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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
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(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 <_ 1.5 - 0 . 3 < Im z < 0 . 3
TABLE OF CONTENTS Introduction Chapter I.
Introduction and motivation: theta functions in one variable
ix 1
§1. Definition of / & \ Z , T ) and its periodicity in z §2. ^ x , i t ) as the fundamental periodic solution to the Heat equation §3. The Heisenberg group and theta functions with characteristics §4. Projective embedding of (E/& + TLi by means of theta functions §5. Riemann's theta relations §6. Doubly periodic meromorphic functions via ^KZ,T) §7. The functional equation of ^ 1 ( Z , T ) §8. The Heat equation again §9. The concept of modular forms §10. The geometry of modular forms §11. /iP-as an automorphic form in 2 variables §12. Interpretation of H/I\ as a moduli space §13. Jacobi's derivative formula §14. Product expansion of t £ and applications §15. Representation of an integer as sum of squares §16. Theta and Zeta §17. Hurwitz maps Appendix: Structure of the inverse limit J-£ §18. Hecke operators References and Questions
14 24 28 33 34 44 53 60 64 66 74 83 92 95 103 116
Chapter II.
118
Basic results on theta functions in several variables
§1. Definition of and its periodicity in z §2. The Jacobian variety of a compact Riemann surface §3. xP'and the function theory on a compact Riemann surface Appendix: The meaning of A §4. Siegel's symplectic geometry §5. tP'as a modular form Appendix: Generators of Sp(2g,Z) §6. Riemann's theta formula and theta functions associated to a quadratic form §7. Theta functions with harmonic coefficients
1 4 5 11
118 13 5 146 162 171 18 9 202 211 227
ix
Introduction This volume contains the first two out of four chapters which are intended to survey a large part of the theory of theta functions.
These notes grew out of a series of lectures given
at the Tata Institute of Fundamental Research in the period October, 1978, to March, 1979, on which notes were taken and excellently written up by C. Musili and M. Nori.
I subsequently
lectured at greater length on the contents of Chapter III at Harvard in the fall of 1979 and at a Summer School in Montreal in August, 1980, and again notes were very capably put together by E. Previato and M. Stillman, respectively.
Both the Tata
Institute and the University of Montreal publish lecture note series in which I had promised to place write-ups of my lectures there.
However, as the project grew, it became
clear that it
was better to tie all these results together, rearranging and consolidating the material, and to make them available from one place.
I am very grateful to the Tata Institute and the University
of Montreal for permission to do this, and to Birkhauser-Boston for publishing the final result. The first 2 chapters study theta functions strictly from the viewpoint of classical analysis.
In particular,
in Chapter I,
my goal was to explain in the simplest cases why the theta functions attracted attention. T) for
-I look at Riemann's theta function
z € (E, x G H = upper half plane, also known as
and its 3 variants ^ i ' ^ f 0 ' ll* to embed the torus
(C/Z+Z«T
We show how tnese can be
1?" ,
used
in complex projective 3-space, and
X
how the equations for the image curve can be found. the functional equation for
We then prove
with respect to SL(2,Z) and show
how from this the moduli space of 1-dimensional tori itself can be realized as an algebraic curve.
After this, we prove a beautiful
identity of Jacobi on the z-derivative of i?*.
The rest of the
chapter is devoted to 3 arithmetic applications of theta series: first to some famous combinatorial identities that follow from the product expansion of •'$'; second to Jacobi's formula for the number of representations of a positive integer as the sum of 4 squares; and lastly to the link between 4?" and
£
and a quick
introduction to part of Hecke's theory relating modular forms and Dirichlet series. The second chapter takes up the generalization of the geometric results of Ch. I (but not the arithmetic ones) to theta functions in several variables, i.e., to ft € .%*
ifHz,Q)
where
= Siegel's gxg upper half-space.
Again we show how <&'
can be used to embed the g-dimensional tori X„ space.
z G (Cg and
in projective
We show how, when ft is the period matrix of a compact
Riemann surface C, /L^is related to the function theory of We prove the functional equation for
C.
and Riemann's theta
formula, and sketch how the latter leads to explicit equations for
X~
as an algebraic variety and to equations for certain
modular schemes. of modular forms
Finally we show how from ^z,fl) a large class can be constructed via pluri-harmonic
polynomials P and quadratic forms Q. The third chapter will study theta functions when ft is a period matrix, i.e., Jacobian theta functions, and, in particular,
xi
hyperelliptic theta functions.
We will prove an important identity
of Fay from which most of the known special identities for Jacobian theta functions follow, e.g., the fact that they satisfy the nonlinear differential equation known as the K-P equation.
We will
study at length the special properties of hyperelliptic theta functions, using an elementary model of hyperelliptic Jacobians that goes back, in its essence, to work of Jacobi himself.
This
leads us to a characterization of hyperelliptic period matrices ft by the vanishing of some of the functions /iP-f 1(0, Q) .
One of the
goals is to understand hyperelliptic theta functions in their own right well enough so as to be able to deduce directly that functions, derived from them satisfy the Korteweg-de Vries equation and other "integrable" non-linear differential equations. The fourth chapter is concerned with the explanation of the group-representation theoretic meaning of theta functions and the algebro-geometric meaning of theta functions.
In particular, we
show how i?(z,ft) is, up to an elementary factor, a matrix coefficient of the so-called Heisenberg-Weil representation.
And we show how
the introduction of finite Heisenberg groups allows one to define theta functions for abelian varieties over arbitrary fields. The third and fourth chapters will use some algebraic geometry, but the chapters in this volume assume only a knowledge of elementary classical analysis.
There are several other
excellent books on theta functions available and one might well ask —
why another?
I wished to bring out several aspects of the
theory that I felt were nowhere totally clear:
one is the theme
Xll
that with theta functions many theories that are treated abstractly can be made very concrete and explicit, e.g., the projective embeddings, the equations for, and the moduli of complex tori.
Another is the way the Heisenberg group runs
through the theory as a unifying thread. the discussion in Ch. I when
However, except for
g = 1, we have not taken up the
arithmetic aspects of the theory:
Siegel's theory of the
representation of one quadratic form by another or the Hecke operators for general g.
Nor have we discussed any of the many
ideas that have come recently from Shimura's idea of "lifting" modular forms.
We want therefore to mention the other important
books the reader may consult: a)
J. Fay, Theta Functions on Riemann Surfaces,
is the
best book on Jacobian theta functions (Springer Lecture Notes 3 5 2 ) . b)
E. Freitag, Siegelsche Modulfunktionen, develops the general theory of Siegel modular forms Hecke operators
(introducing
and the (^-operator) and the Siegel
modular variety much further. c)
J.-I. Igusa, Theta functions, Springer-Verlag, 1972, like our Chapter IV unifies the group-representation theoretic and algebro-geometric viewpoints.
The main
result is the explicit projective embedding of the Siegel modular variety by theta constants. d)
G. Lion and M. Vergne,
The Weil representation, Maslov
index and Theta series (this series, No. 6) discuss the
xiii
algebra of the metaplectic group on the one hand, and the theory of lifting and the Weil representation on the other.
(This is the only treatment of
lifting that I have been able to understand.)
The theory of theta functions is far from a finished polished topic.
Each chapter finishes with a discussion of
some of the unsolved problems.
I hope that this book will
help to attract more interest to some of these fascinating questions.
1
Chapter I :
Introduction and motivation: theta functions in one variable
$ 1, Definition of *(z # T) and its periodicity in z, The central character in our story is the analytic function £(z, T) in 2 variables defined by *(z, T)
s
I
exp (TT in 2 T + 2TTinz)
where z c <E and TeH, the upper half plane Im T > 0. It is immediate that the series converges absolutely and uniformly on compact sets; in fact, if Im z < c and Im T >e then 2 exp(rr in T+ 2TTinz) |< (exp -TTC) hence, if n
# (exp2trc)
is chosen so that n
(exp -TTC)
o
. (exp 2TT c) < 1,
then the inequality I
2 I n(n-n ) exp(nin T+2TTinz)|< (exp-TTG) °
shows that the series converges and that too very rapidly. We may think of this series as the Fourier series for a function in z, periodic with respect to z I—> z+1, 2 *(z, T) * Z an(T) exp (2tTinz), a n (T) = exp (TT in T) neZ which displays the obvious fact that *(z+l, T) * *(z, T)
2
The peculiar form of the Fourier coefficients i s explained by examining the periodic behaviour of 0 with respect to zl
>z + T :
thus we have *(z+T,T)« Z exp(trin 2 T+2tTin(z + T)) neZ o
« E exp(TTi(n+l) T-TTiT + 2tTinz) ncZ «
Z exp(TT im T-IT iT+2TT imz - 2niz) where m = n+1 mcZ
* exp (-TTiT - 2TTiz). *(z, T) so that £ has a kind of periodic behaviour with respect to the lattice A.c
so
we may try the simplest more general possibilities: f(z+l) = f(z) and f(z +T) = exp(az+b) .f(z) . By the first, we expand f in a Fourier series f(z) = Z a n exp (2tTinz), a i C , n ncZ Writing f(z+T+l) in terms of f(z) by combining the functional equations in either order, we find that
3
f(z+T +1) = f(z +T) = exp(az+b) . f(z) and also f(z +T +1) = exp (a(z+l) +b) f(z+l) s
exp a# exp (az + b) f (z)
hence a = 2nik for some ke2Z. Now substituting the Foruier series into the second equation, we find that Z a exp (2TT inT) . exp (2iTinz) n€2Z n = f(z +T) = exp (2TTikz + b) f(z) = Z a exp (2TTi(n+ k)z) . exp b nc2Zn = Z a , exp b. exp (2ninz). ne2Z n ~ Or, equivalently, for all n c 2Z, we have (*)
a n
If k
8
= a . exp ( b - 2 n in f ) . v n-k
0, this shows immediately that a / 0 for at most one n and we
have the uninteresting possibility that f(z) » exp (2TTiz). If k f 0, we get a recursive relation for solving for a n + i r a in terms of a instance, if k • - 1 , we find easily that a
*a n o
exp (-nb +lTin(n-l)T)
for all nfZZ .
This means that f(z) * a
Z exp (-nb-TTin T) exp (TT in T + 2tTinz) nCZS
» a * ( - z - \T -b/2tTi , T).
for all p.
For
4
If k > 0 , the recursion relation (*) leads to rapidly growing coefficients a
and hence there a r e no such entire functions f(z). On the other hand,
if k < - 1 , we will find a |k | -dimensional vector space of possibilities for f(z) that will be studied in detail below. This explains the significance of 0 ( z , T) a s an entire function of z for fixed T, i . e . , £(z, T) is the most general entire function with 2 quasi-periods. S 2. 0 (x, it) as the fundamental periodic solution to the Heat equation. In a completely different vein, we may r e s t r i c t the variables z, T to the case of z = x e E
>
and T = it, t e E .
*(x, it) = Z exp (- n n nez
Then
t) exp (2rrinx)
= 1+2 Z exp ( - n n t) cos (2TT nx) . nclN Thus 0 is a real valued function of 2 real variables. It satisfies the following equations: (a) periodicity in x:
£ (x+1, it) = £(x, it)
(b) Heat equation: - £ _ ( t > ( x , it)) = 2 Z ( - n n 2 ) exp (-irn 2 t) cos (2TTnx) ^X nclN J L (*(x,it)) = 2 Z (-4TT2n2) exp (-TTn2t) cos(2nnx) ^x2 neIN Or >2 _ L ( * ( X , i t ) ) = - i ~ — 9 (*(x,it)) . *t
4TT^X2
This suggests that we characterise the theta function #(x, it) a s the unique solution to the heat equation with a certain periodic initial data when t = 0.
5
T o e x a m i n e the l i m i t i n g behaviour of # ( x , i t )
as
t
> 0 , we integrate
it a g a i n s t a t e s t p e r i o d i c function f (x) = Z a
exp(2TTimx).
Then 1 1 V*(x, it)f(x)dx = C Z a 0 ^ n,m = Z a n, m
m
exp(-TTn t ) . exp(2 TT i(n+m)x) dx
1 exp (-Tin t) 5 0
= Z a n exp n "
(-TT
ex
P ( 2 TTi(n+m) x) dx
n t)
T h e r e f o r e , we get that 1 lim $ * ( x , it) f (x) dx t—>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. ^1^2 ^l b2 i i a2 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 •£
=
(C* = { z c C / ( z | « 1 } )
7
X , a , b )c-ti c ^ stands for the transformation: where ((X,a,b)
<*> 2 = X exp (IT ia T+ 2TT iaz) f(z+a T + b). Hence the group law on -KL is given by (X,a,b)(X',a» ,b») = (XV exp (2TT iba»), B + a'.b+b')). Note that centgr of <^ = (C1 = commutator subgroup [ff,-S] and hence J(A
is a niipotent group.
The group &
and its representation as above are familiar from
4
Quantum Mechanics. Because of this connection, we will call/the Heisenberg group. In fact, the relation (*) is simply Weyl's integrated form of the Heisenberg commutation relations. Now recall that we have the classical theorem of Von Neumann and Stone which says that -fi. has a unique irreducible unitary representation in which (X , 0, 0) acts by X. (identity). In fact, this representation is the following: On our space of entire functions f(z), as in $1, put the norm 2 C 2 2 || f || = \ exp (-2TTy /ImT ) lf(x+iy)| dx dy. C Let Ji
be the subspace of all f(z) such that |] f || < oo. Then,it is trivial
to check that U. irreducible.
.
is unitary on*Jt and it can be shown that j( is
(In fact, the Hilbert spaces $t
and L (B) are canonically
isomorphic as -£* -modules where -<£ acts on L (1R) by
( u X,a,b f JM
= Xex
P <2TT i a x )
f(x+b)
2 xelR,f c L (1R)). Thus we have in hand one of the many realisations of
this canonical representation of -£* . However, for the moment, this is not needed in our development of the theory. To return to £; note that the subset T = t(lfa,b)c§|a,bcZZ } is a subgroup of Q . By the characterisation of £ in $ 1, we see that, upto scalars, £ is the unique entire function invariant under T . Suppose now that t is a positive integer; set IT * [ (1, 4a, 4b)} c r and V = [entire functions f(z) invariant under IT) . x Then, we have the following: Lemma 3 . 1 . An entire function f(z) is in V. if and only if f(z) = such that c
n
= c
m
Z c exp (TTin T+ 2TTinz) ncl/x2Zn
if n-me42Z,
In particular, dim V . = 4 *
2
Proof. For a , b c R , identify T& with ( l , a , 0) e -^ and ^
#
with
( l , 0 , b ) c - £ . If f c V j , then by in variance of f under S^c XT, it follows that E c1 exp (2TT inz). ml/|Z n 2 On the other hand; write c1 = c exp (TTin T) and express the invariance f(z)
of f(z) under T,, n e 2Z, as required.
a short computation shows that c
= c
for all
(Converse is obvious).
c
For mcIN, let u r
4 € IN , let ^
+jfc
For
4
X
• t U,a,b)/\ep
; a,be j2Z) (mod I D
= p x 2 X(i-2Z/XS)X (^ZZ/XZZ) with group law given by ( X , a , b ) ( V , a ' , b ' ) = ( XV exp ( 2 n i b a 1 ) , a+a', b+b'). Now the elements S- / , T.. . e-o. commute with XT (in view of (*)) and hence act on V - .
This g o e s down to an action of 2 , on V . ; in fact, exactly
like
(
E
2 cnexp(nin T + 2ninz)) =
/^nCl/^S a s i s e a s i l y checked.
nC
£
2 - , exp(nin T+2ninz)
c
V
l ^ ZZ
J
This g i v e s us the following:
Lemma 3 . 2 . The finite group £ Proof. Let W c v , be a
a c t s irreducibly on V^ .
~H ~ s t a D l e subspace.
Take a n o n - z e r o element
f e W , say, f(z)
L nCl/xZ
Operating by powers of Sw
2 exp(n in T+ 2 n i n z ) , c_
c
/ 0.
n
on f(z), we find in W:
Z exp(-2TTin p/X) . (S , , f)(z) P/ 0
10
Since c n
"o
f 0, we see that W contains the function £ nen
o
exp (nin T + 2TTinz) . +i2
Now working with T^ / instead, we find that W contains similar functions for every nQ c 1/^2/ 17L and hence W = V^ . In fact, we have also the finite analogue of Neumann-Stone theorem for Oft. , namely, # , has a unique irreducible representation in which (X , 0, 0) acts by X . (identity), but we do not need this at this point. For our purpose, the important point to be noted,is that, because of irreducibility, the action of £*, on V. determines a canonical basis for V^ and <^^ acts in a fixed way.
The standard basis of V^ is given by the so called theta
functions £
.
with rational characteristics a , b e l / j 2Z, defined by : for
a , b c l / x 7L,
lb=\V
= ex
P ( 2 " i a b ) T aV
Explicitly, we have: *a b ( z ' T )
=
c x P( T T i a 2 ' r + 2TTia(z+b)) *(z+af+b, T)
«
Z exp(TTi(a2+n2)T+2TTin(z+aT+b) + 2iTia (z+b)) ncZ
=
E exp (TTi(a+n)2T+ 2tti(n+a) (z+b)) . nc2Z
Now we see that we have: (0)
#o>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
in (lL2Z)2t x
0
Write *. = * . , 0 < i < * 2 - l . For l a^, D^
2 all z e
(*0(*z'T)'---
. (We shall check in a minute that there i s no z, T for which
they are all 0). Since (* Q (z+x, T ) , . . . , * *
( z + x , T)) = (* (z, T ) , . . . , * 2 u
X -1
(z, T))
X -1
and (* ( Z + J & T , T ) , . . . , * 9 2 (z,JtT,t)) = \ ( * (z, T ) , . . . , * 92 (z,T» o x -! ° i -l 2 where X = exp(-TTii T - 2TTix"z), it follows that this defines a holomorphic map x2-l a : E_—»IP
zi
>(
, *(*z,T),...).
To study this map, we first prove the following: 2 Lemma 4 . 1 . Every fe V^, f f 0, has exactly I zeros (counted with
12
multiplicities) in a fundamental domain for
a r e the points (a+ p + i ) T + 0>+q + i ) , P, q « -
The z e r o s of * a
b
On particular. * . , *. (i ft)
have no common z e r o s and s o q>^ i s well defined). Proof. The first part i s by the standard way of counting z e r o s by cantour integration: choose a parallelogram a s shown m i s s i n g the z e r o s of
f:
Fig. 1 R e c a l l that we have
# z e r o s of f
2ni
S
• dz
a + 6+a*+6*
Since f ( z + i ) = f(z) and f (z+XT ) = const, exp (-2TTiXz)f(z), we get that
^ +| 6
= 0 and j + j
6*
a
= 2niX
.
o*
A s for the second part; note that * (z, T) i s even in z and it has a single z e r o in
On the other hand, we have: 2
*i i2 ( - z ' ^'
T) =
Z
e x p
neZ£ =
=
(TTi(n+*)
T + 2TTi
(n+i^"z + *)
L exp(TTi(-m-i)2T+2TTi(-m-|)(-z+|))
ifm«-l-n
E exp (TTi(m+i) 2 T+ 2 n i ( m + i ) ( z + i ) - 2 T r t ( m + | ) >
mtTL 2# 2
13
and hence & , i s zero at z = 0. It follows that £ . has the zeros lA a,b 2 stated and since this sum gives I of them mod 4 A T , there cannot be any more. Next, observe that the group -S, modulo its centre, i . e . , (l/j2Z/jt2Z) naturally acts on both E
and IP* "* and the map qp# is equivariant.
To
see this; let a , b e j 2 , then it acts on E T by z |—> z + ( a T + b ) / i (and this action is free). On the other hand, if
(1 a b)
' '
,2 i
the action on IP*
(z , . . . , z
i
o<j<x2-i
iJ
J
is given by o
J*
>(Ic ,z.,M,,Ic
9
z.) #
Now we see that cp#(z+(aT + b ) / Z ) = ( *
, f ( A z + aT + b, T ) , . . . ) l
(l,a,b)
i
= ( . . . . , I c . . * . ( i z , T) , . . . ) and so cp^ is equivariant. It is interesting to note that, although we have a commutative group 2
2
(1/jZS/iZZ) , i . e . , (TLjlZL)
2
#
, acting on IP
1
" , by Lemma 3 . 2 , the
action is irreducible, i . e . , there are no proper invariant sub space si We now prove that cp * is an embedding: suppose, if possible, cp (z ) = cp f (z 2 ), z f z 2 in E T , or that dcp^z ) = 0 (the limiting case when z 2 —> z )
#
Translating by some (a T+ b ) / i , a , b e l/^ZS, we find
14
a second pair z^ , z^ such that cpx(z») = q> x (z'), or d q ^ z * ) = 0. Take 2 i -3 further points w ^ w ^ . ^ w 2 , , all points so far being distinct mod ^ A T .
Seek an f e V j , f / 0 , sucn that f(z ) =f(z«) - f ( w ) » . . . = f ( w o J = 0. *
i.
1
x
-u
.2
This is possible because, writing f = Z X.* , X .€ (C , we get X -1 linear 2 equations in the X variables X . . . , \ 2 and so they have a non-zero w 2
solution. Since fytej) • fy( 2)*
X
~1
iX f o l l o w s t h a t
f z
( 2* * °- ° r '
if dc
P/zi'
8
°.
f has a double zero at z^. Similarly, we get that f(z') - 0 or f has a c
2 double zero at z». Therefore, f has at least i +1 zeros in
i2-l
isomorphic
. Invoking a theorem of Chow, we can say that it is even
an algebraic subvariety, i . e . , «p^(ET) i s defined by certain homogeneous polynomials.
Since everything is so explicit, we can even determine this
directly.
For simplicity, we will limit ourselves to the case 1-2 and show 3 in the next section that tp (E,) is the subvariety of IP defined by 2 quadratic equations. $ 5. Riemann^s theta relations. Riemann's theta relation is a very basic quartic identity satisfied by
£ ( z , T). A whole series of such identities exist, based on any n Xn integral t 2 matrix A such that \AA = m I . Riemann's identity is based on the choice
15
1 A
I
1
1
1 1
1 -1
"I ,
A = I
I , m = 2 and n « 4 1 - 1 1 -1
The matrix identity
1 - 1 -1 1^
AA M L i s equivalent to the identity between
quadratic forms: « 4 (x 2 +y 2 +u 2 +v 2 ).
