a

,

in:

m, k

k m J T ( l - q 22nu ) - Z

m cIN

am

As soon as k t 2, the term in brackets can be broken up, so we get: (*) k

: L 7- = n c Z < z+n ) (k-l)i

( I

n

exp 2TTinz).

neIN

Thus if k >2, we get: E(T)«2[C(k)+

Z (Xj^rtit , 1 j t Zv n ncIN meIN [k~1K

k-1 exp2TTin(mT))]

=2[C(k)+^~^( Z (Z nk"1)exp2iriNT)] {k lu " NcIN nlN k = [ i ( k ) + i ^ l i L _ ( £ _ 1(n)exp2trinT)] +Vi

k

(where a An) = I d = sum of k powers of all positive divisors of n), K d|n This identity still remains valid even for the case when k = 2 if only E 2 ( T ) is summed carefully, i . e . , sum first over—n and then over m, in which c a s e , this calculation shows that it converges conditionally.

In particular, this

shows that lim

E k ( r) = 2 C (k)

Im T — > o o hence E ( T) has good behaviour at the cusps, and therefore if k > 4,E (T) is a modular form of weight k and level 1. Note that its Fourier coefficients are the more elementary number-theoretic functions o (n). Our plan is ultimately to write £ (0, T) as an Eisenstein series related to E 2 ( T ) : 4 first notice that £ (0, f) is a modular form of weight 2 whereas E g does not even converge absolutely, so our proof of the functional equation for E2

78

breaks dovir '! However, reca^ that »f for r«

9;

is only a modular form

so what we can do is to: (2^.

Modify the Fisenstein s e r i e s E« slightly,

thereby loosing intentionally a bit of periodicity but gaining absolute convergence. Let

EJ(T)1

Z

[

m,ne2Z

?(2mr+ (2n+l))

" 7^ ((2m+l)T+2nr

Since (4m+l)T 2 +(4n-4m)T- (4n+l) (2mT+2n+l) 2 ((2m+l) T+2n) 2

1 1 (2mT+(2n+l) 2 " ((2m+l) t+2n) 2

Am + Bn < C (m2 + n 2 ) 2 S ( m 2 + n 2 ) 3 / 2 ' we get that E 2 ( T) is absolutely convergent on compact sets.

Moreover,

if we sum over n first, then both the series

1

« and

neZZ, ( 2 m *+2n+l) 2 ( > 2/ 3*) for some constant C . Thus o ^ < I exp (2n/n) £ |f(x+i/m)|dx < exp (2TT/n) . CQ . m k < C . m k , Vm > 1 , (*) In fact, it converges if Re(s) >k but we won't prove this.

88

as asserted. Proof of Theorem 16.2. The convergence of Zf(s) is immediate for Re (s) >k+l (by Lemma 16.3). Now we relate f and Z . by the Mellin transform as before: CO

M(f(ix) -a Q )(s) = £ ( Z a m exp (-2TTmx/n) . x1s dx x 0 meIN co s dx = Z a m ^ exp(-2TTmx/n) x -$- . mcIN 0 Replacing x by y = 2TTmx/n in the m" 1 integral, we find: M(f(ix) -a o )(s) =

co s I ^ exp(-y) ( j f - ) y s ^ mcIN 0 m y

=

s< ^ a m , n " , > T e x p ( - y ) y 8 ^ ^n

meIN

0

= (~)SZ.(s)r(s) 2 TT

I

(Here we a r e assuming Re(s) >k+l and we can interchange the o r d e r of Z and J because the calculation shows that CO

-D

/

\

Z j a m K exp (-2TTmx/n) x m elN o Now since f (z) e Mod,

, we see that g(z) e Mod

~xYZ

In view of the elementary divisor theorem >

94

the new maps are all compositions of SL(2, TL)/? acting on H / r

and the

basic maps T#= T : H/I\ t y ' in given by translation by y

=

(n * ) *

>H/r ' n

* Tl

> ^ T#

Hurwitz studied these in the form of the "modular correspondences", i . e # , we have 2 maps:

hence we can consider the image

H/rx

»c je c(H/r 1 )x(H/r 1 )

Clearly C £ i s just the image in (H x H ) / ( r x T J (T, X T ) in H x H.

It i s called the I

of the locus of points

modular correspondence.

If

H/r

i s taken a s the moduli of complex tori, then it i s e a s y to check the following: Proposition 1 7 . 1 . Let

Then

T , T e H#

i

3a covering map TT : E

^^> 1

whose covering

2

group i s translations on E

T

by a c y c l i c group of order I l

Currently, the most fashionable approach to this new structure i s to consider the i n v e r s e limit

#.u»_H/r n n of all the Riemann surfaces in the tower.

This

i s not the same a s

H just

95

a s the real line 1R i s not the s a m e a s the compact abelian topological group (Solenoid) lim "R/nZZ n i . e . , the induced map H -

i s not even bijectivej

The important point i s that the Hurwitz map

T : H/r , Y

x

>H/r

n(ad-bc)

' n

between different s p a c e s p a s s e s up through the tower and induces a b i j e c t i v e map T of ^ to itself. Thus GL(2,Q) a c t s on the space T( . Appendix: Structure of the inverse limit ^y» (1) F i r s t l y , J i has a kind of algebraic structure.

In fact,

H

/r

is

canonically an affine algebraic curve: abstractly this i s because we can compactify it by adding a finite s e t of c u s p s , and a compact Riemann surface has a unique algebraic structure on it.