(x+y+u+v) + (x+y - u - v) + (x -y+u -v) + (x -y -u+v)
Fix a T and write *(z) - *(z, T) and A = A T , etc. Now for all choices of r\ c \ A / A , we form the products *(x + r\) £ (y + r\) *(v + r\) and sum up, putting in simple exponential factors to make the functions look like: B(0):#(x)*(y)*(u)*(v) -
Z exp [ TTi( I n 2 ) T+ 2 t r i ( I x n ) ] n,m,p(qc2
r* 2 2 2 2 2 r» where I n = n + m + p +q and L xr\ = xn + ym + up + vq
B(i) : *(x+i) *(y+i) *(u + *) *(v + i ) 2 I exp[Tri( I n + 1 n ) T + 2TTi ( Z x n ) ] n,m, p, q c2Z B ( £ T ) : exp[TTi(T+Z;x)]^(x+|T)^(y+i
T ) ^ M { T ) * ( V + } T )
S e x p [ , T r i ( I ( n + | ) 2 T ) + 27Ti(i;x(n+l))] n, m, p, q€2Z B(i(l+T)) : exp[TTi(T+Ex)]^(x + | + y ) ^ ( y + i + y ) ^ ( u H
+
y)^(v+i + y )
I e x p [ n i ( I n ) +TTi( I ( n + | ) 2 f ) + 2tti ( £ x ( n + i ) ) 3 . n, m, p, q zTL Calling the exponential factor e
and summing up, we get :
16
e *(x+ *!> ^(y + ^)^(u + n) *(v + n)
Z
*1 = 0 , l , i T , i ( l + T ) s
2
exp[TTi(E n 2 ) T + 2TTi(Exn)]
I
n, m, p, q all in Z£ or all in \+7L
and n+m+p+q e 2 7L. For simplicity,
let us write a
n x = | (n+m+p+q) ,
Xj
m x * | (n+m-p-q)
yj = \ (x + y-u-v)
,
i ( x + y+y+u+v)
Px = \ (n-m+p-q) ,
u
i
s
^ (x-y+u-v)
q x = | ( n - m-p+q) ,
v
l
=
2 (x-y-u+v) .
Note that the peculiar restrictions on the parameters n, m, p, q of the summation above exactly mean that the resulting n , m , p
and q
are
integers. Also observe that we have the identities: = E n.
Zn
and L xn = 1
x n
i i •
Now substituting these in the above equation, we get: I e *(x+Ti)*(y+rO*(u+Ti)*(v + TO tl=0,i,iT,i(l+T)
*\ =2
Z
exp[ TTi(I n j T + 2TTi(I x ^ ) ] .
n1m1,p1,q1«Z Thus we have the final (Riemann's) formula (using B(0)): (RJ :
Z
e^^(x+Tl)^(y+Ti)^(u+Ti)*(v+Ti) = 2 ^ W y J ^ u ^ ^ v J . t
If we had started with another matrix A such that
2 AA = m I n , we
would have found an identity of order n, involving summation over
17
translates of £ with r e s p e c t to all the division points r\ of order m, i. e . , T)€ ^ A/A .
To u s e the identity (Rj), it i s natural to reformulate it with
theta functions £ , a,b
with c h a r a c t e r i s t i c s a , b « ? 2 ; there a r e 4 of t h e s e ,
namely: *
(z, T ) =
*
I(Z,T)
2 £ exp (rrin T + 2 n i n z ) = £(z, T) nc2Z
= Z exp(TTin 2 T+ 2 n i n ( z + | ) = * ( z + | , T )
O* 2
2 ( z , T ) = Z e x p ( n i (n+i) T+ 2 n i ( n + £ ) z) = exp(rri T/4 + TTiz)*(z+| T, T)
*!
i(z,T) =Zexp(TTi(n+i) 2 T + 2TTi(n+|)(z + i)) = expfai T/4+TTi(z+J))*(z+l(l+T), T)
f. 2
# 2
F o r simplicity, we write these a s £
**oi' *10
and
*11
#
** i s
immediatelv
verified that *
oo
(-z,
T)
V-z-T)
s
=
*
oo
(z, T)
%i ( z - T )
*io(-z'T)=Vz'T)
showing that * *11^°' T^
=
° '
i s different from the others, and confirming the fact that
wnile tne
other 3 are not z e r o at z - 0 (cf. Lemma 4. 1).
Riemann's formula g i v e s us:
:
*oo< x ) *oo ( y>*o° ( u ) *oo< v >
+
* U | W * 0 l W *01 * O l W
^10w*10(y)*10
Now
18
where x ^ \ (x+y+u+v), y ^ i(x+y-u-v), etc. Now replacing x by x+1 and using the fact that his changes the sign of *-
(R
3>
:
*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 V
u )
V
v )
- V x ) V ^ V u ) *io(v) + V x ) '„W V u ) V v ) In other words, we get all 4 theta functions on the right hand side by putting a character into the sum on the left hand side. More variants can be obtained by similar small substitutions: v i z . , replacing x , y , u and v by x+ a , y + p,u+Y , v +6 where a , p, Y , & e | A and a + 0 + Y + $ e A . listed below in an abbreviated form all the r e s u l t s .
We have
19
First we make a table containing the fundamental transformation relations between the d' 's that are needed for a quick verification of Riemann's theta formulae. Table 0 (zi
>-z)
(z i
*OO(-Z'T>=*OO(Z'T)
* o o
( z +
>z+i)
-
V"Z'T) = V Z ' T )
VZ+*'T)
^o(-Z,T)=^10(z,
*10(Z+i'
T)
*ll(z+*'
* n ( - » . 1) • - * n ( z , T)
(zi *oo ( z + * T ' T )
= (exp (
T ) =
= T ) =
T) =
> » + £T)
" TTi T / 4 " T T i z ) ) *10
T)
^ Q 1 (z+iT, T) = -i(
"
) * n ( z , T)
^ 1 0 (z+}T, T) «
H
) ^ 0 ( Z . T)
(
^ n ( z + i T , T) = - i (
"
W
\a i
i i • 2/
T) = -i (exp(--TTiT/4- -TTiz))
oo
V " V " *n<
*'
) ^Q1 (Z, T)
"
) =
(
ti
) = -i(
it
) =- (
II
^ 2 1 ( Z , T)
>*10 ( 2 ' T ) ) * 0 1 ( Z . T)
) »OO (z. T)
V
Z
'
T )
*OO (Z ' T) *11(Z'T) "*10 (Z » T)
20
RIEMANN'S THETA FORMULAE I.(RJ:
£
e „*(x+Tl)*( y +Ti)*(u + TO*(v+Ti) = 2 * (x. WyJJfri, W v J
where e - = 1 for r\ - 0, \ and e = exp(ni T+ni(x+y+u+v)) for r\ = | (1+?), and Xj = i(x+y+u+v), y x = i(x+y-u-v), Uj = |(x-y+u-v) and v^- |(x-y-u+v). II. Via Half-integer thetas: x — 2 x r» 2 £ =*(x, T) =Iexp(TTin T+2ninx), * = Lexp (rrin T+ 2nin (x+£)), *XQ= Zexp(TTi(n+i)2 T +2TTi(n+i)*)) and ^ = Zexp(Tri(n+i)2T+2TTi(n+i)(x+i))
+
(H6): C d «
• • - • •
+
-
"
VltflC&^lXo'K ^Xl^O^O 1 ! 1 tf C C
+
.
-
.
+ "
-
-
"
- "
+
"
-
-
"
. »
"
"
"
+ » -
.
--
+
-
"
-
.
»
"
.
00*00 01 0 1 x iAyiAuuvi
9,%
--2*n *n *io*io
-
+
= 2*X1*yi*U1*V1
"
. . .
:
-'•S #*>
•2*£«W£1
-
"
+
"
-
"
=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
=
v. = 0):
* o o ( x ) \ o ( u ) 2 + * l l < A l < " > 2 • *0lW2 %
+
*10(3Ao
^o(x+u)'oo(x-u)*oo(0)2-
Likewise, (R 3 ) + (R_) and ( R J + (R-) (with x = y , u = v) r e s p e c t i v e l y give:
(A 2 ) : ^ ( x + u j ^ t x - u ) V 0 )
2
' *00W\o2 " Vx)2*10(u)2 = *01(x)2*ol(u)2-tfll(x)2*n(u)2
and (A 3 ) : * 1 0 ( r t . ) * 0 ( x - u ) * 1 ( ) ( 0 ) 2 - * 0 0 < x ) 2 * o o ( u ) 2 - » 0 l ( X ) 2 * 0 i < y ) 2 = *10(x)2*io(u)2-*ii(x)2*n(u)2These are trypical of the "addition formulae 1 1 for theta functions for calculating the coordinates of © (x+u), cp (x-u) in t e r m s of those of q> (x), cp2(u) and q>2(0). There are 12 m o r e e x p r e s s i o n s for the products C' (x+u)£ c d (x-u) in t e r m s of * f ( x ) ' s , * f ( u ) ' s , for ab f cd, which we have written down on the next page. All are obtained from the formulae (R ) by just setting the variables equal in p a i r s .
22
III. Addition Formulae (Ax) r f j x + u ) ^ ( x - u ) * > )
= • i « ) # i ( « ) + # 1 2 1 (x)#J 1 («)-^x)^ 1 («) + #^x)#J«)
*oi(x+u> v x - u > *„>
• * i ^ > ) - * > ) vu)-#«J(x)^i(',,-*3x,*S(',)
V**
0
*10< x - u) *10 (0)
^ o o W * o o < » > - ^ x ) * 0 ^ ) = *10' x >^ u ) -»J(*)*i(«)
* o o ( x + o ) ' o i ( x i ) * o o « » * o i ( 0 ) " *oo(x)*01(x)*oo(u)*01 -*oi ( x ) *U ( x ) *01 ( u ) *U < u ) *01 (x+u) *oo< x - u >*oo (0) *01 (0) ' *oo(x)*01<x>*00<">*0lW +* 0 1 (x)* n (x)* 0 1 (u)# u (u) *oo (x+u) *10< x - u >*oo (0 %< 0 > - *ooW*io ( x ) *oo ( , a ) *l(/ u ) 'lO^'oo^'oo^'lO^
+
Vx)*ll(x)Vu)*ll(u)
= d oo( x )*10< x )*oo< u '*10< u )-*01 (x) *ll (x) *01 (u) *U (u )
* 01 (x+u)* 10 (x-u)# 01 (0)* 10 (0) = * oo (x)* 11 (x)* oo (u)* n (u) + # 01 (x)tf 10 (x)* 01 (u)# 10 (u) *10 ( x + u ) *01< x - u )*01 ( 0 ) d l(/ 0 )
•-•oo< x ) *U ( x ) *oo ( ' , >*ll<'* + *01< x > # 10 ( ^l ( u , *10 ( , , >
" * 2 i (x) *oo (u) -*oo(x)*n(u> 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<0) *10 (0)
-*ooM*llW*0lW*10< u > - *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<«'*ilW
+
Vx)*ll(x)*<>>*10(u)
* 0 1 (x+u)^ j (x -u)* o o (0)^ 0 (0) = - * o o M * l 0 W*01 (u)# n (u) + * 0 1 W*! j (x)*oo(u)*10(u) *U (X+U >*10< x - U) *oo< 0 ) d 0l(°) =*oo( x >*01 (x) *10<»)*ll( u ) + * 1 0 ( x ) * l l W * o o ( u ) i , 0 1 ( u )
' u ^ i M i y o ^ W
=-*0o<x>,,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 *oo<°>2 - * 0 l W 2 * 0 l ( ° ) 2
<E2) :
*
n
+
*io«2*10<°>2
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<0)2*o = V 0 ) 2 ^
V°)2x2
* ( 0 ) 2 4=v°> 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
to check that
"i—r *(z-at) l < i < k *(z-b.) is periodic for A-, hence is a meromorphic function with zeros at a.+ \ (1+T) and poles at b.+|(l+T).
(If we use $. - instead of * , we get
zeros at a. and poles at b-). In fact, all meromorphic functions arise like this and this expression is just like the prime factorisation of meromorphic functions on IP :
f(2) = 7J( ! iVVo) lzi*z at z = d./c..
homogeneous coordinates, zeros at z = b^/a^ and poles
25
Method III. Second logarithmic derivatives: Note that log & (z) is periodic upto
addition of a linear function.
Thus the (doubly) periodic function
^ 5 - log *(*) dz 2 is meromorphic.
This is essentially Weierestrass » (p -function.
To be
precise, $>(z) = - - £ - log * (z) + (constant), ll dz 2 the constant being adjusted so that the Laurent expansion of {p(z) at z = 0 has no constant term. Method IV: Sums of first logarithmic derivatives: Choose a . , . , , , a, eC and X , . . . , X eC such that £ X. = 0 .
Then one checks that
Z X. — log # ( z - a . ) + (constant) i
idz
l
is periodic for A T , hence is meromorphic with simple poles at a . + | ( l + T ) , residues X .. Again, this gives all meromorphic functions with simple poles and is the analogue of the partial fraction expansion for meromorphic functions on IP : f(z) = E i
^- + (constant) (z - a^
We give a few of the relations between these functions: for example, to relate Methods I and II, we need merely expand each & h ( 2 z )
as a
product
1
of 4 functions £ ( z - a . ) , times an exponential factor (the a. s being the zeros of ^ 1 _(2z)). For instance, £ _ ( 2 z ) is 0 at the 4 (2-torsion) points ab
0 , 2 <> l
T
» 2(
ll
1+T
)»
and
its factorisation:
26
*„<*«> *oo<°> V 0 ) * 1 0 ( 0 ) " 2*oo<*> V * > * 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: <*11<'>*1>> - * > ) 2 ) * o o ( 0 ) 2 - V 2 ) 2 * o o ( 0 ) O 0 )
-#oo(Al(0,a
(using the fact that ^ ( 0 ) = 0 and *
(0) = 0 since * Q 1 is an even
function and *
2 2 Dividing both sides by * o o (0) ^ ( z ) ,
is an odd function).
we find that the resulting equation is simply
d*2
"
* oo (0)
,
{0)2
hence a J8 (z)
(x)2
# X1
oo
2 *i<°> * (x) 2 = (constant) + — " 5 - . — Q f l _2_ . *nn(0f * (z) 00 11
One of the most important facts about the ^ -function is the differential equation that it satisfies.
In fact, using the obvious facts (from
the above equation) that (i)^(z) = $> (-z), (ii) the expansion of <£(z) at z = 0 begins with z -2 and (iii) the constant rigged so that this expansion has no constant term, it follows that: ^ (z) = J - + a z 2 z2
4 + bz
+# #m ^ n e a r
Therefore, 3 fr «(z) = - - 2 + 2az + 4bz + Z3
z
= o#
27
and hence
$'(z))2=4--^-16b+... z6
z*
z6
z2
But 0
so that (^'(z) 2 -4^(z) 3 + 20 a ^(z) = -28b + . . . Thus the function j§* '(z)
- 4^(z)
+ 20a^(z) is a doubly periodic entire
function and hence is a constant. This means we have an identity: If' (z) 2 = 4 ^ ( z ) 3 + g 2 (T)^(z) + g 3 (T) which is Weierstrass' differential equation for (J*(z).
Differentiating
this twice, we also get the differential equations: $>"(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( ^ajL±L, T)
2 2 exp[TTic(cT+d)y + 2TTicy(a T+b) +TTic 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 +
± i i * 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
I exp(-TTin2d/c) £ exp trie y 2 (T+d/c) dy . l
To evaluate the integral; first suppose that T = i t - d / c . 00
Then we get
00 2
2
J expnic y (T+d/c) dy = f exp(-TTc 2 y 2 t) dy -00
-OO
i i. e . , if u = c t* y,
- —T
1 ^ 2 y exp (-TT u ) du
7J. oo
-1/ct*.
9 using the well-known fact that
2
\ exp (-TTU ) du = 1. It then follows by -00
analytic continuation that for any T with Im T> 0, we have C 2 2 V exp TT ic y (T+d/c)dy J
-00
where (
s
1 j c[(T+d/c)/i]*
) 2 is chosen such that Re( ) 2 > 0. The sum is a well-known
"Gauss sum" S,
= '
which in fact is just c
£
<\ exp (-TTi n d/c)
l£n
times some 8
root of 1. This may be proved either
directly (but not too easily) from number theory, or it can be deduced, by induction on c+ )d| , from the compatibilities of the functional equations. In fact, the exact functional equation is given in the following:
32
Theorem 7 . 1 . Given a , b , c , d e 2 Z
such that ad-be
s
1, ab and cd even, we have
then for a suitable C,an 8 t h root of 1, that (F
:
1> x
* < - T Z T > ^ ) * C ( c T + d ) * e x p ( n i c z 2 X c T + d ) ^ ( z , T) . C T +d
C 1-rQ
To fix C exactly, we consider two c a s e s : first a s s u m e
c > 0 or c
a b d > 0 (multiplying ( ) by -1 if n e c e s s a r y ) , hence I m ( c T + d ) > 0 c a choose (c T+d) 2 in the first quadrant (Re ( ) > 0 and Im ( ) > 0):
s
0 and and
(a) if c i s even and d i s odd, then
C
where
x (-)
"X
(
|dl}
0 i s the Jacobi symbol (to take care of all c a s e s , we s e t (—) - +1),
(b) if c i s odd and d i s even, then C =exp(-TTic/4) (4). Proof. We have only to s e e (a) and (b); we first check it for two special c a s e s (i) and (ii): /•\
,a
b,
,1
b»
d)
(c
d)
- („
x)
,
, b even.
Now (F ) + (a) says the obvious identity, namely, (F2):
* ( z , T + b ) = *(z. T )
(ii)
(c
d
)-(!
o}-
Now ( F j ) + (b) reads a s (F3):
* ( Z / T , - 1 / T ) = exp(-TTi/4). f i . exp(TTiz 2 /T )t> (z, T)
which we have already proved since it i s trivial that S Q
l
= 1.
We get the
general c a s e by induction on | c | + |d\ : if \d\ > | c | , we substitute
T* 2 for
33
T in (F 1 ) and use (F ) to show that (F^) for a , b , c , d
follows from ( F ^
for a , b t 2 a , c , d t 2c. Since we can make jd t 2c| < jd| , we are done. Note that Id t 2c| f l d | or \c\ because (c,d) = 1 and cd is even. On the other hand, if
|dj < \c\ , we substitute - 1 / T for T in (F-) and use (Fg)
to show that (F.) for a , b , c , d follows from (F..) for b, -a,d, -c: this reduces us to the case |dj > | c | again. The details are lengthy (and hence omitted)but straight forward (the usual properties of the Jacobi s y m b o l , e . g . , reciprocity, must be used). It i s , however, a priori clear that the method must give a function equation of type (F ) for some 8 t n root C of 1. § 8. The Heat equation again. The transformation formula for & (z, T) allows us to see very explicitly what happens to the real valued function £ ( x , i t ) , studied in § 2, when t—>0.
In fact, ( F J says: l
o
* ( x / i t , i / t ) = t 2 exp (nx*/t) *(x,it) hence *(x,it) * t"2exp(-TTx2/t) S exp(-TTn2/t-K2TTnx/t)) ntTL = t"T Z exp (-TT(x-n)2/t). neZ In completely elementary terms, this is the rather striking identity: 1 + 2 S cos (2TTnx) exp(-TTn2t) = f2 rexp(-TT(x-m) 2 /t). nelN meZ i
o
But t 2 exp (-TTX /t) is the well-known fundamental solution to the Heat equation on the line, with initial data at|t = 0 being a delta function at x = 0. Thus £(x, it) is just the superposition of infinitely many such solutions, with initial data being delta functions at integer values x = n. In particular, this
34
shows that *(x,it) is positive and goes to 0 as t—>0 uniformly when 1 -x > x > c . $ 9. The concept of modniar forms. Let us stand back from our calculation now and consider what we have got so far. In the first place, the substitutions in the variables z, T for which £ is quasi-periodic form a group: in fact, SL(2, 2Z) acts on C x H b y (z, T) I—» (z/cT+d, (aT+b)/(cT+d)) because z/(cT+d) a'((aT+b)/(cT+d))+b' » V((aT+b)/(cT+d))+d»' C '((aT*b)/(cT+d))+d' ' . / I (a'a+b'c)T+(a'b+b'd) v " '(c'a+d'cjT+fc'b+dd1)' ( C 'a+d f c) T+(c'b+d'd) Moreover, this action normalises the lattice action on z, i . e . , we have an action of a semi-direct product SL(2, ZZ)X ZZ2 a b on ((z+mT+n)/(cT+d),(aT+b)/(cT+d)). Actually; not all of these carry * to itself; we put on the side condition ab, cd even. To understand this condition group theoretically; note that we have a natural homomorphism Y N : SL(2, 7L)
> SL(2, 2Z/N2Z)
35
for e v e r y N.
Its kernel I I - , the s o called "level N-principal congruence
subgroup" i s given by F
N '
£(
c* d
) e S L ( 2
'
2 )
/b'
c
" ° ( m o d N ) , a,d = l ( m o d N ) ] .
Before we study the level 2 c a s e explicitly, let us r e c a l l that the group SL(2, 2Z/22Z) of «ix m a t r i c e s (1
0} (0 1
1 1
(1
1}
1
0)
(0
1}
i s isomorphic to the group of permutation on 3 letters ( 1 , 2 , 3): (1)(2)(3), (12)(3), (23)(1), (123), (13)(2), (132). We define, following Igusa, T
CSL(2, 2Z) to be v~ l
SL(2, ZZ/22Z) consisting of (* J ) and (J ). SL(2,7L) of e l e m e n t s that whereas T
of the subgroup of
2
1,«
Clearly this i s the subset of
( a °) such that ab and cd a r e even. c u
i s a normal subgroup of SL(2,2Z), I*
vi
Note however
i s not; it has
1 , it
2 conjugates: v ' V
°)
C1 *)) and v - 1 ^ 1
°)
I1 ° ) )
described by the conditions c even and b even respectively. groups for which 0
and A
have functional equations.
They a r e the
If we write out
* ( z / ( c T + d ) , (aT+b)/(cT+d)) when ( a
5 ) ^ I*
, we find that it i s an elementary factor t i m e s ^ ( z , T )
or * 1 0 ( z , T ). The s i m p l e s t way to s e e this i s not to try to describe how an
arbitrary
*a b r d)
in SL(2, 2Z) transforms the A.'s - which leads to
interminable problems of sign - but rather to consider the action of 2 generators
36
(* * ) a n d (** "*) of S L ( 2 , Z ) .
Their action is summarised in the
following table: Table V # (z/f, -1/T) = (-iT)*exp(TTiz2/T) * (z, T) *-<»/*'
*oo=Vz'T) 01 #1Q(
oo •"
V '
'
)
=exp(Tri/4)#o(")
*n( " )
=exp(ni/4)* (")
) -
*10< " >
V ">
*10<
)
*11<
From this, the action of any C
- -(
,) can be described.
(The formulae on the
c d
left are verified directly by substitution in the Fourier expansion. The 1st formula on the right is (F^). The 2nd comes for instance by substituting z + | T for z in the 1st and using the functional equation of 0
in z; the
3rd comes from substitutions z/T for z, - T~ for T in the 2nd ; and the 4th from substituting z+^T for z in the 3rd). Geometrically, the reason the funny subgroup Tl
2
arises is that
0(z, T) is 0 at the special point of order 2, namely, \ (T+l) € i / L / A - , and a b it is easy to check that ( J cF. i(T+l)to |(T'+l)mod
2
if and only if z |—> z/(c T+d) carries
A r where f» = (a T+b)/(c T+d).
However, we shall focus our attention in this section on the behaviour of the functions 0..(O, T) of one variable T. Note then that the functional equation of 0(0, T) reduces to: l
*(0,(aT+b)/(cT+d)) = C(cT+d) 2 *(0,f) 8 where C
2 s
1, C as given in Theorem 7 . 1 . This will show that 0(0, T)
is a modular form in T in the following sense:
37
Definition 9 . 1 .
L e t k e 2Z
&NflN.
B y a m o d u l a r f o r m of w e i g h t k &
l e v e l N , we m e a n a h o l o m o r p h i c function
f(T) on t h e u p p e r h a l f - p l a n e H
such that (a)
for all
(ac
T«H&
J}) c T^
,
f ( ( a T + b ) / ( c T + d ) ) = (cT+d) k f(T) (b) f i s b o u n d e d a s f o l l o w s : (i) 3 c o n s t a n t s
c&d
such that | f ( T ) | < c
(ii) V p / q € Q , 3 p o s i t i v e r e a l s
c
if I m T> d
&d
|f(T)|
and
such that
if|T-p/q-idp#q|
T h e s e t of m o d u l a r f o r m s of w e i g h t k & l e v e l N i s a v e c t o r s p a c e and i s (N) d e n o t e d by Mod
(N) .
T h u s a n y f € Mod
i s bounded outside an horizontal
s t r i p ; a n d t h e c i r c l e s of r a d i i d „ c e n t r e d a t p / q + id ( i . e . , touching F ' p, q P, q t h e r e a l a x i s a t t h e r a t i o n a l p o i n t s p / q ) a r e c a l l e d t h e h o r o c i r c l e s for f.
See Fig. 2
Note t h a t S L ( 2 , 7L) a c t s on t h e " r a t i o n a l b o u n d a r y p o i n t s " QUJoo} of H and t h a t if
f ( ( a T + b ) / ( c T + d ) ) = ( c T + d)
f( T)
then t h e bound a t p / q c Q U [ o o ] i s e q u i v a l e n t t o t h e bound a t
+b_ / / \+ d
(the bound a t co b e i n g the c o n d i t i o n (b) (i) : I f(T ) ( < c if I m T > 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 1P*
Y2: H/r defined by
VT)=(*o2o(0'T)'*02l(0'T)'*l20(0-T))As in § 4, the point is that in T (y T), each function picks up the same factor z e (T) and SO f is well-defined. Moreover, Y 0 is equivariant for the finite y 2 « group S L ( 2 , S ) / r
which acts on H/r because r is normal in SL(2, 2Z), 4 4 2 2 and on IP because the 3 functions £..(0, T) are mapped into combinations 4
of themselves by every y cSL(2, 7L). In fact, using the action tabulated in §9, we find: if T2 (T) = (x_, o x 1i # x 0z) , then *2(T+1) = (x 1 ,x Q ,ix 2 ) and Y (~nf) = ( x ^ x ^ x ^ Moreover, by equation ( J ) in §5, the image of Xy lies on the conic
A : x 2 = x? + x 2 o
1
2
but missing the 6 points (1,0, ± 1), (1, + 1, 0), ( 0 , 1 , * i) where the conic meets the coordinate axes x. = 0. The missing points are clearly accounted for by
45
the cusps: in fact, if Im T *ii ( 0 ' T *'
tt i S C l e a r
*
that
(0,T)
>+1, *
oo hence Y (T)
>+ co, then by the Fourier expansions of
(0, T)
>+l and *
01
(0,*)
>0
10
> ( 1 , 1, 0). Acting by SL(2,2Z), the other cusps will map
onto the other missing points. The easiest way to "extend ^
*° * n e cusps"
is this: (a) define explicitly by "scissors & glue" a compactification H/r of 4
the orbit space H/r ; H/ r is an abstract Riemann surface, and then (b) verify the T0 extends to a holomorphic map on all of H/r . For (a), it i s useful to recall the classical fundamental domain for the action of SL(2, 2Z) on H: v i z . , the set F c H defined by F = (T«H/|T|£i (cf. diagram below).
and (Re T | ^ | }
46
47
Let D
=
oo
{TcH/ImT>1^
Since D
ooCU(F+b)= U vF, bc2Z V-(J5>
it follows that : V T J . T c D ^ , if T- ^T T
i
=
V
Thus mapping D
b
'Y
s
^o i J
for some Y cSL(2,2Z), then
a n d Y(OO) =
°° •
to H / l \ identifies only the pairs of points
T
and
T + 4k, k c Z . Equivalently, if w = exp (ItTiT), then H/r =>the punctured disc = {w/0<|w|< exp(-|Tr) } *
D
J
l i
5)/baO(mod4)}.
l
What we want to do i s to glue together H / I \ and the full disc | w| < exp(-£TT), identifying these two on the punctured disc. We can do the same thing at the other cusps. For all p/q e Q , let D p /q be the horocircle: D
Then it i s m
p/q
s
tTCH/|T-p/q-i/2q2| < l / 2 q 2 } .
easy to check that whenever yfoo) = p/q, Y « S L ( 2 , s > .
Y(DJ =D i Thus, if f # T 2 c D p / q conjugate to (*
b
and
T
i * 6 T2
for some
8
« S L ( 2 , Z ) , then 6 i s
), or what i s the same, /1-bpq
bp \
\ -bq 2
1+bpq/
for some b c 2Z Let w
•H
><£ be the function defined by
48
W
p,q(T)
=ex
P ( " T r i / 2 q (qT-p)).
It is easy to see that : V T , T C D / . 1 2 P/q w
r,P#q n( Tli) = w ~p, ^q( T*9 ) ^ Z»= ^7 Tl = 6 T 0* for some 6 as above.
Thus, as before, the image of D ,
in H / I \ identifies only the pairs T
and 6 T for 6 as above with b = 0 (mod 4). In other words, we have: H/r => the punctured disc = {w
/0<|w
|<exp(-jTT) }
2
= Horocircle D , / {(* " b P q P/q -bq* Again, we glue the full disc j w disc.
Clearly, if y(p/q)
=
b
P )/b = 0 (mod 4) ] . i+bpq
I < exp (-£TT ) to H/I*4 along the punctured
P'/q 1 » Y € ^A » then the above operation at p/q or
p'/q' has the same effect on H / I \ . So we need only do it once for each orbit of I \ acting on QU [ oo] • It i s not hard to check that I \ has 6 orbits on QU (oo] , namely! (i)
T4(OO) = l o o } U { p / 4 q | p odd]
(ii) r (0) = ( 4 p / q | q odd] 4
f
(Ui)r4(i)
=Cip/q|p,qodd]
(iv) T 4 (l)
= [ p/q | p,q odd and p = q (mod 4) }
(v) T4(2)
=£2p/q|p,qodd}
(vi) T (3) = £ p/q| p,q odd and p = -q (mod 4 ) ] = TW-1) So we define H/r to be H/T\ with 6 "cusps" adjoined by the above procedure, 4 4 one for each of the above orbits. It is a priori not at all obvious that H/r
49
i s a compact Hausdorff space'
Perhaps, the e a s i e s t tray to s e e this i s to
d e s c r i b e it alternatively by a fundamental domain: for 1 < i £ 6, let
Y. € SL(2, 7L) c a r r y eo to the 6 cusps oo, 0, { , 1, 2 & 3 . F o r instance, we take Y. to be : /l Ox /0 - 1 . ,1 0 1M1 -2M2
(
Let b " ( G ^ fir4
-2v /2 -3 , . ,3 ( -l*'*! -l* ^ 1
-4. .I*"
* ), 1 < j < 4 , be c o s e t representatives in G
where G ^
24 e l e m e n t s v. t H/T
- 3 , ,1 -5K(1
i s the s t a b i l i s e r {(*
b
for the subgroup
) } of oo in SL(2,7L) m
are c o s e t representatives for S L ( 2 , 2 Z ) / I \ .
Then the
Therefore,
i s just the non-Euclidean polygon f l £ i £ 6
with its edges identified in pairs (cf.
F i g # 4 * ' ) # Clearly, the c l o s u r e of
this polygon m e e t s the boundary R U [ o o } at the 6 cusps co, 0, | , 1,2 & 3, and we have added these limit points to H / r
to obtain H / T . Thus H 7 I \ i s a
compact Hausdorff s p a c e , It i s now e a s y to extend Y
****** to Y9 : H/I\
2 >IP : 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
exp(TTin 2 T+TTin) = 1 + 2 I ( - l ) n w 2 n *n1(0,T) = I U1 ne2Z ncK w * l n ( 0 , T ) = £ exp(TTi(ri+i) 2 f) = 2 w* I 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, 4L(«?Awe (o.have: t)/*2(o,f)) 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
T+
1) =f(z,
T)
d) f ( z / t , - 1 / T ) = (-iT)*exp(TTiz 2 /T) . f(z, T) and for all z c
lim Im T
f(z,
T)
= 1.
>+oo
Proof. We have used all these properties of £ (z, T) repeatedly, except perhaps (c) which follows from the identity:
54
*(z+i, T+D = Z exp [TTin2(T +1) + 2TTin (z+|)] ntTL 2 - E ( - D n + n e x p (TTin2T+ 2TTinz) ncZ = * ( z , T)» Conversely, to see that these properities characterise £ (z, T), take any such f: by the results of § 1, (a) and (b) imply that f(z, T) = g ( T ) * ( z , T ) for some holomorphic function g(«r) on H, Now by the results of $ 7, (c) and (d) imply that g(T+D = g(f) & g ( - T _ 1 ) = g(T). Thus g(«r) is a holomorphic function on H/SL(2,2Z). (e) implies that g(«r)
>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
into the
defined by
„ 4 4 4 4 F : x^ + x* = x + x o . o 3 1 2 Proof.
We give the proof in 6 s t e p s : *
1
2
(1). That T i s a normal subgroup of ( j 2Z) K SL(2, 7L) i s a straight> 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 =
• «CX ( f } )
where C
is the elliptic curve defined by the equations: \ V o \
= a X
l l
2
+ a
2x2
2
2
K x 3 " V l - alX2 - (*o2o<0' *>• *021(°'T)' *10(°-
< W V Also, C
T))
'
satisfies the further equations: 2 2 2L X + a293x Q a x -l = a-x 1 2 2 2 / a x" = a^x - a„x 0 C o 2 2 o 13 r
(*.) K
J °
}
°
obtained by combining the 2 above equations. (For certain limiting values of the a.'s, like a I
o
= 0, a
l
= 1, a
z
= i; the first two equations become
dependent, but we can always find two independent ones in this full set of 4 equations).
58
By assumption P c C T flC , . Now look at the Lemma 11.3. For all T , T ! C H , the curves C
and C , are either
identical or disjoint; in particular, we have (by $ 10): c T
nC
T
,^<
>y
T
) =Y2(T') in IP 2
* T1 = Y ( T) for some y * T •
<
Indeed, any point P = te, x-, x 0 , x 0 ) on the curve C O
1
Z
O
determines the T
curve completely because we can solve (*) for the a.'s (upto scalars) in terms of the x. 's obtaining:
a
2
%t =X(x
2 2 . 2 2X oX2 +X1X3)
This gives the a.'s in terms of the x.'s unless all the expressions on the right are 0, e . g . , when x- = X2 = 0. However, if we solve the 1st equations in (*) and (*'), we get: a
<**'>
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 . ,
59
But cp embeds the torus E T , in IP
3
and so we must have z'-z"cA
t.
Thus 3 5 c r * such that 6 (z", T 1 ) = (z», T') and hence by (z, T) = (z«, T ! ), as required. (6). Image l £ Surface F : this is immediate beacuse squaring and adding the equations (*), we get a
i+a2)(xl+X2)' 2 2 2 But from §5, Jacobi's identify (J^) gives a = a +a 2 implying 4 4 x + xQ o o
s
o(x0+x3)
= ( a
4 4 x + x , as required. This completes the proof of the proposition. 1 2
As an immediate consequence of Prop. 11.2, we deduce the following: 3 Corollary 11.4. The surface F in IP has a fibre structure over the 2 conic A in IP . To see this; define a holomorphic map: TT:F
>A, ( x o , X l , x 2 , x 3 ) l
>(ao,ara2)
by whichever of the 2 formulae that gives ( a ^ a ^ a ^ t (°# °# °)# *• «• = x4. +^ 4x_ a s x x2f t -2 x,2 x20 1 2 o o 2 1 3 2 2 2 2 2 2 2 2 a « = x~x* - x « x * or a t s x x 0 - x xn 1 o 1 2 3 1 1 2 o 3 2 2 . 2 2 4 4 a a 2 = x o x 2 + X1X3 2 = Xo * X l 4 4 4 4 (Using x + x_ = x + x 0 , we see that the 2 sets of formulae agree and
a
0
o
that a
2
3
1
A
2 2 = a- + a~ ). It is clear that the individual curves C , c F
recovered as the inverse images under TT of the points ! ( T ) c A , other hand, it is also clear that TT(F ) = A A
where F
can be On the
= Image f and
= (A-6 cusps). In other words, F Q is a fibre space formed out of the
various curves C^ lying over A Q . Furthermore, F0%(
60
We may summarise the discussion by the following commutative diagram:
>(CxH)/r*
7
>F
>FcIP3
> H
•
>A0
>AcIP2
fibres H
/F
This suggests the interpretation of H/r
or A
as a moduli space which
we will take up in the next section. f 12. Interpretation of H/r
as a moduli space
We are led to the interpretation of H/r (or more generally H/r ) as a moduli space when we ask: if T , T' € H, when are the complex tori E
and E , biholomorphic?
Obviously, any biholomorphic map f:E
>E
TI
^ts
to tneir
universal
0*0
coverings <E , i ; e . , it is induced by a biholomorphic map f :
>
such that (for XcA ) ~(z+X) = f ( z ) + T ( \ ) where f^ : A of E
>
^** * 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
a bijection from A onto
61
A
. In particular,
LtA
, , or L = c Tf + d for some c , d c 2Z.
over, L T C A , , i . e . , L T = a T ' + b for some a,be2Z.
T
More-
Thus
af +b " c T'+d "
On the other hand, L and L T must generate A T , which means ad-be = * 1 . But an easy calculation gives that Im(aT> + b ) = M - b c ) I m T ' cT'+d Jc T ' + d r
= Im T #
So since T , T1 c H, we must have ad-be = +1. Thus we have: E » E T
<
* T= Y T1 for some veSL(2, 2Z).
T
The converse is clear: if T= (a T' + b)/(cT f +d), define f (z) = (c t , + d ) z . Note that f (A ) = A-, and hence f induces an isomorphism f:E
>E ,
Therefore, we have proved the well-known fact: Proposition 12.1.
Let T, T'tH.
Then T = V ( T ' ) for some y « S L ( 2 , 2 Z )
if and only if 3 a biholomorphic map f : E T ^ ^ T ' • ^ r » ©Qui^lently, f Set of complex tori E - modulo J K/SL(2,2Z)zl K (biholomorphic equivalence J Now we ask: what then is the space H / r ? Something stronger than "biholomorphic equivalence" is needed and it is done as follows: fix an n and consider the 2 natural automorphisms of
>(z
P n : (z, T)»
>
+
n"* T )
T)
Observe that for each T fixed,a anT, 6 nT r * n and 3n induce automorphisms
62
of the t o r u s
E _1 , i . e . , a
T T and p a r e the t r a n s l a t i o n s on E - b y the n n T
2 g e n e r a t o r s of the group (of n - d i v i s i o n points) *• A J A - .
Now we have the
following: Proposition 1 2 . 2 . only if
Let
T, T ' f H ,
Then
Tsy{t')
for s o m e ytT
*ET»
3 a biholomorphic map f : E ^
if and
S^ 11 ** the c o m m u t a t i v e
diagrams:
anT o f = f o anT
->ET,
i.e.,
n
n
P
;'0f
• » . » ;
-> 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 ) = - 2 T r C q l / 4 - 3 q 9 / 4 11 So the formula (J 2 ) reads a s
+
5q25/4-
].
65
[ q ^ - S q 9 ^ B q M / 4 . 7 q 4 » / 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
+
_. -
] C
•
,f
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 •Jj *oo
But in view of the Heat equation
*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)
2
J T (l-expni(4m) T) mcIN | 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+l -2 2m T~T/* *mx T T f/ix 2m+l 2m+l 22WWt, A, 2m+l 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
m
m € IN
discovered by I. Macdonald whenever k is the dimension of a semi-simple Lie group!
(cf. M. Demazure, Identites de Macdonald, Exp. 483, Seminaire
Bourbaki, 1975/76; Springer Lecture Notes No. 567 (1977)).
73
These results may perhaps be considered more combinatorial than arithmetical.
They have interesting applications in the theory of the
partition function p(n) : we refer the reader to Hardy and Wright, An Introduction to the Theory of Numbers, Oxford University Press, 1945, Chapters 19 and 20.
74
$ 15, Representation of an Integer as sum of squares The most famous arithmetic application of theta series is again due to Jacobi and is this: let r k (n) « # { ( n 1 # . . . . i^) f2Z k / n 2 + . . . + n 2 = n} = number of representations of n as a sum of k squares (counting representations as distinct even if only the order or sign is changed). Thus, for instance, r 2 (5) = 8 as 5 « 2 2 + l 2 = 2 2 + (-1) 2 « (-2) 2 + l 2 - (-2)* + (-1) 2 = l 2 + 2 2 = (-1) 2 + 2 2 = l 2 + (-2) In terms of q
s
= (-1) 2 + (-2) 2 .
exp n i T, recall that we have *(0, T) - E qn2 nc 7L
and hence
*(o.T) k =
Z
....
T.
n + q
i -+»l
n,c2Z n. cZZ 1 k = T r, (n) q ncZ+ i . e . , £ ( 0 , T)
is the generating function for these coefficients r,(n).
k = 4, we have the following: Theorem 15.1 (Jacobi): For ncIN, we have 8 T, d d|n r (n) - / • 24
if n is odd
E d if n is even. d|n & d odd
For
75
Proof. One way to prove this result is to deduce it from infinite product expansion of £ , but a more significant way (the significance being in having 4 more generalisations) is by relating & to Eisenstein s e r i e s following Hardy and Siegel (*) . We proceed in four steps. (1). Eisenstein s e r i e s : the basic Eisenstein s e r i e s a r e the holomorphic functions
E. (T) k
1
-—£
Z
m.ncZB 'm*+nT (m,n)^(0,0)
- Z 4 U A < r
.
Xk
x/0
Here k is a positive even integer, and k > 4 to ensure absolute convergence. In fact, a s the lattice points a r e evenly distributed, the sum
behaves like the integral
tt
(x+iy|~
dx dy
|x+iyl >1 t 1-k dt). Note that if ( a c E (aT+b) k c T + d
h cSL(2, 2Z), then: d Z (m,n)^(0,0)
* [m(aJ^)+n]K_ c T+d Z r =
(*) The proof given here was explained to me by S. Raghavan, and it follows an idea of Hecke.
76
2
9
because the mapping from 7L to 2Z* sending (m,n)t is a bijection.
>(am + cm,bm +dn)
Moreover, its Fourier expansion is easily calculated: we group
the terms as follows: E (T) =
Z
k
TT + E ( E
, HF) m/0 ne2Z (m T+n)
n/*0n =
2 £ "1T+ ncIN n k
-2[C(k)+
2
E ( £ ; meIN n*2Z ( m
E
( I
, xk T+n)
) (since k is even)
^—r)]
meIN n e S
(mT+n)
(where C (s) is the Riemann zeta function). Already the terms in the parentheses are periodic for TI
>T+1. To expand them, start with the
well-known infinite product expansion: sin rr z =
| | (1 - (- ) ) . n neIN
TT Z
Taking the logarithmic derivative, we get: TTCOS TTZ
1
Sin
Z
TTZ
"
+
y-
/
2z
*
1
ncIN^ ^
(the series on the right converges absolutely and uniformly on compact subsets in C ). This can be rewritten in a series which converges if Im z > 0 as: - i n (1+2 S exp 2ninz) - - i n * + e x P 2 " i z ncW l-exP2niz COS TT Z
= zL+ E^T (-4-+-!—) . z+n z- n neIN
(The term in brackets cannot be broken up , otherwise convergence is lost).
77
Differentiating this (k-1) times, we get: -(2TTi)k Z nk'1 nfIN
exp 2TTinz = ( - D ^ V - D l [-*-+ Z ( T ^ k * — ^ J l . z nclN ( z + n > (z-n) K
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)<5
£ n e2Z
((2m+l) T+2n)2
are absolutely convergent and so: EJ(T)=
Z
C Z
1
-* -
m€2Z ne2Z(2mT+2n+l)^
Z
1
ne2Z
((2m+l) T+2n)Z
-
Let us now evaluate the inner sums:
(*) In fact, E 2 defined by conditional convergence, is not a modular form: cf. Weil, Elliptic functions according to Eisenstein and Kronecker, Springer, 1976.
79
1st t e r m ;
( i ) i f m = 0;
_i _ _2 -_ 2 E ,-7-^772 = 2 T l U - P 2 ) » 2(1-1) | [ (1.p , n c Z *zn x' p odd prime p prime
•2(l-l)C<2)-§.sL--£ . (ii) if m > 0 ;
E - =i n f Z S (2m T + 2 n + l ) 2 4
Z n c 2
\ [(mT+})
+
5 nr
= -TT2 E n e x p 2 T T i n ( m T + | ) neIN = -TT
(by (*) )
E (-1) n exp 2TTi(nm) T neIN
(iii) if m < 0 ; changing m, n to - m , - n - 1 , we s e e that the s a m e formula a s above holds with - m instead of m. In other words, for m f 0, we have
Z
1
ncZ
(2mT+2n+ir
7
=TT2
(-Dn
E
exp2ni(|m|n)T
neIN
2nd term:
(i) if 2 m + l > 0 :
E ntZZ
((2m+l)T+2n) z
= -TT 2 E n exp rrin (2m+l) T n c]N
(ii) if 2m+l < 0 : changing m , n to - m - 1 , -n, we find that the s a m e formula holds with - m - 1 instead of m. A s each of these can be summed over m individually, we get: E * ( T ) = J - - 2TT2
= 2-+2TT2 4
E ( E ( - l ) n n exp 2 n i n m T)<-2TT2 E me IN neIN mcZ
E n ex P TTin(2m+l) T neIN
E ( ( E (-l) n+1 n)ex P TTiNT)*-2TT 2 E ( E n ) e x p T T i N T NeIN n|N NeIN n|N N even gT odd
80
=^{1 +8
.2 = -^-{1+24 4
E-n + I[ nn + E n l eex p TTiNt+8 E ( E n)exp TTiN T } E n|N 1 NeIN njN jn|N \n|N NcIN Nodd N odd N even n , N e v e X n o d d L_ n \ n J E
(( E n)expTTiN T )«-8
NCIN\
n|N
E
( E n) e x p n i N T }
NcIN
N e v e n \ n odd
n|N
N odd
(where we have u s e d the identity
2r-2
" -2 " • , # . -2+1 = 3 t o conclude that
Ed2rE E2Sd+ E d = 3 E dfor l < : s * r - l , d|Nx d|Nx d|Nj d|Nx (3) E
N =2 ^
r>0andN
odd).