Concretely, if n = 4 m , we consider

R n the ring of holomorphic functions f

/ 0 and sequences (n(p)} of non-negative integers such that n(p) = 0 for all but a finite number of p's. We embed Q in A diagonally, i. e . , as the subring of adeles (x

; . . , , x , . . . ) such that x

for all p. This makes Q into a discrete subgroup of J\ and

= x = a/b c Q

because if a / b c Q

a/b| is small, then some non-trivial prime occurs in the denominator,

so p-adically a/b t ZZ . The adeles frequently arise in studying inverse limits. The simplest case is the solenoid mentioned above: Proposition 17.2. There is an isomorphism of topological groups: (*) :

lim

B/nZ:«A/Q

neIN Proof. Let n =TTp be the prime decomposition of n eIN. Define a p subgroup K(n) of A by K(n)={(0; . . . , X p , . . . ) / x p e p n ( p ) 2 Z p , V p 3 .

97

Then K(n) is compact and OK(n) = (0), hence n

A » lim A/K(n). Therefore, A / Q « lim A / ( « + K ( n ) ) A / ( Q + K(n))

given by xi

is an isomorphism.

> (X;..., 0 , . . . )

This map makes sense and is injective because

( n ; . . . , 0 , . . . ) = ( n ; . . . f n , . . . ) + ( 0 ; . . , l - n l . . i ) « Q + K(n).

Surjectivity follows immediately from: Lemma 17. 3 (Approximation for Q) : Given a finite set S of primes, integers n(p) > 0 and p-adic numbers

a €$

for peS; there exists a rational number

a e Q such that (i) (ii)

a^cp^'a^Vp.S ac2Z

for all p^ S.

The reader may enjoy checking this. This example should serve as motivation for the more complicated adelic interpretation of jf£ . For this we consider ft'

=GL(2,Q)\GL(2,A)/K 0 0 . Z ^

98

where K

and Z ^ are the subgroups of GL(2,A) of matrices

XMX^.^Xp,...)

X€K

given by


K

and > X ^ ={Xoo ° ) x e B * and X =1 Yp . 0 X co P 2'

XeZ < 00

00

Note that determinant gives a map

det : # '

X l f ^ / H * = A*/**.*" .

Vsing the unique factorisation of a fraction a / b c Q , namely,

./b-tti) T T

n(p) P

p prime where n(p) c 7L and n(p) = 0 for all but a finite set of primes, it is easy to see that

A*/«*.K^ IT

^*

p prime which is a compact totally disconnected space.

Now we see that the connected

components of = det" 1 (l)

Then 2£' is in fact connected as a corollary of: Theorem 17.4. $£z $('

and in this isomorphism T

right multiplication by A

= (I Y

2 JY»...#Y.-.-)

becomes

99

( i . e . , identity on the infinite factor and right multiplication by y on the finite factors). Proof. An easy generalisation of the proof of Prop. 17.2: we need the Lemma 17.5 (Strong approximation for SL(2,aa , Pp nn ^b Vp

\

(i)

X=

(ii)

X € S L ( 2 , Z ) for all p { S.

\ P

n(p)

cp , iy(p) d p f x p f o r

suitable

VVvW»-

(For a proof see Lemma 6.15 in Shimura's : Introduction to the arithmetic theory of automorphic functions, Tokyo-Princeton, 1971). We now analyse Jf'

in a series of steps:

Step I: The natural map SL(2,« > )\SL(2,A)/K a>

(*)

>GL(2,Q)\GL(2,A)°/K o o .Z o o

is an isomorphism. It is clearly surjective because modulo suitable elements ( a '

) in

GL(2,Q) and ( °°

) in Z ^ , we can alter any X in GL(2,A)° until its *co determinant is 1. To see injectivity, say X , X cSL(2,A) have the same

images, i . e . , X, = A X 0 B C for suitable A € GL(2,Q), B c K 1

Z

But then we get:

, CeZ CO '

. 00

100

1 = det A, det C. * > On the other hand, we know that det A cQ , det C elR but these two have nothing common in jf\^ . So det A = 1, i . e . ,

AeSL(2, Q), hence det C = l , i # e # ,

C = ( i ( J J ) ; . . . , I2, . . . ) Thus C € K

. This means that X

SL(2,Q)\SL(2,^)/ K

and X

define the same element in

, as required.

Step II: For neIN, let n =TTpr P . Define a subgroup K(n) of SL(2,A) by P K(n) = C ( I 2 : . . . . X p , . . . )/X p e SL(2, 2Zp) Vp

and

- pr(p)^p \ Then it is easy to check that K(n) is a compact subgroup of SL(2,A) and that 0 K(n) = [ 1 } . neIN Consequently, using (as before) the fact that SL(2,A) = lim SL(2,i\)/K(n), *~n we get: I**):

SL(2,Q)\SL(2,^|/K o o

Step HI, :

^

>

lim SL(2,:-".y5 J ) — )

This is the same as ,1 0 , v

, *> °

which in this form carries SL(2,^) to itself.

,1 0 xY ,1 0,

v

Restricting this to SL(2,1R),

the action is

IXooi —

>((

V - '

o

0 /

-*)Xo°:--'*I2"")

and acting on H this is the map Tf

>Jt T , T= X O T (i)

defining T^ . This comples the proof of the theorem. We give a 3rd interpretation of { : (3). J\ as a moduli space: we state the result (without proof): Isomorphism classes of triples (V, L, cp) where -

I V = one dimensional complex vector space L = Lattice in V cp = an isomorphism : —\—

>(§;)

of "determinant l"

To explain "determinant l" : note that the complex structure on V orients V and enables us to distinguish "orientation preserving" bases of L, i . e . , those bases e , e 2 of L such that if iej = aej+beg, then b > 0. Any two such bases e.,^2

and f^fo are related by f

= ae 1 +be 2 & f2 = c e ^ d e g

with ad-be = 1. Any such basis gives us an isomorphism

103

Now cpoqT o

i s given by a 2 x 2 m a t r i x (

(which i s w e l l - k n o w n t o be A*, that x t - y z = 1. where

L'cQ.L*

z t

) with e n t r i e s in

the ring of finite a d e l e s ) .