(T ) i s a modular f o r m : A s an obvious c o n s e q u e n c e of the a b o v e , w e have a functional equation
for
E 2 ( T ) , namely,
.*.( T ) .
E2
!
E
E*(--L)
2 m , n c 2 Z (- ?H[L+2n+l) T
= -T2(
I m , n c2Z
f
(
(2n T - 2 m - l ) 2
2m+l
2
1
n x z—+2n) J
T
((2n+l) T -2m)2
2 * TZE2(T+2) because
E*(T+2) =
E m, n e 2 Z
and now r e p l a c i n g
2m T + 4 m + 2 n + l )
( ( 2 m + l ) T+4m+2n+2) 2 .
m by n and n by -2n - m - 1 , we s e e that t h i s s u m i s
p r e c i s e l y the p r e v i o u s o n e .
81
Thus
E?(-X) = -f 2 E*( 2 T) 2
T
This shows that E ( T ) has the same functional equation as * (0, T) for the subgroup 1^ g c SL(2, 2Z), namely, (
Ej(a^.)M-l)C(cT+d)2E^T).
*>
Now modulo I \ , there are 3 cusps, i. e . , (p/qlpodd, q even } U { o o } , { p / q | p , q odd} and {p/q[p even, q odd} , or oo, 1 and 0 for short.
Notice that the extra substitution T \
• -— T
in T carries 0 to oo. On the other hand, by its Foruier expansion, 1,2 E^ is bounded for Im «r£ c, so E has the bound for a modular form 2 « at oo and hence at all rational points representing the cusps oo and 0. As for 1, we must expand E 2 (- y r +1) (in a Fourier series ) as we did for E 2 and check that its only terms are exp TTin N T , N > 0. But 1 4.1 * = = r~ -2 E (-JL+1) 2
T
La i— n 2n+1 n , m c 2 Z^L( ( ( 2 n +>1 > T + 2 m )
r((2n+l) T+2m-l) 2
and this can be expanded in a Foruier series just like E 2 (T). (In fact, since (2n+l) T i s never 0, there is no constant term either).
The
conclusion therefore is that E ( T ) is a modular form. The final crucial step is the following: (4)
E 2 (T) = (TT 2 /4)* o o (0,T) 4 ftate that this identity at once implies the
theorem by comparison of the Fourier coefficients.
On the other hand,
to prove (4) itseIndirect verification of Jacobi's theorem^ say the first £ 2 4 10 coefficients r 4 (n), shows that E 2 ( T ) - (TT /4) *OQ(0f T ) has a zero of order 10 at the cusp i oo, i. e . , that the meromorphic function
82
f = E2 9V*
' oo
-n 2 /4 '
0*0
on H/r has a zero of order 10 at the cusp ioo.
But ?*
is zero only at
the cusp 1, and there has double zero: so f can have at most 8 poles which is a contradiction to # poles = # zeros unless f = 0. This might be considered a lazy man's way to finish this argument! In fact, there are more elegant ways to go about: v i z . , (a) it is quite easy to check that the space of modular forms f for T
with
l , <&
the functional equation (*) in (3) above, is one-dimensional. £ knowing that the values of E
2 4 and (TT / 4 ) 0
at the cusp
So merely ioo are equal
OO
&
is enough to conclude that they are equal everywhere; or, (b) one could note that E 0 is zero at the same cusps where £ *# 4 Eg/*
r
z
i s . Hence
oo
is bounded at all cusps, and so by Liouville's theorem it is a constant. Most important point here is to see the underlying philosophy of
modular forms: these are always finite dimensional vector spaces of functions characterised by their functional equations and behaviour at the cusps.
Thus between functions arising from quite different sources
which turn out to be modular forms, one can expect to find surprising identities! In particular, Jacobi's formula has been vastly generalised by Siegel. We shall describe without proof Siegle's formula in Chapter II: it shows that for any number of variables, certain weighted averages of the representation numbers r(n) of n by a finite set of quadratic forms can be expressed by divisor sums c polynomials in the $
bare
e<
^ ual
to
• or equivalently, that certain Eisenstein series.
83
5 16. Theta and Zeta The most exciting arithmetic application of modular forms, however, is one which is partly a dream at this point: a dream however that rests on several complete theories and quite a few calculations.
The
dream has grown from ideas of Hecke, taking clearer shape under the hands of Weil, and now has been vastly extended and analysed by Langlands. The germ of this theory lies in the fact that the Mellin transform carries the Jacobi Theta function to the Riemann Zeta function and that in this way, the functional equation for £ implies the one for J . We want to explain this and generalise it following Hecke in this section, postponing Hecke's most original ideas to the next two sections. M carries a function f(x) defined for x eIR
The Mellin transform
with suitable bounds at o
and co to an analytic function Mf(s) defined by co Mf(s) = f f(x) x S ^ o
, a < Re(s) < b
and it is inverted by 1
f(x) = - —
c+ico j Mf(s)x" S ds, c-ico
ce(a,b).
It is just the Fourier-Laplace transform in another guise because x = expy carries -co < y < co to o< x < co, and in terms of f(expy), we have: co
Mf(s) = J f(expy) exp (ys) dy -co which for Re$-0 is the Fourier transform of y |
>f (expy) and, with
84
suitable bounds on f(expy) as y
>"too, Mf(s) is analytic in s = u+iv
when u is in the same interval (a,b). The usual inversion gives 1
f(exp y) exp cy =r—r~ \ 2 TT 1
J
Mf(c+iv) exp(-ivy) dv
c+ico
f(exp y) = ~—— 2TU
\ J.
Mf(s) exp (-sy) ds,
C-lOO
as asserted. In particular, apply this to f(x) = 2 I exp (-TTn2x) . n€lN Note then that 2 l+f(-ix) = L expTTin x = £(o, x) nc2Z
f (x) = * (o, ix) - 1 . As we have already seen (in $ 9), recall that | f ( x ) [ < C exp (-TTX)
as
x
|f(x)|
as x
>co
>0
Thus provided Re(s) > 1 , we have: M(*(o,ix) -l)(*s) = 2 J ( I exp (-TTn2x)) x* S 0 ncIN
^
Since this integral converges absolutely, interchanging the order of land Z gives: M(
"
) (|s) - 2 Z ( y exp(-TTn2x) x * s ^ ). ncIN 0
85
In the n t n integral, we make the substitution y = rrn x, obtaining M(
"
M i s ) = 2 Z (TTn2) 2TT^S( Z
2
yexp(-y)y*S^
n-s)Texp(-y)y*S^
neIN
y
0
= 2TT"^ S C(s)r(is) . Thus we have the fundamental formula (for Re (s) >1): (*)
-is ?° Ag Hx 2TT 2 C W H i s ) - J (*(o, i x ) - l ) x 2 S 2± . 0
This can be used to prove in one step the most well-known elementary properties of £(s): Proposition 16.1. The Riemann zeta function f(s) = Z n~ s , Re(s) > 1 , neIN has a meromorphic continuation to the whole s-plane with a simple pole at s = 1 and satisfies a functional equation, namely, 5(s) = 5(1 -s) where g(s) = TT 2 C(s) H i s ) . Proof, Recall that we have > I * (o, i/y) = y 2 *>(o, iy), y elR . We use this in (*) above as follows: 2TT- 2 S C(s)r(is) = | W o , i x ) - l ) x 2 S - T + 1
jWo,ix)-l)x2S-~. 0
The first integral converges for all se C and defines an entire function. As for the second:
86
5 <#(o. ix) - 1) **8 £ - ( J *%i/2L)xi s £ , . ( > . l d x X
0
0
x
x?
J
o
= ( f # ( o , i y ) y * ( 1 - 8 ) ^ ) - | ; ( y = l)
= (f(*(o.i y )-i)y* (1 - 8)d / + T y 4(1+ %-| J
1
2 = entire function 1-s
1
2 - . s
Thus 5(S) = T T 4 S C ( s ) r ( i s ) - - 1 - j i j + i
f
(#(o.ta)-l)(xK
X
*
( 1
-
, )
)^
which shows that g(s) is meromorphic with simple poles at s = 0 and 1, and is unchanged for the substitution s i
> l - s . Recall that T(s) has
a pole at s = 0 and hence £(s) is analytic at s = 0# This completes the proof of the proposition. Dirichlet series: The above considerations can be generalised as follows: fix (for simplicity) an integral weight k > 0 and a level n > 1, and suppose that f(z), z c H , is a modular form of weight k and level n. (N. B: we are replacing the usual variable T by z here). Then f(z) can be expanded: first we have a m exp(2TTimz/n) f(z) = E meZ£+ because (*
n
)«T
and hence f(z+n) = f(z), etc. We associate to f, the
formal Dirichlet series, defined by z(S)=
E» meIN
Then we have the following:
m
i»-".
87
Theorem 16.2(Hecke): The Dirichlet series Z f (s) converges(*) if Re (s) > k+1 and has a meromorphic continuation to the whole s-plane with one simple pole at s = k. Modk
= V+1C V
Moreover, there is a decomposition of
such that whenever f e Vfi ( e= * 1), Z (s) satisfies a
functional equation, namely (
2^T)Sr(s)Zf(s)
=€
^k"S^k-s^Zf(k-s)-
Proof. Let us first find bounds for the growth of the coefficients a : Lemma 16. 3. There exists a constant C such that | a m | < C mk ,
VmeIN,
Proof. Let us evaluate the m t n Fourier coefficient of f by integrating f along the line z = x+i/m, 0 < x < n. We have n J f(x+i/m) exp(-2nimx/n)dx 0 = Z
l i
#-77+n
=n a
m
?exp[2TTiX(x^/^)-2TTimxldx n
n
exp(-2Tr/n) K '
On the other hand, by Remark 9.4, we have |f(x+i/m)| < C Q . m k for m > 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
=
<=r>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
~x
g(z) « ( f ) " k « - | ) Z meS
+
b m exp(2TTimz/n), say.
But then we have: (
Ttr ) S Z f (s)
r ( s )
= M(f(ix)
=
^
" ao)(s)
s dx r ,.,. s dx J ( f ( i x ) . a o ) x S f ^ ( f ( i xx ) - a o%) x S ^
89
1
0
1
1
This shows that Z f (s) has a meromorphic continuation as asserted. Since T(s) has a simple pole at 8 = 0 , Z (s) has a simple pole only at s = k. Finally, the map f) • >g obviously defines an automorphism of (n) (n) Mod' of order 2, so decompose Modk into the 2 eigen spaces V + 1 = { g « f ] and V _ 1 = ( g = - f } and the above identity gives immediately the stated functional equation. This completes the proof of the theorem. Examples of Dirichlet s e r i e s . What sort of Dirichlet series do we get as functions Z f (s) ? Here are 2 simple cases: Example 1. (Epstein Zeta function); f (z) = *(o, z)
I
Let exp (TT i (n + . . . + n2k.) z
V"*n2kc2Z r = Z 2k^m^ + me2
e x p TT mz
^
We know that f(z) has a functional equation for T. and z \
•
2
including z \
>z+2
> - •=•; and upto a root of unity in its functional equation, it is a
modular form of weight k and level 2, The associated Dirichlet series is 2 k (s)
Z *
= I r2k(m)m-s= Z _ _ i _ . (nf+...+n|k) mClN nv...,n2ke2Z (a1,...,n2k)^(0,...f0)
90
This is the simplest Epstein zeta function. It is a particular case of the zeta functions
where Q i s a positive definite quadratic form and A is a lattice. It follows from what we have proved that Z 2k^s^ 1 S
a
mesomorphic
function with a simDle pole at s = k, and has a functional equation for the substitution s |
>k-d.
Example 2. (Dirichlet series associated to Eisenstein series); Let f(z) = E k (z) =
Recall that E,
Z (mz+n)"k , ke2IN m, n$ZL (m,n)/(0,0)
is an Eisenstein series introduced in the previous
section. According to the calculations made there of its Fourier expansion, we have Z F (s) = c, k
K
Z ak-i(m) m meIN
where a t (m) = I d , c =2 K d|m
(k
. "1)-
.
On the other hand, we have C(s)C(s-k+l) =
Z m~* m,neIN
n'***'1
z
Z n " . (mn) m,n elN ok^(l)AS0
Z
XelN Thus Z
Ek(s)
=
V
C(s)
C<s-k+1) •
We can now state the central theme of the dream referred to at the beginning of this section: it is to say that the class of Dirichlet series that
91
arise naturally in arithmetic (viz., Artin's L-series attached to finite representations of Gal(Q/Q), Hasse-Weil £ -functions attached to algebraic varieties over ® by considering their points mod p, and generalisations thereof
(*)
) is the same as the class of modular form Dirichlet series Z f
plus their generalisations associated tooerfeun types of modular forms^
'i
Now by their very definition, every arithmetic Dirichlet series is equal to an Euler product: Z(s) = I I (rational function of p" S ). primes p A pre-requisite for the coincidence of the 2 classes is that ZJs) Euler product for a set of modular forms f spanning Mod
has an
: this is the
main point of Hecke's further ideas that we now turn to (in the last 2 sections of this chapter).
(*) cf. J. - P . Serre, Zeta and L-functions, Arithmetical Algebraic Geometry: Proc. of a conference held at Purdue University (1962), Harper & Row, Publishers, New York, 1965. (**) cf. A, Borel, Formes automorphes et series de Dirichlet, Seminaire Bourbaki, 1974/75, Exp. 466; Springer Lecture Notes No. 514,1976. As Serre has explained to me, for the Dirichlet series Zf to be part of this dream, one wants to put some restriction on the eigenvalues of the invariant
differential
operators acting on these forms, e.g. most of Maass' non-holomorphic forms are not included.
92
§ 1 7 . Hurwitz maps Let us describe abstractly the group-theoretic background to theta functions and modular forms: (a) for a complex torus, we have: (i) lattice A acting on C by translation (ii) the orbit space E =
\SZ)> giving
an
interplay
A key fact here is that when the
automorphic equation (*)
f(z+X) = e (z)f(z)
is required only for XeiA, then g(z) = f(z+^i) for n in the larger lattice 1-A again satisfies (*) for all X e i A . lations with respect to all points in
Shrinking A further, eventually t r a n s $ . A a r e incorporated in the function-
theory. (b) for modular forms, we have: (i) SL(2,2Z) acting on H (ii) the orbit space H/SL(2,2Z) (iii) modular forms on H# As before, we can replace SL(2,2Z) by the s m a l l e r groups T . Then in place n of the one Riemann surface H/SL(2, 2Z), we obtain a whole tower of Riemann surfaces, namely:
93
J
I
The group SL(2, 2Z)/T
I
= S L ( 2 , Z / n Z £ ) acts on H / r
function-theoretic entity, namely, the ring Mod
and on the basic of modular f o r m s of
level n. However, notice a difference here: w e are not enlarging the group SL(2,Z). Just as
A c j j . i l , also
SL(2,Z) c SL(2,Q).
Actually i t i s b e t t e r to think
of S L ( 2 , 2 : ) a s G L ( 2 , 2 ) + , the elements of G L ( 2 I Z ) with positive determinant. Then SL(2, 2Z) c G L ( 2 , Q ) + i s a bigger enlargement of "integral" by "rational" e l e m e n t s . To incorporate GL(2,Q)
into the picture; note that y€GL(2,Q)
does not
to itself u n l e s s YT y
map any H / r
) c S L ( 2 , 2Z), a c Q , Y (a 0 a if n = (ad-be) m where
only in the trivial c a s e s , i . e . , i s that v r Y" c r n m
Y(ok'
= (
c d> : a . b ' c ' d
e Z
.
- F which o c c u r s Instead, what occurs
(a.b.c.d) - 1 .
We therefore get a new map which we call a Hurwitz map: T : H/r Y n z1 acting "sideways" on our tower.
r-^H/r > H / v r v"/1 /Ti Y n (canonical ' m covering) >YZ
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
=
>H/r ' n
( 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
/<*oo*0lV 2 k '
feMod n>
l
which can be c h a r a c t e r i s e d a s the T -invariant functions with "finite order n jDoles" at the c u s p s .
Then H / r
i s the maximal ideal space of R n .
If
n = i m , t h e n R m , RX<=R . Let •p ( holomorphic functions f on H invariant for s o m e Tn and 1 (K = U Rnn = | \ n ) \ ( which have finite order poles at the c u s p s . ) Then J{
i s isomorphic to the maximal ideal space of ^
, i. e . , Jjf
is
the s c h e m e Spec 6L - {generic point } , i. e . , Spec (^ minus its unique nonclosed point corresponding the prime but not maximal ideal (0). (Notice
96
that (jf(^ is a "non-Noetherian Dedekind domain M)# (2) Adelic interpretation:
let us first recall the concept of adeles:
the ring of rational a d e l e s ^ . is by definition the subring of the product E x
A^^
Q of elements P p prime
~ ^ (x
>
; x 2 , x „ , . , , , x , . . . ) such that x c Z
for all
but a finite number of primes p where 2Z
is the ring of p-adic integers in the field Q
of p-adic numbers.
Jft. is a topological ring, if a basis of open neighbourhoods of 0 is given by U
cj[n(p)r
((x
a> : —
V"
, l | x
«>l
< e
'V
p n < P )
V
p }
for various c > 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 ) )
:
E/nZZ
> 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
< •••OX oo
=( oo
cos 0, sin 8 . a J and X n = r Yp - s i n e , cos 0 P 2> 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
Define
GL(2,A)° = { X c G L ( 2 , A ) | d e t X e Q * . I R > } 5f£' =GL(2, Q)\GL(2,A)°/K o o .Z ( x > = 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,<8): Given a finite set S of primes p, integers n(p) and matrices X c SL(2, <J ), p c S; there exists an X«SL(2,Q) such that (p / ll+p V (nP( P)>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
/ • A - 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,
The natural map r n \sL(2,IR)/K o t )
>SL(2,Q)\SL(2,^)/K Q 0 . K(n)
induced by the natural inclusion SL(2,1R) C—-> SL(2,A) given by XI
> ( X ; . . . , I 2 , . . . ) is an isomorphism.
To see this: let us write
SL(2,A f ) ={X€SL(2,A) |xTO = I 2 >, i . e . , the 2X2 matrices formed from the "finite" adeles. Then strong approximation (Lemma 17, 5) says that
101
S L ( 2 , A f ) = S L ( 2 , Q ) . K(n) hence the map in (*') above i s s u r j e c t i v e .
A s for i t s injectivity: l e t
X 1 , X 0 c S L ( 2 , R ) have the s a m e i m a g e , then for s o m e A c S L ( 2 , Q ) , B e IC 1
ct
,
CO
C e K(n), we have (X. ; # , . , l o , # . » )
=
" ( A • • 0 , , In « • • • ) • v'Qry' *•»
' ( A X ^ ; , , , ,
Therefore
o
'
).
with p f n . ' o
Thus A c H S L ( 2 , 2 Z ) = SL(2,7L) p P
and hence
Since X, = A X 0 B _ , X- and X 0 define the s a m e e l e m e n t in 1 * <x> l L
A eT . n
rn\SL(2,R)/Ka) Step IV.
as required.
H « S L ( 2 , 1R)/K
= lim
SL(2,1R) a c t s t r a n s i t i v e l y on H and
K^
Thus we get:
T \ H SUm n
Finally:
because
i s the s t a b i l i s e r of i eH. ^i
, # #
= I 2 , hence A = c " cSL(2, 7L ) and A M^mod p r ( p ) )
V p , we haveAC
where n = p r ( p r i
ACpJ„,
2 * • • • ' • ' 2'* • • ' n '
^T" -lim
T \SL(2, HJ/K^ n
SL(2,0j\SL(2, J ^)/K O C ) .K(n)
(by (*')
n S S L ( 2 , <&f^U2,//\)/K00
(by (**))
SGL(2.«)\GL(2,A)°/Koo.Zoo
(by (*))
o as required. L a s t l y , to check the action of T
on J f
to check for the b a s i c o n e s T ^ , i . e . , when y right translation on jx
o
: it i s c l e a r that it s u f f i c e s =
defined by (. . ) , i . e . , 0 *
xo ( 0 i)* J * IN.
Now look at the
102
> X
(X oo ;...,X p ,...).
< a>:-".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(
The r e q u i r e m e n t i s
In t h i s m o d e l of ^ ? , T^ i s the map (V, L, cp) i s given a s the i n v e r s e i m a g e of <X7L\7L X (0)) in
> ( V , L*,cp') (Q/5Z)
under the map
nat
- "*?» Q.L/L
—->{
and cp1 i s given by Q.L/L
^
> (Q/2Z)2
Q.L'/L'
5
> (Q/ZS) 2
§ 18. Hecke o p e r a t o r s We s h a l l now study the action of the Hurwitz m a p s defined in the p r e v i o u s s e c t i o n on functions.
The s i m p l e s t way to define t h i s action i s to c o n s i d e r Mod k
= U Mod^ n) k neIN
.
T h e s e a r e modular f o r m s of indefinite l e v e l : the r a t i o of any two functions h e r e i s a function on < j£= l i m < n
H/rn
#
Now G L ( 2 , Q )
a c t s on t h i s v e c t o r s p a c e by +
f *—>fY w h e r e f Y ( T ) = (cr*-d)~ k « ^ g - )
.
T h u s , in the l i m i t , w e jump up f r o m having an action m e r e l y of a quotient of S L ( 2 , 2 Z ) , to having one of GL(2,Q)
.
T h i s a c t i o n has b e e n much studied of
104
late in the context of the decomposition of GL(2,j^ acting on L (GL(2,Q)\GL(2,A)). action on Mod, into disjoint T
(*)
Hecke looked however at the reflection of this group
: take any ycGL(2., 0), » a n v ncIN ; then decompose I^Y^ left cosets:
r n Y r n = (r nYl )U(r n Y 2 )U... U(r Yt ). Y.€GL(2,
QJ +
.
Lemma 1 8 , 1 . The number t of the cosets in (*) above is finite. Proof, Let
I c IN be such that
and let m = ad-bc. Then we know that yT
Y c T , hence yT
So if T = U IL^ 6., 6. c l \ , then we have: " l<jSr nm J 1 n r r
»» »\ U ' V r j . c J
l£j
c
^nY •
U rnY6
J
l£j
and this proves the lemma. Lemma 18. 2. Let Y#Y. be as in (*) above. For feMod
i
T*(f)= Y
Y
1
I f 3# Then T* (f) e ModJ ^j^t Y
n)
*
, let
(i. e . , T* is a map of Mod[ n) to Y k
itself, called the Hecke operator associated to Y cGL(2,Q) ). Proof. We have only to check the T -invariance of T (f): so let ft c T , then from (*) above, we have:
hence for some permutation a on { l , . . • , t } and for some ft. cT , we get: T
But then we get:
j
iTa(j)
105
(T* (f))6 = ( I fY3 ) ' = S f V =E fV»0> 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<j
j
)((= union of p+1 left c o s e t s )
P
J
Proof. Let GL(2,Q) act on row vectors (a,b) by right multiplication. Then 2 an integral X with det X" = p, c a r r i e s index p.
Moreover, 7L . X
X1X"1eGL(2,2Z).
- 7L # X
7L
2 onto a sub-lattice
if and only if 2Z
LC2Z
- 7L , (X..X
of ) or
Since det ( X ^ ' ) = det X /det X o = l , X 1 X < " 1 e S L ( 2 , 2Z).
Thus we have an i s o m o r p h i s m S L ( 2 , Z ) \ { X e G L ( 2 , Q ) / X integral & det X = p ) ~ [sub-lattices L c Z
2
of index p } .
2 But such an L n e c e s s a r i l y contains p2Z , s o it i s determined by a 1-dimensional subspace
2 L of ( 2 Z / p S ) .
There are p+1 of these and the corresponding
L's are the span of
{(i,o),(o,p)},{(i,i),(o,p)},...,{(i,p-i),(o, P )}& {( P ,o),(o,i)}. These a r i s e from the X's respectively given by
(i \(i O p O p
b
<* P - 1 ) * ^ ° > . O p
0 1
This proves the equality of the 1st and 3rd s e t s above. On the other hand, for any L c Z Z 2
of index p, 3 Y 1 e S L ( 2 , Z S )
such that L . Y
= span of ((1, 0), (0,p)}
106
o and hence, if XcGL(2,Q) is such that L = 7L .X, then we get: 2Z 2 .X = 2Z2(* p ) . Y ^ or X = Y ^ ° ) Y~* for some Y 2 cSL(2,2Z). Thus the 1st and 2nd sets are equal. This proves the lemma. This lemma enables us to explicitly compute the Hecke operator * * (1) T - n ( = T for short) : let f e Mod' , i. e . , f is a modular form for the (I U» p K 0 p' full group SL(2, 2Z), and let the Fourier expansion of f(«r) be given by f(T) = Z a exp(2TTinT). nc2Z + * Now by definition of T (cf. Lemma 18.2), we have: (T*f)(T)=f(()1(T)+
Z f°P(T) 0*j
=f(pT)+pk[f(T/p)+f((T+D/p)+. . .+f((l+p-l)/p)] -k anp . ^ Z a n exp(2mnpT)+[ Z neZZT ne2Z ^ Z exp (2TTinj/p). exp (2TTin T ) ] 0<j
( Z a exp 2nin T) ne^pn
= Z b exp(2nin T1 ne2Z + n where p (*):
. apn
if pfn
b P1_k»pn+an
if
Pln
P An immediate consequence of this formula is the: Corollary 18.4. For all primes P j , p 9 , the operators T
and T
commute.
107
Indeed, if (T* (T* f))(T) = Z A c exp(2TTin T ) P + 2 Pi " n .e2Z then we have
,P2'\,„ c
U
*2<»
= ,' 2
p
2
^ 2 " k p i " k a pPlP nr l P o2 '^'
k
^'
k
V/
a
tt
P
P
2
2^n
|-
)
P2" k ( Pl" k a P l P 2 n
+
1 2
V
2
l^ n
lf P
2^»
^
«
i
1 -k # 1 -k \ . 1 -k >2 ( 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 "
*
1
T T
s ,
K-i
-*» "
k 2S D ~ \. pD ( 1 - a. pis + p )
•
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 -' + p k - 1 .p 2 ')z f <.).. 1 .
I p D Drime prime as asserted.
J
This completes the proof.
We do not want to develop Hecke's theory at g r e a t e r length but only to give a few e x a m p l e s and to state his main result and the dramatic conjecture that has been made in this connection.
F o r full proofs and d e t a i l s , cf. A. Ogg,
Modular forms and Dirichlet s e r i e s , Benjamin, 1969,
To state Hecke's main
result, we need s o m e m o r e notation: let TT: SL(2,2Z)
->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,
>f 6 where 6cSL(2,:Z) and 6 = (* ) (mod n), at (2Z/nZ)* # 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* <_*_-s i ) • I I d - a n P + e( P )p .p &)
=
where a p P
-(k-1)
* * = eigen value of T = T , rt B P (1 Ox 0 p'
and C(p)= eigen value of R ( p
mod
n)
What are these f-'s? The prime example is the Eisenstein series: we saw k in % 16 that Z E (s) = c k C(s)C(s-k+l) where c k - 2 i g l i l . = c
k
TT ((l-pV^l-p^.pV 1 ) p prime -2s-1
= c
k
TT p' + P k - 1 . p V . p prime
Hence, by Prop. 18,5, it follows that
Ill
T
p
(E } = ( 1
k
V " k ) Efc= (l+P*"1) p"*"" E k .
In fact, it can be shown that the f 's in the theorem break up into 2 disjoint groups: (1) cusp forms and (2) generalised Eisenstein series
f a (T) =
Z mnm^Z (mrm2)/t(0,0)
c(m1tm9) l Z — (mT+m9)k 1
where the c ( m - , m 9 ) ' s depend only on the m.(mod n)# The latter forms fa have Dirichlet series Z f (s) of the type Z f (s) = L(x , s ) . Lfy^ s-k+1) where L is the Dirichlet L-series (cf. Ogg for details). In particular, we get a direct sum decomposition
Mod,
= (Cusp forms) ©(Generalised Eisenstein series).
We can use the results of §S 13-16 to fit *
4
into this picture. In fact, we
saw in $ 15 that Z
4
oo
(s) = 8
Z ( Z d)n" S + 24 T ( Z d ) n" nclN d\n n cIN d|n n odd n even d odd
= 8 Z ( I d ) n " s - 3 2 E ( Z d) (4nf nclN d|n neIN d|n using the easily verified fact that if n is even
112
£
d
if 4 f n
dln
/ " 1 S d - 4 I d if 41 n
din
L d l"
dodd
d
lf
Thus Z (s)= 8(1-4 *oo
) I (Sd)n' nelNdln
- 8(l-41"S)C(s) . C(l-s) and hence, by Theorem 18.6, we get: T
pP ^ o°o° )
= (1+
4 P " 1 ) t f oo '
Unfortunately, the generalisations of Jacobi's formula to other powers £ are in fact not so simple; e . g . , paper 18)
2k
S, Ramanujan guessed (cf. his collected works,
and Rankin proved (Am. J . Math., 1965) t h a t :
(subspace spanned by the") *oo €"\ (JEisenstein series The identification of the eigen functions f
f J
< === > k
-
4
•
of the Hecke operators as poly-
nomials in the functions r\
>& L (0,nT) seems to be quite hard to describe a,b of weight k = 1 or 2, For example, take the case:
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
*\
and these traces and determinants are integers in K. We say that a is odd if P»(c) is conjugate to (
) where ceGal(Q/Q) is complex conjugation.
Such compatible families of representations (at least in GL(n), some n) arise in great abundance from the theory of etale cohomology of algebraic varieties. For any such family, we can form the Dirichlet series: -1 (s) = T T (1-Tr (F )p" S + det (F )p" 2s ) p p p p W] x x P*s We may be able to supply suitable p-factors for pcS. Now building on partial Z°
results of Kuga, Sato and Shimura, Deligne proved the following: Theorem. Let f e Mod • Hecke operators T
> 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. <& converges absolutely and uniformly in z and
Q in each set c
max I* Im z. I< . 1• Im
Q
>
-
l 75—
2TT
and
c0I
2 g
hence it defines a holomorphic function on
(Eg xi
.
119
Proof; expOjri nftrtt-27ri nz)
<_ exp(-7rc 2 (In i )+c 1 2;|n i |) = n e x p f - i r c ^ + c j n . | ) ;
hence the series is dominated by( Y exp(-irc0n +c.n) ) v l l n>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 — ?<3 — L^-quasi-periodic of weight SL if_
Definition 1.2. is
f (z+m) = f (z) , f(z+ft-in) = expC-iriJl^m-ft-m - 27ri£ • tz -m) . f (z) for all
m € 2g .
Let
R«
be the vector space of such functions
f.
As in the previous Chapter, one of the applications of such functions is to define holomorphic maps from the torus projective space. the same weight f. (a)
5* 0
In fact, if
f.,-••,f
are
E?/L0
LQ-quasi-periodic of
£ and have the extra property that at every
for at least one i, then
to
a € (Cg,
123
z I
> (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 £~ • a-6) • f+
h-> ^ = Y exp(2irik a b ' S^t mod k
Proof: in
z.
• 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 €
(2E2g)X
= 2E2g ,
(ii) (iffi2g )
L
= n2Z2g, n € K .
In fact, more generally, for lattices L,L.,L2 in Q g , we have: (^L)1 = nL 1 , ( L 1 ) 1 = L, L x c L 2 < In particular, if L cffig index s.
is of index s, thenffig c L
Further, notice that in this case
is also of index s in L Q . representatives of L /2Z g .
> L^ £ L^ , etc.
a
L
L
Q^ ^ E o
=
is of a
o ^ ^^
Let a.,b.€L , 1 <. i <. s, be coset Let us call the set
127
B Q (L) = jz € CEg| *&[
1
j (z,ft) = 0 , 1 < i < s}/c^(L) i
the set of "base points" in the complex torus <E /ou (L) .
Now the
"rational morphism" tp is given in the J_i
2 Proposition 1.2. For all L c 2Z g
ra
of index s, via the
<;]•••
we have a canonically defined holomorphic map
V
[*9An
"*
s-l
namely cpL(z) = (..., ^ [ ^ j c z ^ J f . - • ) .
We have only to check that follows:
let (a,b) € L
cpT is well-defined.
and (a',b')€ L.
We do this as
Then by the quasi-periodicity
Of * [ £ ] : tf-Mcz+ao'a' ,b') ,fl) = l^fa] (z+fta'+b' ,fl) LbJ " tbJ = e x p [ 2 7 T i t a - b , - 2 7 r i t b - a , ~ T T i t a , f i a , ~ 2 7 r i t a , z ] # [ b ] (z,«) = X(a' , b ' , z ) . e ( ( a , b ) , (a* , b ' ) ) . ^ [ * J
But e ( ( a , b ) , ( a ' , b ' ) ) and t h i s p r o v e s t h e
= 1 and X ( a ' / b ' / z ) result.
i s i n d e p e n d e n t of
a,b
128
The main result concerning this map is this: Theorem 1.3
(Lefschetz):
and assume that
L c r.L
Let
L c 2E "
be a lattice of index s
for some r € ISf.
Then:
(1)
if r > 2, B 0 (L) = 0, i.e., cpT is defined on all of (E g /a n (L),
(2)
if_ r >_ 3, cp is an embedding and the image is an algebraic
subvariety of P
, i.e., the complex torus (Eg/a0(L) is embedded
as an algebraic subvariety of P (3)
s-1
;
every complex torus that can be embedded in a projective space
(or, more generally, whose points can be separated by meromorphic functions) is isomorphic to (Cg/aQ(L) for some Q, £$v
and some L.
For a full proof, the reader may consult §3 of the author's book, Abelian Varieties, Oxford University Press (1974).
Here we
shall skip completely the proof of (3) and outline the proofs of (1) and (2). Note that it is (3) which explains why we have focussed attention on the special type of lattice L
in CCg:
these and
their sublattices are the only ones which lead to complex tori which are also algebraic varieties.
It would be impossible to find entire
functions f(z) quasi-periodic for more general lattices because of this result. The first step in the proof of (1) and (2) is: Lemma 1.4:
Let f(z) be any holomorphic function such that
f(z+fia'+b') = exp(-7Tita,fia'-27Tita'z) -f (z) for all (a',b') £ L.
Then f(z) is a linear combination of the
functions
tf-LMtz.n), i < i < s.
129
This is proven by a variation of the argument used to prove that
1/
is characterized up to a scalar by its functional
equation:
one makes a Fourier expansion of f for the lattice
aQ(L) fl S g
in (E , and expresses the remaining functional equations
as recursion relations on the Fourier coefficients.
These leave
only s coefficients to be determined and it's easy to see that the s functions
z i
> \7 |.
(z,ft) are linearly independent.
The 2
step is a little symplectic geometry over ffi: Lemma 1.5: L
For all L c 7L g
with L c L
and
such that L <= rL , there is a lattice
L, = rL7". Such an L, has a standard basis:
( o , e i ) , • • • , (o,e ) , (f{,fp,••*,< f g' f ") with A((o,ei),(f!,fV)) =-r-6 ij A((f^,fl')/ (f^fV)) = 0. In fact, we can even find 2 such enlargements: such that
L = L x n L^
and
L
L c L , L
= rL^, L' = rL' 1 .
Thus e,,---,e and f_ = Qf'+f",•••,f =^2f'+fn 1' g -1 1 1 ' '-g g g
are a basis
of aQ (L.. ) . We then define a linear map S:
(Eg
>
<E g
by requiring that: S(e.) = (0,•*•,1,•*•,0), the i Let
s(f.i) = <« u ,---,n g i >.
unit vector.
130
In matrix notation, if we write EjF^F" for the matrices whose columns are e.,f! and f'.', then the lemma says
^ . F ' = rl t t F'.F" = UF"-F'
t F'.F" symmetric.
or
Thus S is given by the matrix E
and
fl' = SF = S(flF'+F")
= -fo'OF9 so
+ tF'-F")
Q1 is again symmetric with positive definite imaginary part.
Note that the linear map S carries the lattice lattice
L
Q»
#
Tn
aQ(L ) to the
e purpose of this construction is to take the
function l9"(z,ft') quasi-periodic with respect to the lattice L Q , and form from it the function W'iSz,Q*),
quasi-periodic for a~(L ),
We check: Lemma 1.6:
For suitable
S € Q) , the function
f(z) =
^(Sz+bjA1)
satisfies:
for
a)
f(z+e±) = f (z),
b)
f(z+ftf|+fV) = e x p ( - ^ tf|fif| - ^ i tf^z)-f(z)
1 <_ i <_ g.
Therefore for all
I a. = 0, the functions i=l 1
g(z) =
a.. , • • • ,a € <E9 such that
n f(z+a.) satisfy 1 i=l
131
a)
g(z+e^)
= g(z)
b)
g(z+flf|+fV)
= exp(-iritf^f^-2TTitf:[z)g(z) ,
for 1 < i < g, hence g satisfies the hypothesis of lemma 1,4, The proof is quite straightforward.
Without any b, we use the
functional equation for v* to find the law for f, but come up
with
a root of unity in (b). Adjusting the b, we get rid of these roots of unity.
The second part is an immediate consequence of the first.
This is now Lefschetz's central idea:
that there is a related
Of-function f such that all products f (z+a1) (for a-+-«-+a
^[^(Zffl). Lemma 1.7;
f (z+ar)
= 0) are linear combinations of the functions We next show that:
a)
If_ r >^ 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) • A.
1
fa 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: We can find a normalized basis
Corollary 2.2.
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) a or, if we use the basis per':
i
> < the linear map GO I-
u>., •••,(*) of 1 9
H1(X,Z) a »
T(X,ft ) :
> <Eg >
(/•v-'h) 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 Theorem of Abel: Given cycles [ P., I Q• of the same degree, 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 #[§]
fr[§](Nz,n) Likewise, i f
£a. = £a! ,
\h.
= £b!
mod ffin , then
#[*i](z,fl) 1
1
ntfr*il
is a meromorphic function on X . )
i
(the 1 s t one because ^[jjj] (NCz+m+fim') ,Sl) £[{Jj (N(z+m+nm') ,8) exp(27ritaNm)exp(-2TTitbNrrf) [exp (-TriN2tm*ft m'-^TTi^m' z)] T^t^KNzfl ) exp(2iTitaINm)exp(-2Tri1blNm') [exp (-7riN 2t m' QmI-2TTiNtm' z) ] l&^'.KNz ,Q) and ( a - a ' ) N ,
(b-b')N € ffig ; s i m i l a r l y t h e 2 n d o n e ) .
148
Method II:
3z
8
-
1
"'
a$-(z+a,ft)
= JL_ ^J^I+LiU 3x i
#(z+B,oj
is a meromorphic function on X 0
( b e c a u s e t h e r a t i o of t h e 2
*V*(J+6,n)' 3
/ &Q
as is -£— log( ^ ",
(z.S), •
-1
l9"'s i s m u l t i p l i e d , by a c o n s t a n t
when
z i s r e p l a c e d by z+ftn+m).
Method
III:
3z.3z .
3* 3z. 3
3z.
^2 i s a meromorphic f u n c t i o n ( i t i n c r e a s e s by
on X
32 t t r—- l o g e x p ( - 7 r i mftm-27ri mz) when
-r
dZ . dZ .
z |—> z+^Hn' and t h i s
is zero). This is the Weierstrass &-function when g = 1. Now let ft be the period matrix of the compact Riemann Surface X again.
The applications of XT to the function theory on X are based
on a fundamental result of Riemann who computed the zeroes of P f(P) =
-& (z +
| Zf0\ p
(Z fixed).
o
Note that f(P) is a locally single-valued but globally multivalued function, which is invariant around the A-periods, but, on prolongation around a B-period
B,
f is multiplied by
P exp[-uiftkk - 27ri( f w k + z k ) ] . Pn
149
Theorem 3.1 (Riemann) :
There is a vector
A € (E , such that for all
P
z
€ (Cg, f (z) = v (z +
uj/ft) e i t h e r v a n i s h e s i d e n t i c a l l y , or has p
g zeroes Q,,•••,Q
o
such that
°i a) =
J Jp
i=l Proof:
- z +
A (mod L Q ) ,
io
This is another application of Green's Theorem.
cut open the Riemann Surface X as before. Q. € X n
and
Pfi € X 0 .
Let
consider the 1-form df/f on
We may assume that
A. be a small disc around Q.. X n ~UA..
We
We
It is holomcrphic, hence
closed, so
I
'<*§>
X 0 -UA.
J
a(xQ-uAi;
H
- l \ d4 + i 3 A.
I d4 • i J 2
^
V
r
^k-Bk)
Now f is invariant under the A-periods, hence it has the same value on B, ,B, .
And f increased by -2iTia) on B, which joins A, to A, .
df f
150
Thus the middle integral equals g
I 2*i Juv = 2 nig and the last integral is zero.
Since
—j = 2 7ri(mult. of zero Q.),
this proves that the number of zeroes of f is exactly g (counted with multiplicity if necessary). Next, let
oi, = dg, with g, (P ) = 0 on X n and repeat the same
argument with the 1-form g,»—=-:
|
d < 9kv ^ f' >
x 0 -u A i v
f
df
x
?
f
df
?
x
f
df
Taking these terms one at a time:
°i
f
J'k7= 2*i W 3A
Next
g,
= 2TT1 } P
i
on B
is g, on B
plus
k-
0
6, . because the path A
leads from B„ to B . So
f df _ . J + g k -7 - 6kl
VV
B
f df "7
*
=
6,
(change in value of log f around B ) *1 r
=
6
ko [-7T±f2
/ some
-2TT± J u -2TT1Z
m
l
P
0 (where P, is the base point of the paths B
\"|
+2Tri^integerJJ
and A ).
151
Next g
on A. is g, on A. plus (~fy,n) because the path B
leads
from A, to A, . So
l + g k ^ = L ^k-°k£> ( T- 2Titt i> - ( v ¥ > A Cv A 0 * =
-\i J+ ¥ - 27ri ^ + g k ^ + 2 * i Q k* f ** A
A
£ some
(
A
*
v
&
/•
i n t e g e r J - 27ri AI* g ^
+ 2irift k £ .
Putting all this together, we find: P 1
Q. p
p
o
A
o
*
+ (ia +
k £ 0 k*V
which proves the theorem.
QED
To exploit this theorem, we Symm
make a few definitions:
let
X be the compact analytic space constructed by dividing
X x---x x (n factors) by the action of the permutation group in n letters, permuting the factors: Symm
X =
X x • • • x x/g
152
This is well-known to be a manifold, even at points where (& not acting freely.
is
In fact, if (Xl,---,xn) € X n
and 1
k
k+1
n
(this can always be achieved by permuting the x.'s), then let U be a neighborhood of P disjoint from an open subset X n c X containing x, ,,•••,x . Then
Symm
X contains the open set:
* <x!T,V6n-k) • Then coordinates on U /(§. are
given by the elementary symmetric functions in z; (xn ) , * * * ,C (x,) , V
k
hence that
U /(g, is an open subset of (Ev. Symm
By induction this proves
X is a manifold.
Points of Symm
X are in 1-1 correspondence with so-called
"divisors" on X, positive and of degree n. formal sums:
y k.p.
i=l of points of X, with k, > 0, 1
i I k = n. i=l x
These are finite
153
We will usually write divisors as: n i=l
x
allowing the P.'s to be equal.
As in §2, we have a canonical map
I : Symm n X n given by
> Jac(x) p
I (Jp )
n i = ( J J S m ° dL «)P
Clearly, I
0
is a holomorphic map from Symm
X to Jac(X),
To exploit Riemann's theorem, define fe c Jac(X) to be the proper closed analytic subset: P £
=
{z
OMA-Z
+
f w) = 0, all P
Let U = Jac(X)- £. Corollary 3.2: P.
and if by z by:
0
Then I claim:
For all
u> = z mod L
f 1
P , • • • ,P € X, z € (Eg: P. > 1$* (A-Z +
P z
P € xj .
ui) = 0
for all i
P
$fe,, then the divisor J P. " i 1 P. (o = z mod L .
JUo p
is uniquely determined
154
Kence I : is bimeromorphic.
Symm g X
> 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
<_ g-1.
If dim D = g-1, it follows that dim D = dim (D+X ),
hence D = (D+X ). But then P
D+X
P
+X
P
+•••+ X = D . P
Bv the
156
Corollary, I
is surjective, i.e.,
=
Jac(X).
Together these imply D = Jac(X), which is a contradiction. dim D <_ g-2.
Lemma 3.4:
Thus
QED
Let
e e
(Cg
there are 2g-2 points
satisfy
R.,•••,R g
A.
E_^(x,y) = 0 e
^ (e) = 0, E_^(x,y) f 0. Then e ,S ,--',S e X such that j.
l.
g
a)
x = y
or
b)
x = R.
or
c)
y = Si
±
.
More precisely, including multiplicities, the divisor of zeroes of E
+ e
is the sum of a)
A , the diagonal
b) {R.} x x, c) Proof:
Let
R € X
1 £ i £ g-1
X x{s. } , 1 <_ i <_ g-1.
be any point such that
by Riemann's theorem, there are g points 1>1e+
w) = 0
and by the Corollary:
E_^(R,y) ^ 0. e y,,•••,y such that
Then
157
unique unordered g-tuple such that
(y ,-•-,y
)
1
=
{ g
g
y
P
i^
o
f m = A - lZ + f
«• V
j M)
mod L Q
J
R Since
to) = 0, we may assume y. = R. *1
It follows that:
unique unordered (g-1)-tuple such that
e i=2
Therefore Y7,''',Y
P,
depend only on e:
There is a finite set of points
set S. = v • _-, •
R € X such that
(A)) = 0 . R
To investigate their number, choose S 0 ^ S-,---,S _, .