Hom( ( V , L*,cp') (Q/5Z)

under the map Y

i

i

= £ fY0<J>

j

= T * (f).

j

Y

a s required. This procedure i s best illustrated by the following basic example: Lemma 18. 3.

Let p be a p r i m e .

Then the following 3 s e t s a r e the same:

[ X e G L ( 2 , Q ) / X integral & det X = p ] the double c o s e t S L ( 2 , S ) ( * °)SL(2, 2Z) ~\SU2,ZZ)(P

L

)

°

U

SL(2,2Z)(*

0<j2 ( Pl a P l p 2 „ ) + Pl \ 1

and P

.r

P

l'

, 2»n-

P

n

P

> Jn

n} + Pi"' ap n - _ n _ P l ^ PlP2 P l

*W

»

= symmetric in p and p 0 , a s required, * 1 Therefore, we can expect to find simultaneous eigenfunctions f for all

T#

In fact, suppose that ,i

;'-p1"kV'Yp

Then substituting in our formula for T f, we find:

a a = ^

if p f n

n

p n

k-1 a

pn

+P

a

a

if

Pln

Clearly, these formulae enable us to solve recursively for all a n a s a polynomial in the a 's times a , They are best solved by going over to the Dirichlet series

108

Z.(s) = I a n f neIN n

8

.

More precisely, summarising the discussion above, we have: Proposition 18. 5. (Hecke): Let feMod

with its Fourier expansion

f(T) = E a n exp(2TTin T ) . neZ+ Then the following statements are equivalent: -(k-1) (1) f is a simultaneous eigen function of eigen value a . p for the * Hecke operators T , p prime, (2) for all n eIN, we have

fapn

if

pf n

a a = ) P n < k-1 .. , a +p a if pK n [ pn ^ n ' P (3) the associated Dirichlet series has an Euler product expansion, namely, a n n S = Z f (s) = a. .

I

TKT "

*

T T

s ,

K-i

-*» "

k 2S D ~ \. pD ( 1 - a. pis + p )

1

PP

n eIN p prime (In particular, for such an f, the Fourier coefficients a , n > 0, are determined completely by a«), Proof, We have seen that (1) = = = = = =>(2)

and it is a straightforward

verification to see that (3) = = = = = = => (1). Assuming (2), (3) follows once we show that for any prime p and any q c IN, p {q , we have: (l-aDpS + pK-\pZS)( p

I a ns) = I a nelN,(q,n) = l n n eIN, (pq,n)=l

Let us calculate the expression on the left hand side, i . e . ,

109

L a n (q,n) = l n

I a a (pn) (q,n) = l P n

+

T p (q,n) = l

a (p n)

Z ann'SZ apan(pn)"SZ apan(pn)~ (q,n) = l (q,n) = l (q,n) = l p{n p|n =

I apn(pn)"SI (a Z ann_S(q,n) = l (q,n) = l (q,n) = l pfn p)n

Z a n"S (q,n) = l =

a

E

Z a n n" (pq,n)=l

(pn) pn

(q,n) = l

F

P

+

T P ~ am(pm)"S m=np P~ (q, m) = l

+p k _ 1 a a )(pn)" S + Z p ^ a ^ p m ) " 8 (by (2)) P (q,m) = l F^~ p|m

"S

, a s required.

F r o m this it follows that

[

T T SL(2, TL/nZL)

be the natural map and let T We have

n

= TT"1 (Diagonal m a t r i c e s in SL(2, Z Z / n S ) } .

110

T n = ker TTcr (1) and T^ /Tn = diag SU2,ZZ/nZZ) i.e., r

/r

by f |

>R

KfZ/nZ)*

is a finite abelian subgroup of SL(2, S ) / r n and acts on Mod, a(

f

= fY

'

where TT(Y) = a I 2 , a e (Zi/n2Z) . Hecke's main result

is: Theorem 18.6 (Hecke): Consider the operators on Mod*11' : * a) T

, y

b) fi

=

a b ( c H)» a,b, c,de2Z, ad-bc > 0 and gcd (ad-bc, n) = 1,

) (mod n), at (2Z/nZ)* # >f 6 where 6cSL(2,:Z) and 6 = (* u a

Then (1) all these operator commute, (2) Mod

has a basis ( f } of simultaneous eigen functions for all of them,

and (3) the Dirichlet series Z z

f

f a> )

has an Euler product: a rational function -rnr _s k-1 -2s - 1 s ofp,pln „,„ P* k

-

4

of the Hecke operators as poly-

>& L (0,nT) seems to be quite hard to describe a,b of weight k = 1 or 2, For example, take the case:

nomials in the functions r\ except in the case

(4) M

°

d k

T space of homogeneous polynomials of 5£ ,#A* ~ degreekkiin A modulo = )>degree nin*,,2 * , #* .4 .4 .4 of $ - A*. - A' u multiples r oo 01 10

Then in this space, the subspace of cusp forms is the set of multiples of 2 2 2 £ A*„ A . But it seems hard to describe in any reasonably explicit and oo 01 i o elementary way the subspace of Eisenstein s e r i e s , let alone the set of eigen functions f . (cf. B. Schoeneberg, Bemerkungen zu du Eisensteinchen Reihen und ihren Anwendungen in der Arithmetik, Abh. Math. Seminar Univ.

113

Hamburg, Vol. 47 (1978), 201-209; for s o m e calculations for s m a l l n).

In

the c a s e of degree 1 or 2, i . e . , polynomials in the £ (0,n T ) of degree 2 a, b or 4 , the eigen functions f

can be more or l e s s found - modulo a knowledge

of the arithmetic of suitable quadratic number fields, r e s p e c t i v e l y quaternion algebras.

This i s because the theory of factorisation in imaginary quadratic

fields K and in certain quaternion algebras D allows one to prove Euler products for suitable Dirichlet s e r i e s E x(o) . N m ( a ) " S acM where M i s a free

7L-module

of rank 2 in K, respectively of rank 4 in D,

and y i s a multiplicative character, just a s one does for the usual Dirichlet L-series S y(n)n~s. neZ But Nm(a) i s a quadratic form in 2 or 4 variables in these c a s e s , and s o there a r e Epstein zeta functions. e x p r e s s these a s polynomials in £

Taking the inverse Mellin transform, we can , (0,n«r) of degree 2 or 4 .

F o r the c a s e

of quaternions where everything depends on the s o called "Brandt m a t r i c e s " , cf. M. E i c h l e r , The b a s i s problem for modular f o r m s , Springer Lecture Notes No. 320(1973). In connection with Hecke's theorem, we want to conclude by describing a daring coniecture which arose from the work of Weil, Serre and Langlands, a s s e r t i n g which Dirichlet s e r i e s a r i s e from modular f o r m s . Conjecture.

Their conjecture is this:

Let K be a number f i e l d , K^ its ^p -adic completions. Consider.

the continuous representations P x : Gal(Q/Q)

->GL(2,K X ).

114

Suppose for almost all X, a p,

is given.

We say that the p. f s are compatible

if there is a finite set S of rational primes p such that a) if X lies over a rational prime t, p^ is unramified outside S | j { i } , i . e . , p* is trivial on the p t h inertia group I c G a l (Q/Q), hence p. (F ), F the * P X p p p t n Frobenius element, is well-defined and b) for all X l# X2 lying over ty

J*2 and all p ^ S U t i j , lj

,

Trp^CF^^Trp^Fp) det

P\ (Fn> K \ P

= det

*\ pfn.

and suppose that f

is an eigen function for the

o Let Z

be the product of the p-factors in Zf a " a for p \ n. Then there exists a compatible family tp* } of odd 2-dimensional representations with S = [primes dividing n] , such that Z° (s) - Z° (s) . f a tPxl

115

The case k = 1 is analysed in Deligne-Serre, Formes Modulaires de poids I, Annates de Sci. Ecole Norm. Sup., t. 7(1974), 507 - : precisely in this case, all the p. 's coincide and come from one p : Gal(Q/Q)

> GL(2,K)

with finite image. The conjecture in question is the converse to this theorem, i. e., every series Z« \(s) is Z. (s-m) for some a , m^O I

116

References and Questions

For many of the topics treated, especially for several treatments of the functional equation (§7) an easy place to read more is: R. Bellman, A Brief Introduction to Theta Functions, Holt, 1961. For a systematic treatment of the classical theory of elliptic and modular functions, nothing can surpass A. Hurwitz, R. Courant, Vorlerungen uber Allgemeine Funtionentheorie und Elliptische Funktionen, Part II, Springer-Verlag (1929). For modular forms, a good introduction is: B. Schoeneberg, Elliptic Modular Functions:

An Introduction,

Springer-Verlag (Grundlehren Band 203) (1974). We have avoided, in this brief survey, the algebraic geometry of the objects being uniformised elliptic curves and the modular curves. Two general references are: S. Lang, Elliptic Functions,

Addison-Wesley (1973) where analytic 2 and algebraic topics are mixed (but the series Z exp(-rrin T) is scarcely mentioned), and A. Robert, Elliptic Curves, 326 (1973).

Springer-Verlag Lecture Notes

117

There are many open problems of very many kinds that could be mentioned.

I want only to draw attention to several problems

relating directly to theta functions, whose resolution would significantly clarify the theory. CD

Which modular forms are polynomials in theta constants?

More precisely: Is every cusp form of wt. n _> 3 a polynomial of degree 2n in the functions *$- b (0,i), a,b e Q? (II)

Can Jacobi's formula be generalized, e.g., to vJ(°'T)

(S)^

=

(cubic polynomial in /fr

a,D

oZ

for all

a,b e Q?

's} C,Q

Similarly, are there generalizations

of Jacobi's formula with higher order differential operators (see Ch. II, §7)? (Ill)

Can the modular forms if

,(0,ni) be written, e.g., as

Quadratic polyn. in tr" J'S Linear polyn. in ijh d's (IV)

Can all relations among the

«fcP" ,(0,T)'S be deduced

from Riemann's theta relation, or generalizations thereof? A precise statement of this conjecture is given in Ch. II, §6.

118

Chapter II:

§1.

Basic results on theta functions in several variables

Definition of ^

and its periodicity in z.

We seek a generalization of the function ^ ( Z J T ) of Chapter I where z G (E is replaced by a g-tuple

z = (z-,*--,z )€ (Cg, and

which, like the old 1?* , is quasi-periodic with respect to a lattice L but where Lc(Cg. The higher-diinensional analog of x is not so obvious. It consists in a symmetric gxg complex matrix Q whose imaginary part is positive definite: will appear later.

Let Kr

open subset in (Cg g

*'

.

why this is the correct generalization be the set of such Q.

Thus )?v

is an

It is called the Siegel upper-half-space.

The fundamental definition is:

A9*(Z^) =

I n£Eg

expU^nftn + 2i\£n-z)

-> ->

. t+

(Here n,z are thought of as column vectors, so n is a row vector, t-n»z > •* is the dot product, etc.; we shall drop the arrow where there is no reason for confusion between a scalar and a vector.) Proposition 1.1.

-

l 75—

2TT

and

c0I

2 g

hence it defines a holomorphic function on

(Eg xi

.

119

Proof; expOjri nftrtt-27ri nz)

0 ' Y expC-irc^n + c . n ) n>0

= const.

"

\ exp| -TTCJ n~2 n>0

L

J

2TTC

which

and

converges

2

00

f

I e ' X dx.

like

Q.E.D.