\?(e + fuS) = 0
But
&{-z)
if and only if
# (e + |w) = 0, all y.
= l9-(z) , so x
S0
•#(e +
x = S0 or
Then
Jw) =
$* (-e + fu>)
x
Sc
has g zeroes by Riemann's theorem. R ± e X such that
XT' (e +
f u>) = 0. R.
So there are g-1 points QED
158
With these functions E , we can now prove Abel's Theorem (see §2): given cycles
of tne
J P., I Qi=l 1 i=l 1
same degree, assume
V Z V = id(ZQi). Then we want to construct a meromorphic function on X with poles £p., zeroes £Q. .
In fact, we choose &(e)
€ (Cg
e
so that
= 0
E_>(Pi,y) £ e
0
E^(Qi,y) % o. e Consider the function on X:
(3.5)
f(y) -
i=l e
i-1
e
We fix the sheets of the multivalued functions
E e
a little more
precisely as follows: Choose paths
a. from P. to P.,
T. from P Q to Q. so that
f f! • f f
0)
159
Note that for any
CL, T^, by our assumption I^P.^) =
I
( j^
Q
i^'
the above sums would be congruent mod L~ * Hence altering one of them, we can achieve equality in
CC .
Define f as:
K #(e + f 3) f(y) -
n #(e + j 2) n i=l
p. 1
where the paths from Q., resp. P., to y are —r., resp. -a-, followed by the same path from P. to y in all integrals.
If we do
this, let us examine the effect of moving y around a path in X or of altering the path from P n to y. happens.
If the change is by A, , nothing
If the change is by B, , f is multiplied by: d
y
l
n exp [ - i r i \ k " 2iri(J
u>k + e k > j
d
r
Y
r
i
II exp - 7 r i ^ k k - 27ri(f u>k + e^) P. l
a
y
d
y
exp [-2,1 ( J ^ j o>k - . ^ | Wjc )] Q±
P i
160
Thus f is single-valued.
By Lemma 3.4, its zeroes are precisely
the Q. and its poles the P..
This proves Abel's Theorem.
The beautiful function
E_^ plays the role for X of the ® 1 1 function x-y for the rational Riemann Surface P : namely on 3P , every rational function can be factored
iKy-C^) f(y) = c
TKy-P±) (3.5) is a generalization of the formula to all compact Riemann surfaces.
E_^ is called the "Prime form" because of this role in e factoring meromorphic functions. We conclude the section with one last consequence of Riemann's
theorem: Corollary 3.6:
For all
e £ (Eg
l9(S) = 0 ^ = >
3P
1'
,Pg _l n e X
such that
xJ Proof:
s
II -
In fact take any P 6 X and apply (3.2) to It follows that P.
t»i(A
- I
l
[ Z) = o
i=l I
161
Conversely, to prove " = > " note that \?(e)
= 0 defines a codimension 1
subset of Jac(X) and the right hand side defines a closed analytic subset of Jac(X).
Moreover by (3.3), \
y
&(e + J 2) = 0, all y € X J p
has codimension 2.
'
o
So if we prove " = > " for
e's
such that
l>(e) = 0 y
3
y € X, Cr(e (+ | 2 ) V. 0, P
0
it follows for all e. Take such an e.
By the surjectivity of I ,
we can write it as: P.
A - I J S. 1=1
P
Consider
f (y) = &le + J P
Z\.
0
By (3.2), the divisor of zeroes of f(y) is exactly
o. *
I P..
On
i=1 X
the other hand, 17'(e) = 0 so Pfi is a zero of f(y). Therefore some P. equals P., and e* has the form required by the Corollary. QED
162
Appendix to §3;
The meaning of
In the preceding discussion,
A
A
comes up as a strange
bi-product of an elaborate Green's theorem calculation. like to apply the Riemann-Roch theorem on X to give
We would
another point
of view on the Corollaries to Riemann's Theorem (3.1), leading to a determination of
A
from another point of view.
We need some of
the standard terminology connected with divisors on X: Definition 3.7:
2 divisors D,,D2 on X are linearly equivalent
if eguivalently I(D -D2) = 0 or
3
a meromorphic function f such
D
that n-D 2 = (zeroes of f) - (poles of f).
This is written D 1 = D 2 .
An equivalence class of linearly equivalent divisors is called a divisor class. called Pic X
Under +, the set of divisor classes is a group, (thus the Jacobian Jac X is isomorphic to the
subgroup of Pic X of divisor classes of degree 0 ) .
Definition 3.8: P € X, let z
Let u> be any meromorphic 1-form on X.
For all
be a local coordinate on X near P and write a) = z ^»u (z) -dz
where of
n € ZZ , u(z) is holomorphic and
u(0) ? 0.
Then the divisor
a) is:
(u>) =
I P€X
np -P.
Note that if W, ,w2 are 2 such Informs,
OK/GO^ = f, a meromorphic
function on X, so (a) )~(co ) = divisor of f = 0.
163
Thus these divisors (u)) all lie in the same divisor class K , called the canonical divisor class on X. The set Z
Definition 3.9:
2D
of divisor classes D such that
B Kx
is called the set of theta characteristics of X. Z
Note that
is a principal homogeneous space under (Pic X) ~, the
group of 2-torsion in Pic X
(i.e., V D 1 ,D ? € Z, there is a unique
E € (Pic X ) 2 such that D± = D 2 + E ) , hence card. L Moreover, all
(Pic X ) 2 =
= card. (Pic X)
D € (Pic x) o have degree 0, so
(Jac X ) 2 = | L Q / I ^
and both £,(Pic X ) 2 have 2
2g
=
(Z5/22Z) 2g
f
elements.
The main result of this appendix is: Theorem 3.10: ^ ( z ^ ) = 0.
G c <Eg/LQ
Let
We consider all translates e+0 g
by a point of CC /L^. a)
be the analytic subset defined by of the subset
0
Then:
The map
Di
> f l o c u s of p o i n t s for all P ,..,P
I(P 1 +_ # +P
,-D) \
c
CEg/L
Q
€ X
is an isomorphism set of translates e+0
I
— = — >
which are
I symmetric, i.e., invariant under J z »
> -z
164
b)
The map
locus of zeroes of\ (
n»> '
*
(
« n'
-
)
c
"
/L
n
is an isomorphism , 2g ~ 0 l a /zz 2g _ * _ _
/Set of translates e+0 which £ Iare symmetric
Proof:
Note that iJ'(-z) = ^(z) (this is immediate from the
formula defining i9- ) , so
0 itself is symmetric.
But V[ n „3(z) = 0
n is the translate of g
ftTT+n" € G /L
0
72
by ftn'+r)"- If n'»n" € J ^'
is of order 2 and a
translate of a symmetric subset
of a group by an element of order 2 is symmetric. translate of a symmetric subset
tnen
0
In fact, a
by an element a
is also
symmetric if and only if 0 is invariant by translation by 2a: -(0+a) = 0+a «=s> 0+2a = 0 . Thus to prove (b), we need to check that 0 ± 0+e, for all This is proven similarly to Lemma 1.7:
say
e € G
e ^ 0.
satisfies:
#<J) = 0 « = * #(z+e) = 0 and we must show
e e L^.
Consider
a
—
-:
the zeroes of
4G)
numerator and denominator cancel out*, hence this is a nowhere zero holomorphic function on (E .
If f(z) is its logarithm, we have
Note that by (3.6), u(z) = 0 is an irreducible analytic subset of G g / L o an<3 that by (3.1), 1/ vanishes to 1st order on it: otherwise the f(z) in (3.1) would always have multiple roots and by (3.2), we can take the P.'s distinct in general position.
165
"& (z+e) = exp f (z) • # (z) . In t h i s s u b s t i t u t e
z+ftn+m f o r z and use t h e f u n c t i o n a l e q u a t i o n
$* on b o t h s i d e s .
We f i n d
exp(-iritnfin-2Tritn-(z+e)
for
that + f (z)) . 1^ (z)
= exp (f (z+fin+m) - iri nftn-27ri nz) v ( z ) hence f(z+ftn+m) - f(z) = -2-rri ne + 27ii • (integer)
(*)
for all z € (Cg.
Therefore, 3f/3z. is invariant by z I > z+ftn+m,
i.e., is a holomorphic function on (Cg/L0.
But then 3f/3z. must be
a constant, hence f is a linear function.
If
f (z) = c +2-rri c*z, o
then (*) says ^fi-n = -Si-e + (integer) , c-m
= (integer),
By the 2nd formula,
e
all n € ZZg all in € Z g .
c £ Z g , hence by 1st
=
-fi-c + (integral vector)
e
V
This proves (b). To prove (a), we use the following consequence of the RiemannRoch theorem:
For all divisors E of degree g-1,
166
3P
l'"'Pg-l€
X S - t
3Q
'
*=- p i Therefore i f
l'"'°g-i e
Kx-E .
X S t
' *
I Q.
D € Z, V P , , * • / ? , € X, 3 Q , , « * , Q , 6 X such t h a t l g-i l g"*i 2D - ( P ^ - . + P ^ )
E
Q^.'+Q^
K P ,1+ " + P „g-l ,-D) = »I(Q11+ --+Q„ ,-D). g-l This proves the symmetry of the locus in (a). By (3.6), this locus is a translate of 0.
Finally, Z
has 2g elements in it and by
part (b), there are 2g symmetric (0+e)'s.
This proves (a). QED
Corollary — 3.11: Proof: and 3.10:
A = I(Do-(g-l)P o) for some D o € Z.
Let D € Z map under (a) to 0 itself.
Compare 3.6
it follows that
< locus of pts I(P +--+P
,-D )[= -{locus of pts I (P +--+P ^ - ( g - D P
Therefore translation by
X+I((g-l)P -D ) carries o o
hence is 0. Corollary 3.12:
0 to itself,
QED Let D € Z and i^l) € -^Z2q/7Z2q
to the same symmetric translate of
0 in (3.10).
Then
correspond
)-l\
167
(^[n,,l] (0,ft)=o ^*=^(i3'(fin,+n,,,fi)=o) < = > (3ie1,
Proof:
• • /?
^
X
such that>
Immediate from 3.10.
The set of theta characteristics has an important 2-valued form defined on it: Definition
3.13:
For all
£ = (JJ,') € |s 2 g /2Z 2 g , define
e*
^
^ .
c = ([!), n = (Jjl) € |z 2g /2Z 2g , define
e2(Cfn) - (-l)4 . We want to think of e* as the exponential of a quadratic form on (E/2E)
g
with values in 2L/27Z, and of e 2 as the exponential of a
skew-symmetric form (2/2E) gx(2Z/2ZZ) g
>S/2ZZ.
Since +1 = -1
in 7L/27L, a skew-symmetric form is also symmetric and these 2 are related by:
The importance of e* rests on Proposition 3.14:
For a l l
£ € jffi 2 g /ZZ 2 g
^ [ • r l l (-z,fl) = e^(c)-l9-[^,1] ( z , f i ) .
168
Proof:
This is an easy calculation:
iS'lJij (-z,B) ^
=
I e x p t T T i S n + c ' W n + c ' ^ T r i ^ n + c ' ) (-z + c " ) l n € 2Zg
I exp[?ri m€ ZZ-
(m+c 1 ) ft (m+£ ' )+2TTI• t ,(m+c ' ) ( z - c " ) ] if
m = -n - 2c1
= e x p ( 4 7 r i t c , . C , , ) - 1 > [ ^ J (z,ft) .
Corollary 3.15:
For all
QED
c € |s2g/S2g
e # (c) = +1 <=
^the does N /tiie divisor aivibur $^[L£ ] c,(z,ft)=0 J vz/«;=u auesv «« (not contain 0, or else has J ^a point of even mult, at 0 '
/0 /U does not contain x aoes not contdinv f Qn'+n" or else ) ^has a point of ' even mult, there
e * U ) = -1 * = >
/the divisor *&[£,) (z,ft)=0 v f contains 0 and has a point J x of odd mult, at 0
/0 contains ftn'+n" (and has a point ^of odd mult, there
Here a divisor D on a complex manifold is a locus defined by one equation f(z_, #, ,z ) = 0 in local coordinates, and 0 is a point of multiplicity k if all terms
k k l k2 n (az] z 2 ---z ) in f of total degree
k1 +••+ k
< k
vanish while at least one term in f of total degree
k_ +••+ k I n
= k
does not vanish.
The Corollary 2 comes from the fact
that if f(-z) = f(z), all terms in f have even total degree, while if f(-z) = -f(z), all terms in f have odd total degree. Corollary shows that 0 order 2:
The
always passes through all odd points of
i.e., fin'+11" € yL Q /L 0
such that e^(n) = - 1 . One can
169
count how many points of order 2 are even and how many odd, by induction on g.
Thus they divide up like this: even pts
odd pts
g=l:
4
=
3
+
1
g=2:
16
=
10
+
6
g=3:
64
=
36
+
28
2 2g
_ 2g-1(2g+l) + 2
general genus:
This is a nice exercise.
^
For large g, there are nearly half
of each kind. Using the Theorem, we likewise divide up odd parts: I division of rest:
U£_.
The middle lines in (3.15) show that this
E does not depend on the choice of A.,B. and all the
it depends merely on the multiplicity at 0 of the locus of
points I(P.+---+P follows that I_
£ into even and
i-D) in Jax X.
D E P . + ---+P _,
In particular, if
for some P. €X.
D € E _ , it
In fact,
E + and
have a simple meaning in terms of the function theory on X,
which we describe without proofs.
This depends on a further
theorem of Riemann which complements (3.6): Theorem 3.16:
For all
P, , I
,P
.. € X, let
g _x
e =
A - V L
co. j p
Let
o
j£(IP.) be the vector space of meromorphic functions on X with
at most simple poles at P.,•••,P _, (or higher poles if several P.'s coincide, the order of pole bounded by the multiplicity of P . ) . Then g-1 /multiplicity of the zero of x dim im # J *i> = ( #$) at t = 2
170
Corollary 3.17:
For all D € Z
D
<*=> € Z +
( either 2 P.,--,P ., € x such that D = Pn + -«+P _\ i 1 g-1 — 1 g-11 x or if such P. exist, dim £(P.+«»+P ,) is even ' l l Q-l
D
e z
/ 3 P. ,««,P n 6 X such that D E P.+..+P \ I i g;i i g-i) x and dim ^(P-^-'+P _±) is odd '
For "almost all" X, it can be shown that in fact if D € Z , the D € Z _ , ^ (P +««+P
P.'s don't exist and if
) contains only
constants, hence is 1-dimensional. Corollary 3.18:
D
and hence
Z
can be determined from the function
theory of X by the property: V E (*)
1(E) = ftn•+n", n',n" € | s g .
(Pic X ) 2 , let
€
dim
Proof:
^ (DQ+E)
=
4tri' -n" (mod
Then
2) .
In the bijections of 3.10, we have: (D
+E)
<
>
zeroes of
o
VHn„](z) n
so we have just expressed in (*) that even and odd elements should correspond.
This characterizes D 4t(n'+a')•(n"+a")
for all
n',n"; then one sees that
because if =
a 1 ,a" € y ^ g
and
4 t n'-n" (mod 2)
a',a" € S g
in fact.
QED
171
§4.
Siegel's Symplectic Geometry. The other direction in which the theory of theta functions
develops is the analysis of iJ(z,Q), esp. 1^(0,ft), as a function of ft. Before describing these results, however, we must understand the Siegel upper half space ^v
better.
It is convenient to view ^jv
in several ways, e.g., both as an explicit domain in (Cg g a
coordinate-free abstract way too.
*
and in
We base our analysis on a
very useful elementary lemma in linear algebra: Lemma 4.1:
A: JR g x n
Let
g
> JR
be the skew-symmetric
form t
A ( ( x 1 , x 2 ) , (y 1 ,y 2 )) = t x 1 ' y 2 Then the following data on M a)
g
*2'Yi'
are equivalent:
> JR g (i.e., a linear map a complex structure J: JR g 2 with J = -I) such that A = Im H, K a positive definite Hermitian form for this complex structure.
(The existence
of H is equivalent to:
b)
A(Jx,Jy) = A(x,y)
all
x,y € lR2g
A(Jx,x) > 0
aJLL
x € 3R 2g - (0)),
a homomorphism
i: 2Z g
> V, V a complex vector space,
plus a positive definite Hermitian form on V such that Im H(ix,iy) = A(x,y), c)
a g-dimensional complex subspace
P c I
g
such that
A^tx.y) = 0, all x,y € P iA (x,x) < 0, all x G P-(0) (A
= complex linear extension of A ) ,
172
d)
a
g x g complex symmetric matrix ft with Im ft positive
definite. The links between these data are: a — > a H(x,y) = A(Jx,y) + iA(x,y) S 2 g c (JR2g, J) = V
a —» b
i:
a —> c
P = locus of points ix - Jx
b —> a
i induces by structure on
® JR: 1R g 3R
> V, hence a complex
g
b —> c
i induces by ® (E:
b —> d
Coordinatize V so that ife+k) k
unit vector.
i (k c —> a
^> v with kernel P
Then k
unit vector) =
column of ft =
unit vector).
2c t h e c o m p l e x s t r u c t u r e on 1R -' comes 2
3R ? and i
c
(E 2g
ft
from
(E2g/p
>>
i^ 2Z 2 g c (E 2g
c -> b c —> d
(E "
i s d e f i n e d by
(E 2 g /P
»
(0 , • • • . 1 , • • • , 0 ; - f t . , , - f t . 0 , • * - , - f t . J € P .1-n
.
1 i.
1z
i1-11 spot d —> c
P = s e t of p o i n t s
d —> a
t h e c o m p l e x s t r u c t u r e on 3R requiring
(x, ,-ftx.. )
x = ftxn + x?
H((0,x2), (0,y2)) d —> b
We may s e t
g
i s i n d u c e d by
t o be complex =
t
x 2 - (Im
V = (Eg i (n1,n2)
=
Qn1 + n 2 .
coordinates
ft)~1-y2
.
ly
173
The reader is advised to study this lemma until the different facets of this structure are quite familiar.
The proofs of the
equivalences are easy. It is clear from a, b or c that the symplectic group Sp(2g,H)
acts on this set of equivalent data.
Y 6 Sp(2g,]R) , then
y
carries > Jf = yJy" 1
J i
> (V,i')
(V,i) i
i 1 = ioy~
where
> P 1 = y(P) .
P l
Now we can write such a
Thus if
y
y = (r
n
as:
) /
A,6,0,0
nxn
real matrices
such that:
tJs)t;3Gs)- (-;;). Therefore
»• - *»> - {(c ? ) u ; ) | x x e a*} r/Ax,-Bfix.vI
n
g
^
- {(cx^)|*i *' }•
(
y
\
n1 = (Dfi-c) (-B a +A)"" 1 . This defines therefore an action of Sp(2g,]R) on & , every symplectic matrix ( r
n
j
.
Note that
acts by a bi-holomorphic map
174
We may make the formula look more familiar if we compose with the automorphism
>
y i
y
of
Sp(2,]R).
In fact
so
v 1 • (.; ;>e -s) V-B
A /
Thus after composing with this automorphism of Sp(2g,3R), the action is: ft I
> ft' =
(Aft+B) ( C f l + D ) " 1 .
In particular, this shows that (Aft+B)(Cft+D)
is symmetric.
Another way of phrasing the result is this:
Lemma 4.2: to —
Q
Let
in (d) . —
i: 2Z g Let
> V
be data as in (b) above corresponding
e1 .. , • • 2g • ,e~
e! = l e' = l+g
>A..e. + ^ i] ] )C. .e. + i] ]
L
be a new symplectic basis of
7L g .
be the unit vectors in 2Z ^ and let
>B..e._, ID D+g
L
Y D . .e.^
ID D+<3 Let
i'
be the composition
175
2Z2g
> E2g —±>
V
e. i1
Then
i':
E2g
> V
corresponds to (Aft+B) (Cft+D) "~1.
We reprove this to get another handle on the situation: we may let V = CCg and let i be
i(n1,n2) = fin1+n2. Then i(e!), = 1 k
T A. .Q, . + B.. h 13 kj lk
= (A£HB)ik
by the symmetry of
Q . Likewise, i(e! + g ) k = ( O H D ) i k .
Therefore i' (n lf n 2 ) =
J(n1)i(Afl+B)ik + (n 2 > ± (CG+D) iR .
We change the identification of V with (Cg by composing with the automorphism
(z l f --.,z g ) I
(Cft+D)~ :
> ( I (Cfl+D)"^,---, I (Cft+D)"^).
176
Composing, we
find
i,(S1,n2)k
^(n1)i[(AQ+B) ( C ^ + D ) ~ 1 ] i k
=
Because of the symmetry of
(Afi+B)(CQ+D)
+
(n2)k .
, proven a b o v e , this can
be w r i t t e n
i* (n ,n 2 ) =
(Afi+B) (Cft+D)
-n
QED
+ n .
Here is a diagram that summarizes in terms of m a t r i c e s
the
situation relating i and i' constructed via ft,fif:
(x1,x2)
J2x1 + x 2
H
7Z 2g
D -C
*$l)
(4 3
K
-B
' (Cft+D)
A}
t
(C^+D)
= -ft'C+A
2g
zz
(x1,x2)
fi' =
9,' x 1 +x~
I-
(Aft+B)(Cft+D)
-1- te°)
€ Sp(2g,ffi)
f We verify
a)
are inverse to each other f
J being symplectic
means
because
177
t
(Cfi+D)""1 = -8'C+A
b) that
because
t
fiI = Q'
so this means
t
(Cfi+D)""1 = -t(Cfi+D)"1-t(Afi+B).C + A
which reduces to fc
fc
AC =
CA and
fc
DA - t B C = I
g which in turn follow immediately from (*). The commutativity is straightforward. J We note for later use the following consequence: Proposition 4.4: G9
x
hjt
The group Sp(2g,]R)
acts on the space
by the maps: (z,fl)
Proof:
I
(t(Cfi+D)~1-z/ (Afl+B) (Cft+D)"1).
>
For elements of Sp(2g,2Z) , the fact that this is a
group action follows from diagram diagrams for i and i', i
1
and i".
(4.3), by putting together the In fact, the same diagram holds
if 2Z is replaced by 3R, by the same argument, so we need not restrict to elements of Sp(2g,E).
QED
We can use the action of Sp(2g,JR) group-theoretic definition of ^ Proposition 4.5. stabilizer of
to
il
Sp(2g,]R)
on
to give a purely
as a coset space.
acts transitively on X\,
In fact: and the
is isomorphic to U(g,(E), embedded in
Sp(2g,]R)
178
/ Re X > ( V -Im X
X »
Thus
Im X^ Re XJ
= Sp(2g,]R)/ U(g,(E) .
Proof:
The transitivity is checked quickly by introducing
2 elementary subgroups in Sp(2g,]R): ,K
(I)
Y=
(II)
y
In fact,
=
y
0
v
U V" 1 )' ,1
I ^0
A € GL
Bx
B
) / I'
any
g *g
real symmetric matrix.
as in I acts by (x.. ,x?) i > (Ax1 , A
A( ( Y X 1 , Y X 2 ) , (Yy1/YY2) )
=
* ( Y ^ ) " (YY 2 ) "
x2) , and
(Y* 2 )-fyy^
= t(Ax1)-(tA"1y2) -
t
(tA~1x2).(AYl)
t 1'Y2 " X2"Y1 = A((xlfx2),(y1,y2)), t
wh ile
Y
X
as in II acts by (x..,x2) i >
(x + B x 2 , x 2 ) , and
A ( ( Y X l f Y x 2 ) , (YyJL/YY2)) = t ( x 1 + B x 2 ) . y 2 - t x 2 . ( y 1 + B y 2 =
tx
i'Y2
=
+
ti
2tBy2
~
t x
A((x1,x2),(Y1*Y2))•
Acting on 4£L , these maps carry ^g' and
t x
ft I
>
Afl^A
I
>
ft+B.
2
# y
l
~
x
2By2
179
Hence together they carry il
to
i(A« A) + B.
positive symmetric matrix can be written element of ^v
Since any
A* A, this gives any
. This proves transitivity.
The stabilizer of a point of *v in version (a), lemma 4.1.
Here
(JR
is most easily identified ,J) is a complex vector
space and A = Im H, H positive definite Hermitian. automorphisms of TR ^
The symplectic
that also commute with J are the complex-
linear automorphisms of (3R g ,J) . Since A = Im H, these must preserve H. g
(]R ,J,H) .
So this is the group of unitary automorphisms of In particular, if ft = il , then z, = i(x,), + ^x2^k'
1 <_ k <_ g, are complex coordinates on JR g , so:
Then V
C D7
X
C D'
if and only if A = D, B = -C and as
( A ^-B
B\ /n,\ 1
) F)
A
7
^nj
n +Bn n = (/ AAn,+Bn l 2\
^-Bn1+An27
,
it carries in,+n2 to i(An,+Bn2) + (-Bn,+An2) = (A+iB) (in..+n«) . This proves that the stabilizer of il is as claimed. g
OFD -—
One can now proceed to build up a detailed "symplectic geometry'• on -^y : first one defines a metric on Jhr invariant by the action of
Sp(2g,2R).
which is
In this metric, one can
describe geodesies, compute curvature, investigate totally geodesic subspaces, etc.
A good reference is
180 Siegel, Symplectic Geometry, Academic Press, 1964. This is a generalization of the non-Euclidean metric on the upper-half plane K, which is SL(2, 1R) - invariant, hence has constant curvature.
Our interest however is in the action of
the subgroup Sp(2g,2Z) of Sp(2g,]R) on & a)
. Because
Sp(2g,ffi) c: Sp(2g,JR) is discrete, i.e., 3 a neighborhood U =
{X
X..-6..
of the identity
<
1}
meeting Sp(2g,ffi) only in
Iy
,
zg and b)
The stabilizer of a point of -wy
c)
Sp(2g,S) acts discontinuously on -far : S
= {y € Sp(2g,ffi) yx = x}
is a compact Lie group, (1) V x € J2v
is finite and
3
a
neighborhood U x of x which is stable under Sx such that J V Y € Sp(2g,2Z) :
Y U X n ux f 0 and (2) , v x,y € yqra
<*=>
e Y
sx .
such that x ^ YY
Y E Sp(2g,ffi), ^ neighborhoods U ,U U flYU
= 0, all
Proof of c : TT:
Because
Sp(2g,3R)
f
° r any
of x,y such that
Y € Sp(2g,ZZ). U(g,€) is compact, > Sp (2g^)/U (g,(C) -^—> £v
is proper, i.e., the inverse image of a compact set is compact (in fact, in this case, we get a section by considering
Y' S of
181
form (0
)( Q A -l)
of 4.5-
hence as a topological space
;
A positive definite symmetric
-see proof
Sp(2g,]R) = U(g,(C) x [Sp (2g, 3R) / U (g,(E) ] .) Let TT
be a relatively compact neighborhood of x: (U
then
) is compact, hence
is compact.
Hence the intersection
and discrete, hence is finite.
W fl Sp(2g,ffi) is compact
But this is the same as
F = { Y € Sp(2g,ffi) jy(U^)n U^ yx ? x.
Now F = S UF , where if y e F , then let U ,V
? 0} . For all
be disjoint open neighborhoods of x, x, yx.
(n
X
One checks that
the set
y e F. ,
Let
(u n Y"\))nu^ .
Y€F •vfv X
Y
(2) (2) yU' n U ^0
Y
/
only if
x
yx = x.
The sought-for
Ux can be taken to be
u(3) =
n
yes x
yd2)
.
This proves (1). We leave the proof of (2) to the reader.
One then considers the orbit space
a
g
= ^- g /S P (2g,S)
QED
182
By definition, a subset of C? A
^Q
g
is open.
By (c), each of the local quotients U /S is an open x x
subset of Ot
and the induced topology is immediately seen to be
the quotient topology. in CEg g
'
Each of these local pieces is an open set
modulo a finite group of analytic automorphisms,
hence it is an analytic space. . Q
structure on Q g
—
is open if its inverse image in
Together these give an analytic
is called the Siegel modular variety g
i.
2
a very special but most interesting space.
that
(32
(c2) tells us
is a Hausdorff space.
We conclude this section by tying
(JL
together with the
theory of projectively embeddable complex tori, generalizing the ideas of Ch. I, §12»
We need some preliminaries on cohomology.
For our purposes, we will use DeRham cohomology:
on any oriented
compact manifold M, k ix, ™\
TT (M,3R) H
~ =
space of2 closed exterior k-forms -r -r i i—z space of exact exterior k-forms
—*-
and in this isomorphism, the following subspaces correspond:
H k (M,E)/ torsion
/space of closed k-forms co with \ \integral periods around all k-cyclesJ exact forms
In particular, if V is a real vector space and lattice, then for any k elements k-cycle
X , » * » ,\.
L c V
is a
€ L, we get a
a(A.,•••,A, ) on V/L consisting of the image of the
k-dimensional cube:
183
r k < I li=l
t.A. 1
x
i € V O £ t . I
Then taking periods on , HK(V/L,E)
£ l ,
for a l l
1 i^ J
>
V/L.
a (A,,•••,A, ) , we get an isomorphism
-^->
r multi-linear alternating forms^ | | kjg > 7L A: Lx- • • xL
On the other hand, for complex projective space P :
H k (P n ,ZZ) =
,0 if k odd or k > 2n [ 2Z if k even, 0 < k < 2n
where if k is even the identification is given by integrating over linear subspace L c p , dim L = k/2.
For our purposes, we need
only to have an expression for a 2-form n
the generator of H (P ,ZZ).
42*
I i
(where zn,--»,z in (C
The simplest is:
3 a_
are homogeneous coordinates:
with the canonical map IT!
representing
log f X | z , | 2 ) dz. A d z . x a Z i az. Vi£o' l ! /
a
jas0
u on P
and what we have written here is
> IP" TT w) .
i.e., coordinates
184
Now if fl €#v , let
l: be data
ZZ
2g
(b) associated to
Q as in Lemma 4.1.
g
may set
Qn-,+n«.
V = (E , i(n ,n~)
lattice LQ,
Explicitly, we
Then Image (i) is the
and V/i(2Z 2 g )
*fi •
Moreover, the alternating form
A:
ZZ
g
* ZZ
g
ZZ
gives u s ,
as we have just explained, a class [A] €
IT(X
Note that if an element
,E) .
y € Sp(2q, ZZ) acts on fi , then
is unchanged and the alternating form on i(ZZ -)
i(2Z
g
)
is unchanged.
This gives us a well-defined m a p :
V»:
a,
set of pairs
(X,[A]),J
X a complex torus, [A] € H 2 (X,ZZ)
modulo holomorphic isomorphisms preserving the cohomology classes
We can easily prove that y is injective: let -> be an isomorphism such that
X<|> ([A ? ]) = [A..].
Write X~
= (EJ/L(
Q.
and lift cf> to an isomorphism of universal covering spaces, and of fundamental groups:
185
) : CC J
where >(z+a) = £(z) + 4>*(a),
(*)
all
a e L^
Then for all i, 34) dz.
8z. (z)
(z+a)
1
Hence -—
is a holomorphic function on (E^, periodic with respect
to L n , hence bounded.
Then -^—
al..
must be a constant, so
$
is
dZ .
linear plus a constant.
We may throw out the constant without
affecting (*) . Note too that
<J>* ([AJ ) = [A ]
implies
A 2 ((f>*x,<{)*y) = A x (x,y) Thus we have isomorphisms: S2g
^
> <Eg
2
ti
> <,*
Z5 9 where
* is symplectic. g
This means exactly that an element of
Sp(2g,S ) carries (CC ,i) to ((Eg,i'), hence Q, to is injective. definition:
ft'
Thus
To describe the image, we make the following
y
186
Definition 4.6.
Let
2 be a complex torus, [A] G H (X,Z) . Then
X
[A] is a principal polarization of X ^f a)
Expressed as a skew-symmetric integral matrix, det[A] = 1,
b)
there is a holomorphic embedding f:
X*
such that if
> pn
[co] 6 H (P
,2Z) is the positive generator,
then f*[w] = N.[A], some Theorem 4.7.
N >_ 1. The map
\x
is bijective between
Q
and the set
of isomorphism classes of principally polarized complex tori (X,[A]).
Having described the projective embeddings of X Q
in §1,
we can easily show that [A] is a principal polarization on X . We shall omit the proof that all principally polarized tori are isomorphic to an X : it is a variant of Theorem 1.3, part (3) and, like that result, is proven in the author's book Abelian Varieties, §§2 and 3.
P
n
To prove [A] is a principal polarization, we embed X n 2g " by the method of §1:
zt
> (
, # M L iJ
(n?,ft),
)
in
187
where [,1 ]
run over cosets of E
g
in — 7L g
and n a fixed
(This is the embedding of §1 for (Eg/nL~
integer, n >_ 2. adapted to <Eg/LQ
by the remark at the end of §1.)
We have a
diagram of maps: 2g •(0)
->
x
n
Thus
c
~^
-> i^-i
TT*f*[co] is represented by the 2-form on (C^ 11 =
i o g ( l l ^ [ ^ ] ( n 2 ^ ) | 2 ) dz iA dz.
2? I J^¥T '3
We n e e d o n l y c o m p u t e t h e p e r i o d s of n Lemma 4 . 8 .
n a(fin,+n 2 ,ftm..+m J
Proof:
The f u n c t i o n a l
to complete the
proof.
2 _ , t _ __ _ t „ n • ( n , - m1U0 n »mn). "2 " V 1 2
equation for
^L
1
]
shows
that
i
,, a . tf^1]
t t 2 l o / ~ 27T n.. . Imftn.,-4TT n , (Im z K n . a. ,2 iz)| 2 (Mz+fir^+r^)) = (e ) ^[b1](n*
hence (*)
log
H &[.
] (n(z+fln,+n?) )
=
2irn
( n , • Im
ft-n1
+2n,-Imz)
I
+ log
a. l\ VL2-] b i i.
12 (nz)
188
We set
< = 5? I air io g (lj^[^ i ]( n S)| 2 )dz i Then d£ = -n
.
(the -~— terms cancel out in d £ ) , and writing I
Im z = (z-z)/2i, we find:
afr< Im
z)
= -h
hence differentiating (*): 2 ,t C (z + fin + n 2 ) =-n ( n..-dz) + £(z) Now t h e r e c t a n g l e
a
is
fi mi+m2
fi(ni+mi)+(ii2+m2)
fini-hi2 so by G r e e n ' s
theorem
J n = J c-c> = -C> =
= n
J(-0 +
2 Jf t
J (-c)
,
__ , 2 Jf
t n,
m.*dz + n
= n 2 [ t m 1 * (-ftn1-n2)
+
2 ,t t x = n ( n1-m2~ n 2 ' m i ^ *
t
• dz
n1•(fln^+n^)] QF.D
189
§5. 17 as a modular form. We want to consider now the dependence of the function v(z,fi) on ft. As in the one-variable theory, the fundamental fact is a functional equation for \r on both the variables z and ft . tricky 8
for the action of Sp(2g,2Z)
As before, there is a rather
root of 1 in this equation.
Without working this out,
we can state the functional equation as:
^(Ncft+D)"" 1 -?, (Aft+B) (Cft+D)""1) (5.1) = C •det(Cft+D)1/ -exptTTi^. (Cft+D) 1 C- z] • & (z ,ft)
where
£ Y = 1, and
Y = {* I) €
Sp(2g,5Z)
satisfies diagonal ( AC) even diagonal ( BD) even. This set of elements of Sp(2g,ffi) may be described as those such that, modulo 2,
y
y preserves the orthogonal form
Q(nlfn2) = tn1.n2 G as well as the alternating form A: this is a group, which we call
(E/2E)
see the Appendix.
In particular
T1 ~ following Igusa.
In the appendix, a set of generators of using these, we may prove (5.1) in 4 steps:
rl
is 2
found and
190
a)
Showing that if (5.1) holds for holds for
Y-,/Y2
G
Sp(2g,2Z) , then it
Y-,Y2'
b)
verifying (5.1) for
£
c)
verifying (5.1) for
(Q
d)
verifying (5.1) for
(
t °__ 1 V
A 6 GL(g,S)
J, B symmetric, even diagonal, J
We may check (a) by a direct matrix computation, but perhaps a more interesting way is to reformulate (5.1) in terms of a closely related function
v
a
which is then to be shown to be
Sp (2g, 7L) - invariant. Since invariance by
y-,
an
^
a
implies invariance by
Y-iY?' ( ) becomes obvious.
of this approach is that the new function importance due to its invariance:
Yo
obviously
The advantage
a
XT must have a certain
this will be explored in
Chapter IV.
To prove (5.1) for all z, it certainly suffices to do so for z = ftn,+n2, n. € Q g .
So as a first step, we substitute
Qn,+n 0 for z and rewrite (5.1) for 1 z
\9" [ ] (0,ft) . n~
We claim that
(5.1) is equivalent to: Dn -Cn #
X
(0, (Aft+B) (Cft+D) X )
L-Rn+An. -Bn,+An2 -I (5.2)
= C ^ d e t f O H D ) ^ - exp (-TriSi^ t BD-n 1 +2TTi t n 1 t BCn 2 -Tri t n 2 t ACn 2 W [ n 1 ] (0,ft) J, n o t on
n^,n2,Q.)
191
This calculation goes like this:
you substitute ftn..+n2 for z,
use the fact that t
(C^+D)'"1(fin1+n2) =
^ , (Dn 1 -Cn 2 ) + (-Br^+Ar^)
by (4.3), and then use the definition of &[
](0,fi). (5.2) follows 2 except that one has a messy exponential factor, viz. exp of n
Tri( t n 1 ^+ t n 2 ) (CQ+D)" 1 C(^n 1 +n 2 ) r,
•t -
in
0
.t
n, ftn.. - 2TTI
n,n 2
+ 7Ti( t n 1 t D- t n 2 t C) (Aft+B) (Cft + D) ~1 (Dr^-Crij) + 2iTi( t n 1 t D- t n 2 t C) • (-Bn1+An2) . In this, you separate the 4 terms TT± n2*(
)n , Tri n2-(
)n 2
iri n 1 . (
)n. , ni
)n 2 '
n,«(
and simplify each one, using the basic
A B facts on (r n ) expressing that it is symplectic: t
t
D-A - t B-C = I g D-B = t B-D, ^X-A = ^ - C
For example, t a k e t h e f i r s t .
It
TTi t n 1 { QC^+^)~1((CQ+D)-D)~ TTi t n 1 {^-fi(Cfi+D)~ 1 -D
.
is Q + t D(A^+B) (C^+D) _ 1 D -
- ft+ t D(Afi+B) (C^+D)" 1 D -
TTi t n 1 { ( - Q+ fcDAQ + fcDB) (Cfi+D) " 1 D 7 r i t n 1 { ( t B-Cfi + t BD) (CQ+D)" 1 D T r i ^ {-tBD}-n The o t h e r s a r e
similar.
.
2tDB}n,
2tBD}n1
2tDB}n1
2tDB}n1
192
However, if we examine the above calculation we see that it would have come out even simpler if, instead of n, l H n ] (0,fl) = exp[iri n1^n1+27rin1n2] i9,(fin1+n2,fi) we use a modified v that we will call 17 a :
Written out,
(£/30
^tn1]^) = J
If we use v a
t t gexp[7Ti (n+n1)^(n+n1)+2TTi (n+^)-n2]
instead of v , the "messy exponential factor" is
rather: 7ri(tn1^+tn2) {CQ +D) ~ 1 C (ftn-^n^ -
.tn 0Qn - T.t Ti n 1 n 2 1
TTI
+ 7ri(tn1tD-tn2tC) (Aft+B) (Cft+D) ~ 1 (Dr^-Cr^) + iTi(tn1tD-tn2tC) . (-Bn1+An2) , which, treated as before, turns out to vanish identically! Thus the functional equation becomes: Dn un,1 -- C n i
(5.4)
[
^ L n-Bry . + A n J ((Afl+B)(OHD)
X
)
C det(OHD)1/2-£a[
r
n,
- 1 ](fi).
"2
193
How about the factor in a sense.
det(CJHD)
1/2
?
This too can be "eliminated*
Let w = t(CQ+D)
1
.z.
Then -dz,A-"'Adz . 1 g
dw, A--"Adw = det(CSHD) 1 g Hence (5.4) says: Dn^Cn,
|((Aft+-B) (Cft+D) L
) . /dw A" ' 'Adw
-Rn.+An_ -Bn 1 +An 2JJ
g
a 1, e . #L r[n l(ft). /d Z l A...Adz c 2
Proposition 5.5: a) on ffi g ,
by
b) on -b, c) on
Let
, by
(Eg,
by
Sp(2g,ffi)
(n ,n 2 ) i ft z
i •
>
act as follows: (Dn..-Cn2 ,-Bn,+An 2 )
> (Aft+B) (Cft+D)""1 >t
{CQ+D)"1-z.
Then the functional equation for v asserts that, up to an 8
root
of 1, #a[
](fl). /d Z;L A---Adz
n
g
2 is invariant under
r..
?
<= S p ( 2 g , E ) .
Next, to prove the functional equation, we must consider the 3 generators of
ri
9.
194
Case I :
Y=
,A 0 (Q t A ~ l ) ,
A G GL(g,2Z).
Then (5.1) r e d u c e s t o : z, d e t ( A ) " 1 / 2
^ ( A ? , A.fi^A) = which i s immediate , w i t h l9(Az, A^A) =
I
£ a 4
&{z,a)
r o o t of l , i n
fact:
e x p t T r i ^ A ^ A - n + 27ri t n-A«z]
=
I exp[7iit(tAn)fi(tAn) n€Eg
+ 2Trit(tAn)z]
=
I e x p t i r i mftm + 27ri mz] m£Z g
= ^(z,ft) and since det A = +1, det(A) Case II:
I B y - (0 T ) /
B
'
is a 4
root of 1.
symmetric, even diagonal.
Then (5.1) reduces to l9-(z,ft+B) = C- #(z,ft) . Here we may take £ = +1, because in fact t9*(z,ft+B) =
J exp[iri t n(fi+B)n + 27ri t nz] n€2Zg
I e x p t u i nBn]-exp[TTi nftn + 2TT1 nz]
because
nBn i s always an even i n t e g e r .
195
y = (j
Case III;
Q)
.
Then (5.1) reduces to:
# (ft 1 z,~^" 1 ) =
(5.6)
c-det(ft)1/2-exp[TTitz.ft"1z] • ^(zfft) . 1/2
0 ^"'^
c«det(ft) '
In fact, this is true with
replaced by det(")
where the branch of the square root is used which has positive value when ft is pure imaginary. We could prove (5.6) along the lines of the proof of Chapter I, but instead we will give a different proof based on the Poisson Summation Formula: (5.7)
f a smooth function on ]Rg, going to zero fast enough at
°° t
f its Fourier transform: f(U
=
•i
I
f(x)
exp(2TTitx.£)dx1---dxc
then
l
n€2y
f(n) =
I
f(n).
n€Sy
We a p p l y t h i s w i t h f (x) = e x p ( 7 r i xftx + 2-rri x * z ) .
I
n€ZZ J
Then
f(n) = # ( z , f t ) .
To c a l c u l a t e
f,
we n e e d t h e f o l l o w i n g
Lemma 5 . 8 :
F o r a l l ft €&. , z € (E g ,
integral:
e x p ( i r i xftx + 2iri x . z ) d x , « « « d x
= (det
ft/i)
e x p ( - 7 r i zft
z) ,
196
Proof:
Rewrite the integral as
e x p ( - T r i zft
z)
exp(Tri
(x+ft
z)
ft(x+ft
z))dx,,•••,dx
As both sides of the equality to be proved are holomorphic in ft and z, it suffices to prove they are equal when ft and z are pure imaginary.
Therefore, we may assume ft = i A-A
,
A real positive definite symmetric
z = iy
,
y
real.
Then the integral becomes:
exp(-7ritzft~1z) J
exp[-7rt(x+(tAA)~1y)tA«A
Replacing x by x+( AA)
(x+ (tAA) ^ y ) ] d x ^ • «dx
y, this is
exp(-Tri z ft z )
exp[-iT x A»A x]dx,««'dx . Kg
9
Substituting w = Ax, this becomes
e x p ( - i r i zft
t
z)
-1
e x p ( - i r i zft
= exp(-7ritzft
1
exp[-TT w w ] • ( d e t A)
_i t+• -1/2/-> z) ( d e t ^A-A) x / •
z) (det
ft/i)
1/2
.
g
9 n i=l
dw•.*••'
+00 +00 _ 22 f _7TWi e dw.
J
1
QED
197
We may now c a l c u l a t e f(£)
f:
e x p ( i r i xftx + 2iTi x - z ) e x p ( 2 i T i x« ^ ) d x 1 • • - d x ^
=
J
J a
**
= (det fl/i) 1/2exp(-7rit(z+C)«
1
(z+^)).
Therefore J f(n) = (det Q/i)~1/2exp(-7ritz«~1z) £ exp (-TTi^Q-1!! ~27ritn^"'1z) n€ZZg n€ZZg = (det fi/i)~1/2exp(~7ritzfi"1z) a9'(fi~1z,-^"1) (replacing n by -n in the sum in the last step).
This is (5.6) .
This completes the proof of the functional equation.