Note that Vfi, 3 z r TT.i. nftn 2Tri nz > e e . . Cr.. coefficients S7

\9(z,ft) 7* 0

„ . £ i s a F o u r i e r e x p a n s i o n of

Tri nftn e

Q

=

because

0 . .. „ \7 , w i t h F o u r i e r

, ~ ^ 0.

may b e w r i t t e n more c o n c e p t u a l l y * a s a $UrQ>

where

such that

series

J.„ exp(Q ( £ ) + £ ( £ ) ) nfcZ?

is a complex-valued quadratic function of

complex-valued linear function of n.

n

and £ is a

To make this series converge,

it is necessary and sufficient that Re Q be positive definite. Then any such Q is of the form Q(x) = -rri x-ft-x, and any such

I

n € -|w

is of the form £(x) = 2iritx-z,

z € (Cg

0 was explained this way in a lecture by Roy Smith.

120

hence any such

-v (&/Q) equals $(z,ft).

(This gives a formal

justification for the introduction of -i^ )• To ft , we now associate a lattice

L

e (Cg:

L^ = 2 g + QS g i.e., L~

is the lattice generated by the unit vectors and the

columns of ft. The basic property of ~$ for

zi

>z+a, a € L Q .

Here quasi-periodic means periodic up to a

simple multiplicative factor. § -$ (z+ftm,ft)

is to be "quasi-periodic"

In fact,

(z+m,ft) = S

(z,ft)

= exp(-iri t mnm - 2Tri t mz)^(z,ft)

Vm € Z g

.

121

Proof. of

&

The 1st equality follows from the Fourier expansion

(with period 1 ) ; the 2nd one holds because of the symmetry

of ft: e x p [Tri nftn+27Ti n(z+ftm)] = e x p ( i r i nftn+iri nftm+Tri mftn+2iTi n z ) = exp[Tri and. as I

(n+m) ft (n+m)+2iTi n+m

(n+m) z-7ri mftm-2iri mz]

v a r i e s o v e r 2E, . n d o e s t o o ;

exp[TTi nftn+2-rri n(z+ftm)]

so

= e x p (-iri mftin-27ri m z ) , £

n€ffi g

exp(7?i nftn+27U

n£2g Q.E.D.

In fact, conversely, if f(z) is an entire function such that

f(z+m)

= f(z)

f(z+ftm)

then f(z)

= e x p ( - 7 r i mftm-27Ti

raz),f

(z)

= c o n s t . $ * ( z , ft) .

Proof:

Because of the periodicities of f(z) with respect to

2Z^ , we can expand f(z) in a Fourier series:

f(z) =

[ c

exp(2Tri n z ) ;

now the second set of conditions gives us recursive relations among the coefficients c :

nz)

122

f (z+£l ) =

(ft.

= k

I c exp 2-rri n(z+£l ) r€X? column of

=

J c exp(27ri nSl)exp(2Tri n*z)

Q) , h e n c e

exp(-7rift, -27Tiz, ) \c exp(27ri nz) = \c exp(27ri n£l)exp(27ri n*z) .

f(z+fi,) , we obtain:

Comparing the two Fourier expansions for

c+ n+e,

Thus

f

= c+e n

k

KK

e

k

= k

unit vector.

is completely determined by the choice of the coefficient c n . Q.E.D.

This result suggests the following definition: (Cg

Fix Q € 4* . Then an entire function f(z) on — ? (f 0 (z),...,f n (z))

defines a holomorphic map >*n.

* g /L f l By a slight generalization of $

known as the theta functions

v L 3 with rational characteristics, we can easily find a basis of

R..

These are just translates of -$ multiplied by an

elementary exponential factor: S K (z,ft) = e x p U ^ a f t a + 27rita-(z+b) )-^(z+^a+b^) for all a,D € ffig. Written out, we have: x?[*](z,n) L J b

=

I exp[irit(S+a)fi(n+a) n€S^

The o r i g i n a l A 9 i s j u s t $ integral vectors,

^f!+!l L

Finally,

\/L]

n

= expU^t-Z)

b+m J

9\l\(l+m,Sl) L J b ->(z+ftm,ft)

a,S

.

are increased by

hardly changes:

-0[*]<J,n>. L

the q u a s i - p e r i o d i c i t y

$L

and if

+ 2 f r i t (n+a) ( z + S ) ]

of

bJ

~$tvJ

^ s given by:

= exp(27Tit2.S) .£[ l ( z , n ) L bJ -> = exp(-2?Ti S-m) 'expC-Tri mfim-2iri m ^ z ) - m

l(z,ft)

124

which differs only by roots of unity from the law for-9

. All

of these identities are immediately verified by writing them out, but should be carefully checked and thought through on first acquaintance.

Using these functions, we prove:

Proposition 1.3:

Fix ft e/y. . Then a basis of

Br: is given

by either: i)

f+(z) = -$ [ a

0 < a.< I 1

l(£-z,£.ft) ,

L 0 o J

-

or r

If

0 -,

g£(z) = A9 L

ii)

L£/£J

-,

(2/ )T -ft),

0 < b,< £

2 £ = k , then a 3rd basis is given by h+ £(z) = -$ a D '

iii)

ra/k-, I (JUz,fl) , LK/kJ

0 < a.,b. < k. x 1

These b a s e s a r e r e l a t e d by jg =

£ a

exp(27ri £~

h-> ^ = Y exp(2irik a b ' S^t mod k

Proof: in

z.

a-6)

f+

• c£)f->

As above, we expand functions in

c

R

By quasi-periodicity with respect to

a function f lies in

R

if and only if

f

as Fourier series Q^

, we check that

can be expressed as

125

f (z) = X x

where

T x (n) exp {-nil nez9

• n-fi-n + 2-rri n*z)

i-s constant on cosets of

characteristic function of

a+kZ

£*ZZ . Taking

g

f

becomes

x to be the f->; taking Y

X

a

*

A

t o be t h e c h a r a c t e r n \ >exp(2iri£ n*b) , f becomes g^; 2 and if £ = k , taking x to be the restriction of n I > exp (2Tri&~ • n*S) to a+k«Z g , f becomes h-> £. QED X a, D Let us see how these functions can be used projectively to embed not only

£ g / L n but "isogenous" tori

(Eg/L, L

a lattice

in L n -Q. First some notation:

and identify IRg x TR^

fix ft £h

with

g

(C viaftby a~: IRg x iRg Note t h e n t h a t

aQ

>(Eg, (x,y) ^ Z g x 2Zg

identifies

> ftx+y = z.

with

L~

= Z g + ftZg.

Define e: where

A

m2g xm2g

(C*,

e ( x , y ) = exp 2iriA(x,y)

is the real skew-symmetric form on

IR g *JR

g

defined by

A(x,y) =

x x .y 2 -

yi'x2'

x =

(x

l'x2}'

y =

(y

l,y2)*

126

It is immediate that e(x,x) = 1, e(x,y) = e(y,x)~ and e(x+x',y) = e(x,y)e(x',y).

Thus e is bi-multiplicative and so we can talk of the perpendicular V1

of a subset V c l

g

, namely,

V 1 = {x €]R2g |e(x,a) = l,Va € V}. We shall be particularly interested in the perpendiculars L withinffig of lattices L c (Q g , i.e.,

L1= {x € L^ £ L^ , etc.

a

L

L

Q^ ^ E o

=

is of a

o ^ ^^

Let a.,b.€L , 1 ^ 2, then for all u € (E^, there is a product

g(z) as above such that

g(u) ^ 0.

r >_ 3, then for all u,v € (Cg with u-v ^ a o^ L i^ ' there is

b) If

a linear combination

h(z) =

Ic.g.(z) of products g.(z) as above

h(u) = 0, h(v) ? 0.

such that

c) lf_ r >^ 3, then for all u € (Eg, and tangent vector I d i Hz~ ^ 0,there is a linear combination h(z) =

£c.g.(z) of

products g.(z) as above such that h(u) = 0 , \ d. -r—(u) ^ 0. 2

1

oZ.

132

by (a), B Q ( L ) = M f

This clearly finishes the proof: By (b) , if x,y € CC^/an(L) and

tp_ (x) = tp_ (y) , then provided r > 3,

bl

x-y € aQ(L /L).

Li

Li

Applying the same argument to L

—

1

and products g'

constructed similarly, we deduce x-y € a0(L,'/L) too. By (c) , the differential of

r > 2.

Thus x = y.

cpT is one-one if r > 3 too.

of the lemma is not difficult:

The proof

we take r = 3 for simplicity of

notation and see how (b) is proven.

For the other parts, we refer

the reader to [AV, pp. 30-33].

To prove (b), take u,v € £ g

assume

Then there is a complex number y

h(u) = 0 => h(v) = 0.

and

g

such that for all a,b€(E , (*)

f(v+a)f(v+b)f(v-a-b) = yf(u+a)f(u+b)f(u-a-b).

This is because the linear functionals which carry the function f(z+a)f(z+b)f(z-a-b) to its values at u and at v must be multiples of each other if one is zero whenever the other is zero.

Now in

(*),

If u)

take

logs and differentiate

with respect to a.

is the meromorphic 1-form df/f, we find Go(v+a) - a) (v-a-b) = u)(u+a) - a) (u-a-b), all a,b G (Eg. Thus

a) (v+z) -oj (u+z) is independent of z, hence is a constant

1-form 2iri £c.dz..

But

GJ (v+z) -u) (u+z) = d log f (v+z) /f (u+z) ,

so this means that 9

f(z+v-u) = c e o for some constant c . l

* f (z)

In this formula, you substitute z+e. and

z+ftf!+f'.' for z and use Lemma 1.6. l

. t + •> c z

It follows that

133

t

c.e i e E

^'.(u-v) — = Now write u-v =

c- (flf±+f 7)mod 2 .

Slx+y, x,y € 3Rg.

Take imaginary parts in the

2nd formula, to get f!.Im Q.x —-

. =

c-Im fl-f!, all i.

r

Hence c = x/r.

1

Putting this back, we find:

This means that ( —,—) € L , or (x,y) € rL

= L... , hence

u-v = ct^(x,y) £a (L ) . This proves (b) . For further details, we refer the reader to [AV,§3].

that

Finally,

CC /a0 (L) for

L c "2, J

L ^zf rL , r >_ 2, or even for arbitrary lattices

L c Q g?

what can we say about the complex tori

y

In fact, we do not get more general complex tori in this way, because of the isomorphism: (Eg/c^(L) given by

«

>

<Eg/a^(nL)

such

134

Because of these isomorphisms, the theorem has the Corollary: Corollary:

A complex torus

<Eg/L

can be embedded in projective

space if and only if A(L) c fl(Dg + 0>g for some

g x g complex matrix A, and some Q ^-KT •

135

§2.

The Jacobian Variety of a Compact Riemann Surface. It is hard on first sight to imagine how the higher-dimensional

generalizations of /\!r of the last section were discovered. result showed that the complex tori (Eg/L, (L c L 0

<Er/LQ

The main

and their finite coverings

of finite index) could be embedded by theta functions

in projective space, but no other tori can be so embedded.

This

justifies after the fact considering only the lattices L Q

built

up via ft €$v .

If g > 2, these are quite special: ft has ^ | —

complex parameters, whereas a general lattice is of the form Zg

+ ft-2Zg

ft any gxg complex matrix with det(Im ft) ^ 0, hence it depends on 2 g complex parameters. However, these particular tori arose in the 19th century from a very natural source: compact Riemann Surfaces.

as Jacobian Varieties of

Much of the theory of theta functions

is specifically concerned with the identities that arise from this set-up.

The point of this section is to explain briefly the

beginnings of this theory.

In Chapter III, we will study it much

more extensively in the very particular case of hyperelliptic Riemann Surfaces. Let X be a compact Riemann Surface.

As a topological space,

X is a compact orientable 2-manifold , hence it is determined, up to diffeomorphism, by its genus g, i.e., the number of "handles".

It is

well known that the genus g occurs in at least 3 other fundamental roles in the description of X.

We shall assume the basic existence

136

theorem:

that g is also the dimension of the vector space of

holomorphic 1-forms on X . An extensive treatment of this, as well as the other topics in this section, can be found in GriffithsHarris , Principles of Algebraic Geometry.

The first step is to analyze the periods of the holomorphic 1-forms, by use of Green's theorem.

To do this we have to dissect

the 2-manifold X in some standard way:

i.e., we want to cut X

open on 2g disjoint simple closed paths, all beginning and ending at the same base point, so that what remains is a 2-cell.

Then

conversely, X can be reconstructed by starting with a polygon with 4g sides (one side each for the left and right sides of each path) and glueing these together in pairs,in particular all vertices being glued.

The standard picture is this, drawn with g = 3:

*These are the differential forms u> locally given in analytic coordinates by co = a(z)dz, a(z) holomorphic.

137

-homologous to A3

homologous to B,

138

& r©i

K

G,

'©,11.

X K *

flk

:h A. = l e f t s i d e of A. A?1 = r i a h t s i d e of A.l B"!" = left side of B. B. = right side of B. ax

V o - X - U A , ->>§,•

= -yAt-yBt+YAT+YBT

139

Note that if I ( O , T ) is the intersection product of 2 cycles <J,T, then I(A. ,A.) = I(B.,B.) = 0 1 3 l 3 I(A. ,B.) = 6... i 3 ID

Theorem 2.1.

(Bilinear relations of Riemann):

Let X be a compact

Riemann surface of genus g, with canonical dissection X = X Q JL A x iL • • • _1L A

_1L B x JL- • • JL B

as above. a) for all holomorphic 1-forms

a), T),

g

1 i=l

/w A.

•

" 1 /«•h

/n B.

i-1 B.

= 0

A. l

b) for all holomorphic 1-forms a), /

g

Im I 1

/OJ

Jo) •

Vi=1

A.

> 0

l

l

Note that if we let

)

B.

T(X,ft ) denote the vector space of holomorphic

1-forms, and if we define per:

the period map

nx,^1)

> Hom(H 1 (X, 2Z) , CC) = H1(X,(C)

by

a) »

> < the co-cycle a i

> /a) >

then (a) can be interpreted as saying that with respect to cup product, the image of per is an isotropic subspace of H (X,(E) .

140

In fact, to

U

on H

is dual to I on H 1 .

Thus if we associate

a) the 1-cycle g

d = ^ ( f . ) . A, - X(\ B

„) B, A

i

i

then d(u>) satisfies

I(d(o)),c) =

a), all 1-cycles C.

I (d (co) ,d (n) ) , which is exactly

So per(o>) Uper(n) is equal to

the thing which (a) says is zero. To prove the theorem, since X_ is simply connected, there is a holomorphic function f on X n such that

a) = df.

Then f•n

is a closed 1-form, so by Green's theorem

I

x

d(fn)

o

- I W + f n + L f n - 1 + fn +1.fn A. i

A.

l

B.

B. l

l

[-(f

on A + ) + ( f

on

A.)]n

[-(f

on

on

BT)]TI

A. l

if

+ 1

|

B. l

B t ) + (f

141

As df has no discontinuity on A. or B., f on A. must differ from f on A. by a constant, and likewise for B.,B..

But the path B.

leads from A. to A. (see dashed line in diagram above) and the path A.1 leads from B. to B.. ^ 1 1

Thus

0 = I J (- I » )n + I \ (• { „) n A.

B.

l which proves

J d(fu>)

(a).

=

As f o r

J fa>

Xn

%Xn

0

0

A.

n

l

B.

l

(b)

= " I X

[ w

J a)

B.

A.

1

+

I 1

f a)

fa

A.

1

B. 1

1

as before . The right-hand side is

2i I m f I

a) • A. l

and

d(fa)) = df Adf.

a) j B.

l

Whenever f is a local analytic coordinate,

let f = x+iy, x,y real coordinates.

Then

df A df = (dx-idy) A (dx+idy) = 2i dx A dy. Since dx A dy is a positive 2-form in the canonical orientation Im [ df A df >

X

0.

^°

142

If we now introduce a canonical basis in

r(X,ft ) , a matrix

ft in Siegel's upper half space appears immediately: Corollary 2.2.

We can find a normalized basis

CJ. of

r(X,ft )

such that f a). = 6. . . J 3 ID A. l

Let ft. 13. =

Then ft.13. = ft., positive Jf a).. 3 31 and Im ft.13. is c B i

definite. st Proof: The pairing between GJ'S of 1 kind and A.'s is nondegenerate because of b) in 2.1. By applying a) to u = u). , n = w. we get ft.. - ft.. = 0; finally, in order to prove that, for any a,,..., a real, 1 g

Im

T a.ft..a, > 0, we let u> = LTa.a).. . % l lk k l l

By b) J

1 ,K

0

< Im J cuf I « k ^ k i )«

QED

We may understand the situation in another way if we view the periods of 1-forms as a map: per': H1(X,ffi) > Horn (T ( X ^ 1 ) ,(C) i

a or, if we use the basis per':

> < the linear map GO I-

u>., •••,(*) of 1 9

H1(X,Z) a »

T(X,ft ) :

> <Eg > a

)

I")

143

Corollary 2.3.

per' : H, (X , 7L)

The map

its image is the lattice

L~

> (Eg is injective and

generated by integral vectors and

the columns of ft. The fundamental construction of the classical theory of compact Riemann Surfaces is the introduction of the complex torus: Jac(X) By Corollary 2.3, if P

g-j

a*/Ln.

is a base point on X, then we obtain a

holomorphic map X >

P i

This is well-defined:

> Jac(X) ,P P (f ^i'"""' f " ) m o d periods Po Po

pick any path y from P

all the integrals along

y.

to P and evaluate

If y is changed, the vector of integrals

is altered by a period, i.e., a vector in Lfi. More generally, if

(K = I k.p. L

ii

is a cycle of points on X of degree 0, i.e., £ k. = 0, then we can associate to tt a point I ((A)

€

Jac(X)

given by I(M) = C

u)

w ) mod periods,

144

a

do =

a 1-chain on X so that

£k.P. .

The map (A »

> I(Ot ) plays

a central role in the function-theory on X because of the simple observation: Proposition 2.4:

Ijl^ f is a meromorphic function on X with poles

d I P. and zeroes i=l X

d £ Q. (counted with multiplicities), then i=l 1 d i=l

Proof:

d x

i=l

For all t € (E, let D(t) be the cycle of points where

f takes the value t, i.e., the fibre of the holomorphic map of degree d f: over t. of t.

> nP1

X

If P is a base point, consider I(D(t)-d.P ) as a function o o Because the endpoints are varying analytically, so does D(t) dP

hence

t »

> I(D(t)-dP ) is a holomorphic map 6:

But

np

o

3P1

> Jac(X).

is simply connected, so this map lifts to £:

IP1

> (Eg.

Since there are no meromorphic functions on 3P_ without poles, except constants, and

6 and

6

6(°o)-5(0) = K y^P .1- T Q.). ^ 1

are constant.

In particular 6(0) = 6 («>) QED

145

The beautiful result which is the cornerstone of this theory is: d d [ P., I Q• of the same degree, Theorem of Abel: Given cycles 1 i=l 1 i=l then conversely if f on X with poles

I ( £ P . - £ Q . ) = 0, there is a meromorphic function £p. , zeroes

JQ. .

We shall prove this in the next section.

146

&

§3.

and the function theory on a compact Riemann Surface.

We continue to study a compact Riemann Surface X. we fix a basis {A.,B.}

As before,

of H 1 (X,2Z), obtaining a dual basis

OJ. of

holomorphic 1-forms, a period matrix ft € & , and the Jacobian Jac(X) = (C g /L Q . of

We also fix a base point

§1, we have the function

respect to L~.

By the methods

1/(z,ft) on (Eg, quasi-periodic with

We now ask:

Starting with v^{z,Q.),

1)

PCX.

what meromorphic functions

on Jac(X) can we form? 2)

Via the canonical map X

> Jac(X) P P i > [ oj P o

what meromorphic functions on X can we form? Starting with (1), we may allow ft to be an arbitrary period matrix in

$v

.

Then there are 3 quite different ways in which

we can form L^-periodic meromorphic functions on (E , from the L n -quasi-periodic but holomorphic function "ft n n _>. _,. n \7(z+a. ,ft)

Method I: f(z) =

i=1

n #(z+S. ,ft) i=l where

a. ,b. € (Cg 1 1

1

are such that

I a. = **

l

Tb. mod 2Z 9 , is a L

l

meromorphic function on X~, since the denominator doesn't vanish.

147

i d e n t i c a l l y and t h e c o n d i t i o n

£a. = J&. p r o v i d e s us with

ft-invariance: exp(-7[7ri mftm+27ri in(z+a.)]) f(z+nm) =

f(z) = f ( z ) . exp(-£[TTi mfim+2iTi m(z+b.)]) i

A v a r i a t i o n of t h i s method uses t h e t a f u n c t i o n s w i t h c h a r a c t e r i s t i c : If a,S,a* ,&' 6 | z z g , then #[§] Jac (X)

More precisely, it is surjective and i" 1 (U) g

res I : g

> U

is an isomorphism. Proof:

Let

W

c Symm g X x Jac(Xj be the closed analytic

subset defined by both conditions P. 1

l js

z mod L 0

p

o

and

P

f(P) = $ (t-z +

\ S) P

is zero on

J P.. i=l X

0

Consider the projections

Symm g X By Riemann'stheorem, p^TU)

Jac(X) > U is an isomorphism.

In particular,

p 2 (W) is a closed subset of Jac(X) containing U, hence equals Jac(X). Thus dim W _> g. So

But p, is injective because the P. determine z.

dim p, (W) >_ g, hence p, is surjective, hence p, is bijective.

Therefore

W is nothing but the graph of I , i.e., the first condition g

implies the second.

This is the first assertion of the Corollary and

the rest is a restatement of Riemann's theorem.

QED

155

We next investigate the function y E^(x,y) = & (e + J

where

e € (Eg is fixed and satisfies 17(e) = 0, and x,y € X.

As with f, E is locally single-valued, but globally multi-valued, e being multiplied by an exponential factor when x or y are carried around a B-period. Lemma 3.3. For any P C X , y D

g

= je £ (E /Lfi|^(S + f 2) = 0, all y € xj P

is an analytic_subset of Jac(X) of codimension at least 2. Hence for any finite subset P..,---,? of X, there_ is an e l>(e) = 0, f±(y) = #(e + J 2) K °

Proof: X

for all

Let D be an irreducible^component of D

and let

a), all y £ X. Consider

c Jac(X) be the locus of points

the locus of points a+b, a t D, b 6 X D+X

such that

and call it D+X . Then

is an irreducible analytic subset of Jac(X) containing D

and contained in the locus of zeroes of "& . Hence dim D + X