A
Corollary of our proof which is useful is that: (5.9)
If
y 6 r 4 , i.e.,
y = 1^
(mod 4 ) , then in the functional
equation, c = ±1Proof: In fact, by the Appendix,
T.
is contained in the group
generated by matrices
(o 2\)'
Gc i)'
B c
' symmetric.
For the first of these, the functional equation holds with C
= +1-
But
V-2B 1/ so the 8 (
_)
\I
root of unity
0/
.\0
lj[l
OJ
C involved in the functional equation for
cancels out, and C = ±1 in the equation for (_2R T^
^we
cannot say £ = +1 unless the appropriate branch of /det(C°.+D) is chosen).
QED
198
We now introduce the general concept of a modular form on 4^ Definition 5.10:
Let
r
c
Sp(2g,Z)
:
be a subgroup of finite index.
Then a modular form of weight k and level
r in g variables is a
holomorphic function f on Siegel's upper half-space Jky
such that
for all, Y - (c D ) € r we have f((Afl+B)(Cfi+D)"1) = det(Cft+D)k.f(fi).
If g s 1, we put an extra boundedness hypothesis on the behaviour of f at the "cusps".
If
g > 1, it turns out that this
boundedness is automatic (the "Koecher principle"):
for example,
f will be bounded in the open set of fi's such that
Im ft > c.I for some constant
c > 0.
modular form of level n.
If r =
r , then f is said to be a n
If g > 2, then by a result of Mennicke
(Math. Annalen, 159 (19 65), p. 115), any such subgroup
r , so a modular form of level
level n for some n. that -j) (0,ft)
r
r contains some
is a modular form of
The functional equation for -\) states then
is a modular form of weight 1 and level 4. More precisely,
if we introduce following Igusa the intermediate levels (n,2n) by C
Tor,
2n
r
r.
On
n,2n
C
r
~
n
where n is assumed even and v € r 0 ~ i f ' n, 2n Y -
(
C D> = X 2g
mod
n
199
and 2n divides the diagonals of B and C, then we prove:
Corollary 5.11:
Let n be even.
n,
Then for all n ,n2,m ,m
€ -2g,
m1
# L ± ] ((>,«)•#[,/] (o,n) n2
m2
2 2 is a modular form of weight 1, level (n ,2n ) . Proof:
This follows immediately from (5.2).
In fact, Igusa has shown that the ring of all modular forms 2
2
of level (n , 2n ) , of all integral weights, is just the integral closure of the subring generated by these thetas and that they have the same fraction field.
This is the final result in his book Theta
functions, Springer, 1972. It is an open problem, however, of considerable interest to understand exactly what subring of the ring of modular forms is generated by the thetas.
Geometrically, we can proceed
as in Chapter I and define a holomorphic mapping: P N "1
Mn*., vn ,2n )' B2> by
„a fli
>
(
,-£[ a](0,Q).&[ n 2
a m
a](0,fi), 2
)
1 1 where ( a),( a) runs through all sets of 4 elements in a system of n m 2 2 coset representatives of - S g modulo 2 g . The main result n geometrically is that this is an isomorphism of the analytic space -$yyg /L P
2 2iw ^ t h (n ,2n )
N~l
type.
a
" c ! u a s i"P r o J e c ' t i v e " variety, i.e., a subset of
defined by polynomial equations minus a smaller set of the same
200
Corollary 5.11 suggests that we extend the definition of modular forms to half-integral weights as follows: Definition 5.12.
Let r c r.
be a subgroup of finite index.
2
a modular form f of weight k € =50, and level
Then
r is a holomorphic
function f on JL^ such that f((Afl+B)(Cft+D)-1)
for all
=
f(fl)
(£ ®) € r.
With this definition, we can even extend 5.11 as follows:
Corollary 5.13:
For all
n
!' n 2
e
®9' £ e
z
'l 1
1
'
is a modular form of weight 1/2 for a suitable level Proof:
Consider 1
f(fl) = -#[ n
Substituting if
n
T.
n
i/ 9
e
] ( 0 ^ Q ) / / 9 [ Q ] (0,n) .
2
(Aft+B) (CSHD) ~
for
o, and u s i n g ( 5 . 2 ) , i t f o l l o w s
g
l / n 2 , n e v e n , and /
A
£Bv
€ l.-v ^ C DJ v fS then f ((Afl+B) (Cft+D)
-1)
=+f(fl).
that
201 The sign
e(
)
gives a homomorphism
^ ^ r Let
r be the kernel.
1
Then
- ^ - o]—>™ • r is a level ford I
1
(0,Afl).
QED
Another way to describe the situation is this: let 8(® ) =
§ vector space of functions f: Q g
>(C
such that for some k,£ >^ 1 if a $ ^2Zg
f(a) = 0
f(af£) = f (a) if £ € £ Z g (£ Af
is also called the space of Schwartz functions on the group A g , being the finite adeles). \9[f] {m)
=
Define
J f (n)exp(TTitn^n) n"GQg
Then f I
>$
[f] Ufl)
is a map w£:
S(Q g )
> {v .sp. of modular forms of wt. —, any level} .
The image is the same as the span of the modular forms A/[ 1 ](0,£^) / all n 1 ,n 2 €CD g
because
-$[f]
the characteristic function of w. form
becomes -$ [ni] a+Z
g
if f is taken to be
times the character defined by h
is known as "the Weil map associated to the 1-variable quadratic lx2".
202
Appendix to §5: Generators of Sp(2g,2Z) In the last section and in the next Chapter, we
need at
various places lemmas asserting that various subgroups of Sp(2g,Z) of finite index are generated by such and such elements. We group together all the results of this ilk that we need. is nothing very difficult in any of these.
There
First, the subgroups
we shall consider are: Tn =
{ y e Sp(2g,E)
y E I
and also an intermediate subgroup F
C
2
In fact, if
r
i 2 C ri
mod n}
r.. ~:
= S
P(2
Sp(2g,ffi) acts by reduction mod 2 on (S/2ZZ) g , it
preserves the skew-symmetric form A((x1,x2) , (y1,y2)) =
x 1 «y 2 -
^2'Y1
which, because the characteristic is 2, is also symmetric.
In
fact, over 7L/ 2TL , consider the quadratic form Q((x1,x2)) =
x
i* x 2'
Then A(x,y) =
Q(x+y) - Q(x) - Q(y)(mod 2).
Therefore, the orthogonal group over 2Z/2 2Z (the maps preserving Q) is a subgroup of the symplectic group over Z5/2ZZ (the maps preserving A ) ! r
i,2
=
Let {y
e
S
P ( 2 9 ^ ) | Q(YX) = 0(x) mod 2} .
203
We rest our sequence of generation assertions on one dealing with the fewest generators: Proposition A.l:
T cSp(2g,2Z) be the subgroup generated by
Let
the elements .1 2BX , I Ox „ _ . ^ . .. (0 -,-) , (2C j) / B * c integral, symmetric . Then where
r4 c r c r2-
In fact r
is the group
7
of (^ p)
4|A-I , 4|D-I , 2|B, 2|C. Proof:
We use Let d = (n,m).
Lemma A.2: Let n,m € 2Z, not both zero.
Then a
sequence of the elementary transformations (x,y)
i
>
(x±2y,y)
and
(x,y ± 2x)
carries (n,m) to either (d,0),(-d,0),(0,d),(0,-d) or (d,d). Proof:
Since everything preserves
divisibility by d, we Given (n,m),
may as well divide by d and prove this for d = 1. either |n| <|m|/ |n| > |m|
|m|.
If
(x,y) i so as to decrease |n|.
|n|= |m| . If 0^| n | < |m| , make the map
> (x,y+2x) or (x,y-2x)
(x,y) • so a s t o d e c r e a s e
or
| n | > | m | ^ 0 , make t h e map > (x+2y,y) o r
(x-2y,y)
If |n| = |m|, then n =
(n,m) = 1, |n| = |m| = 1.
and since
If (n,m) = (-1,1) or (1,-1), one of the
elementary transformations carries it to (1,1). we need 2 of them:
±m
If (n,m) = (-1,-1),
204
(-1,-1) . (x,y)
> (+1,+1)
> (-1,+1) I
I
> (x,y-2x) (x,y)
*
>(x+2y,y).
QED
To prove the Proposition, let y € f. Consider y(l,0,"',0;0,...,0)) = (a1,...,ag;b1,...,bg) . Here
4I1a,-l,a~,•••,a ,2 b.,•••,b and g.c.d. (a.. , • • • ,b ) = 1. 1 ' 2' g |1 g ^ 1 g y by a sequence
We shall follow
transformations (Q
),(,
VN-1""
&^ ,'»•,&
of elementary
) until
S 1 Y(l'0r---'0;0f-'0) = (1,0,...,0;0,...,0).
Note that we may have at our disposal the transformations: a.,b. » > a.±2b.,b. i
1 1
a. , b . I i'
i'
I 1
, other a, ,b. left alone
l
k
k
> a. ,b.±2a.
I
a.,a.,b.,b. i j i j a.,a.,b.,b. i j i D
l I
'
I
> a.±2b.,a.±2b. ,b., b . , i 3 D 1 1 3 > a.,a.,b.±2a.,b.±2a., 1 3' 1 3 3 1
"
"
We proceed in stages like this: Step I: Let d = ( a , , ^ ) . Apply (a1,b1) 1
Note that d is odd because a, is odd.
>(a^b^b^
or
By the lemma, we eventually achieve a
l
= ±d
'
b
i= ° •
(a^b^a^.
205
(The other possibilities are excluded because d is odd and b, always remains even at each stage.) Step II:
If a, \ b. for some i, we
We want to decrease |a 1 |.
apply a,,a.,b.,b. \ 1 1 1 1
> a +2b.,a.±2bn,b.,b. 1 1' 1 1 1 1 or a,,a., b +2a. ,b. ±2a., . 1' l _1 l i 1
Again because a, is odd, b. is even, if d' = g.c.d.(a.,b.)* we eventually reach a±
Step III;
= ± d \ b ± = 0.
Repeat Step II until a.. |b. , all i. We also may repeat
a, ,b.. i > a, , b +2a 1 |a,| further.
until
b.. = 0 again.
If a, ]( a. (i >_ 2) , w e first apply
a
l'ai'bl,bi
*
>
a
1'
a
i'b1+2ai'bi+2a1
so that b, becomes 2a., then repeat Step I. decrease la, I 1 1' Step IV:
W e want to decrease
until
a, |a.,b.. l'i'i
Kill b 2 ,---,b a
In this way, we
Then as g.c.d. (a.,b.) = 1, a, = ±1. ^ I'I ' 1
by maps
l,ai'bl'bi '
>
a 1 ,a i ,b 1 ±2a i ,b i ±2a 1 .
Make b, = 2 by maps a
S t e p V:
Kill
l,bl '
>
a1,b1±2a1.
by maps a9,*»»,a Z. g
a,,a.,bn,b. i 1 i l l
> a +2b. , a . ± 2 b w b _ , b . . l i i 1' 1 I
206
(These don't affect a., because b. = 0, i > 1; and since 4|a., i >^ 2, a. is a multiple of 2b.. .)
a
l'bl *
Finally kill b.. by
>a
ifbi~2ai'
N e x t , consider ^•••6.^(0,--^D;!,*),•••,()) = (Cl,---,c , d l f - - - , d ) 6*,6 N NBecause
6 • • -y
must have d, = 1.
is symplectic, and maps Moreover, as
2|c,,*»*,c , 4|d ? ,«««,d
.
(1, •••/)) to (1, •••,()), w e
6 •••y € T, w e have
W e choose more elementary
transformations
6. until l
6„•<$*/, , •••6T,6 .T ,•••6,7 IM1 N-l 1' M M-l fixes
(1,«««,0) and
Step V I :
(0,•••,0;1,•••,0).
Kill c 2 ,**«,c
and make
(c1,ci,d1,di) <
Step V I I ;
Kill
d 2 ,«««,d
c, = 2
by maps
> (c1±2di,ci±2d1,d1,di).
by maps
(c1,ci,d1,di) i
> (c 1 ,c i ,d 1 ±2c i ,d i ±2c 1 )
and finally kill c, by (c^d^ |
> (c 1 -2d 1 ,d 1 ).
The Proposition now follows by induction on g, because (6 •••y) preserves the direct sum decomposition
207
2g ffi
[2(1,0, • ..,0;0,. .. ,0)1 9 Unirn2)
U
2Z(0, • ..,0;lf. • • ,0)J
and is the identity on the first piece. an element of Sp(2g-2,E).
On the 2 n
piece, we have
QED
There are 2 useful ways to get generators of T~ is generated by
Proposition A3:
| ( n ^ = (n^ = 0
I
T2:
either of the following:
a)
(o 2 ?Kc l)C V 1 ) ' ^^
b)
the transformations
AS
V"** 2)
or
1 <_ i <_ 2g
b,)
x i
> x+2A(x,e. ) «e. ,
b2)
x i
> x+2A(x,e.+e.) (e.+e.) ,
1 <_ i < j <_ 2g
e. €ffi^ are the unit vectors.
where
I
Proof:
Both of these contain the generators of Al f in (b),
use the maps b, and b 2 with 1 <_ i < j <_ g and g+1 <_ i < j <_ 2g J, hence a subgroup containing On the other is an abelian group, which may T. be described as hand, T 2 /r 4 generate
V
r
4
" J 2g
+
2
A B {(c° D ° ) m o d
(Check this by examining
2
| B 0' C 0
s
y ™ e t r i c , DQ =
- % }
the condition
A( (I+2X)x, (I+2X)y) = A ( x , y ) modulo 4 ) . It suffices to check that the generators in (a) and (b) generate
r ? /f, i.e., contain elements
208
/I+2A„
I for every gxg
integral A n .
In (a), take upper and lower triangular
A's with +l's on the diagonal. form.
mod 4
1-2 . A. )
In (b) , we put the maps b ? in matrix
Thus if i = 1, j = g+1, it is 3
°\
0 -2
1.
0
0
0
-1!
0
0
0
0
'1
2 0
•J
so that if j = g+i, we get diagonal A 's. And if i = 1, j = g+2, we get 1 -2 0
0
1
2
0
0
0
1#
0 0
0
0
0 -2 0
0
0
0
"•1
0
1
0
2
1
0
0
"'I,
These give off-diagonals An's
Proposition A 4.
QED
r.. 9 is generated by
i::). c vo. c 5 all
A € GL(g,E),
synimetxic i n t e g r a l B w i t h even d i a g o n a l .
209
Proof:
It suffices by A 3 to prove that
generated by images of these elements. show that an orthogonal map of (ZZ/2ZZ)
T,
"*"s
2 /^?
This means that we need only g
to (S/2ffi)
g
is composed
of maps: a) b)
>y V"''Xg'yl'"*'yg ' l'-"'yg'xl'"#'xg other x, ,y, fixed x.,x.,y.,y. • > x., x.+x.,y.+y.,y.
c)
x.,x.,y.,y.
d)
l 3 1 3 x. , x . , y . , y . 1 1 3 1 3
i'
3
1
3
i
»
3
l
J
i
3
k,jrk
3
>x.,x.,y.+x.,y.+x.
"
1 3 1 3 3 1 > x . +y . , x . + y . , y . , y . 1 3 3 1 1 3
"
Let y
This can be done exactly as in the proof of Al.
be an
orthogonal map and say Yd/0,•••,0;0,•••,())
=
(ax, ••-,a
,b1, ••-,b
)
(note:
First, use map (a) to ensure that not all a.'s are 0.
a^b/s
are
0 or
Use maps (b)
to make only one a. equal to 1, and then to make in fact a.. = 1, a
=...-
a
= 0.
Use map (c) to make b« =••• = b
=0.
Then because
Q(l,0,---,0;0,••-,()) = 0 we have
Q(a..,»**,a ,b , • • • ,b ) = 0 too, so in fact at this stage
b 1 must be zero too, i.e., 6 N 6 N _ 1 ...6 1 7(l,0 / ..-,0;0,--.,0)
= (1,0,••-,0;0,- -•,0) .
Next look at 6N---7(0,---,0;l,(),••• ,0) = (c^--- ,c g ;d 1 ,---,d g ) . Because its inner product with (1, • • • , 0 ; 0 , • • • , 0) is 1, d.. = 1. Use maps (b) to kill d o , , , 0 ,d and maps (d) to kill c0,***,c , z g z g
1)
210
while not moving (1,0, • • -,0,0/ * • *, 0) . Then because Q(c, ) = 0, we find 1', • • •g ,c1 ;d, ,•••.
Thus
VM-I,,,VN-I,,,Y fixes (1,•••,0;0,•••,0) and (0,•••,0;1,••• ,0). induction, this proves the result.
As in Al, using QED
Finally: Proposition A5.
Sp(2g,Z) is generated by
(-i all
A
IJ> (O v
1
), Q P
€ GL(g,ffi) , B symmetric, integral.
Proof:
We prove this exactly as we proved A4, except that at
the 2 points where we used the invariance of 0, we use instead maps
or
e)
x i ,y ±
»
> x i / y i +x i ,
f)
x ± ,y i
I
>
X
other x k ,y R fixed
i + Y i' y i '
derived from diagonal B's.
QED
211
§6.
Riemann's Theta formula and theta functions associated to a quadratic form. 1
](z,Q)
We have described how the functions \/[
can be used
112
i) ii)
Q, z variable, to embed complex tori in IP » N for z = 0, variable, to embed ^v /I\ in IP . for fixed
N
Since these maps are not surjective, there must be polynomial identities between the various functions vI
1
](z,fi).
With only
n2
a few exceptions, all identities that I know of are deduced from the theta identities of Riemann.
These are generalizations of
the Riemann identity given in Ch. I for the one-variable case. We will conclude this chapter by describing these. We start with any rational orthogonal h*h matrix T. Theorem 6.1. (Generalized Riemann theta identity): h
i=l
(z,a,$
h j=l
x
t
t
expfcri tr( AfiA+2 A(a.+B))] TT^(z.+fia.+6.) 3 3
A/B€K
i = 1
i
i
i
gxl column vectors, A = (a.. , .. . ,a,) , B = (B.,.,.,3,) gxh
matrices, Z l g '
K =
h
V
n{*'h)
= group of integral g*h matrices, .?/Z{*'h)
-Tl\m{*'h)
,
and
d = [ofV 1 : T"^hnzzh])
212
i _i _ i \
A The main example is:
h = 4, T = -H 1
_
.
. J , so that
\l -1 -1
T =
t
T = T
1
t
=
T
1
.
1/
Note that a
T2
1'a2'a^'a4
i€
2Z
a.+a.G S i D a, + cu+a^+a„e 2ZZ 1 + l a2+a3+a4
4 4 4 so that coset representatives for T2Z n a in T 2 (0,0,0,0) and (o"/o"/o"/y) ?
tne
identity becomes
^(x+^u+v)^(x+y"u"v) »(x-y;u"v)
(R)
n r
= 2 9-9 = ^ ^—
B€—— 7Zg
are
^(x-y-u+v)
exp [47^^^+2711t a ( x 4 y + u + v 2 # ^ a ^ W ^
^-
a€^— 72?
The exponential factor simplifies if we use
I?'-functions
with characteristics: r h
«&> cn
IT* i=l
(it..,.)
h
I t. .6. j=l ^
3. h
= [T_1Eh:
T~1ShnEh]_g. ^,-,Bh o , - A h
LP i=1
I
IJ
213
We can derive this from (6.1) as follows:
y*iy ji <».*> "Et^Y,
1
^
t i j
6
:
= T7Texp(iTit(ZtijYj)fi(EtijYj)+27rit(EtijYj)
( ^ . ( Z j +Sj))) •
•^(Ztij(zj+ftYj + 6j)) = e x p U i tr[ t CfiC+2 t C(Z+D)])TTi?*(^t i . (z.+fl Y .+6.))
because
i
T*T = I ; now apply t h e theorem t o ^ ( Z t . .
l
(z.+tty- + <5•));
exp(7ri tr[ t CfiC+2 t C(Z+D)])exE/!ri tr [tA^A+2tA(Z-»QC+D+B) ] ).
l
TTT9'(z i +^Y i + 5 i + fia i + B i )
_ fI
L
J
j
I j;exp(iri t r ( t (A+C) ft (A+C) +2fc (A+C)(Z+B+D)) exp (-7ritr(2 t C-B)TI^(-«) A B i
= L[
] J
I
A
£exp(-27ri t r
B
t
a + i ^in C B ) n 5 ' f L * ^J
i
3,+6.
i'
w
i
(z j
i) .
X
-L
214
Thus in the main example
a+b+c+d 2 1 /x+y+u+v\ c+g+h]V 2 ' 2
... #f
a-b-c+d 2 le-f-c 2
2"g
I
o^pe Js g /2z g
i aai a+a T . ra-i-a -I Arrda + " l e x p (-2711^3 ( a + b + c + d ) ) - # [ e + 6 J (x) • • . 1 j [ h + B j (v)
This is the formula used in most applications.
Proof of the Theorem; gxh
matrix variable.
LHS = + n
I l
/ #
'
+ h
, n
Then
exp(7ri I ^ . f r n . e S
exp^iri t r ( t N « N ) (g
'
I
(%•?.)-t..)
i , j
+ 2iri
tr^N-Z^T))
h)
e x p f i r i t r ( t (NT)-fi.NT)
I
V
.(g,h) N € 7L I M €2Z
+ 27fi
i
£ N€2Z
Let Z = (z. , • • • ,2^) € (C g '
exp(iTi tr^MflM) (g h)
'
-T
+ 2fri t r ( t (N-T) • Z) ) '
+ 2iTi
tr(tM-Z))
be a complex
215
. re RHS
L J
Bj...,^
oj...^
expUi
= [ ] " | L J 5 .
I L
1
I a. 1
I J
1
I
R
ry..^
t
I
/ ( g
M
'
i
(n.+ai)^(ni+5
expUi
n j
i
t h e sum o v e r A and N t o g e t h e r ,
I
I
tr(t(N+A)fi(N+A)
^
3 (g h
'
,T
fc
(i^+c^) (z"i+li) ) /
+ 2iTi t r (fc (N+A) • (Z+B) )) /
j TO ZZ ( g '
.
Collecting
we g e t
expf-ni tr( t MfiM)+2i T i
I (g h)
) + 2TT1 J i
v
w h e r e A and B a r e summed o v e r 2ZT3' J T / Z Z ( g '
=
^
+ 2-ni £ t n . (z^+fta. +t±) )
J ^.ftn.
I exp(iTi I n ^ i
A NPffi
V
M€(2Z
( g
h
g
' W '
h )
)
tr(fcM-Z)^ ;
^
(g h)
zz ' >Trm '
•exp[2iTi t r
Note
M-B]
that B »
>exp[2iTi
i s a c h a r a c t e r of M € E ( g ' h ] T such c h a r a c t e r s occur for
trSl-B]
+ ZZ(g,h)
some B.
£exp(2iri trSl-B)
Thus
= 0
trivial
on S ( g ' h ? T ,
and
all
.
216
unless If
TL for all B€ZZ (g ' h
tr^M'BK
M 6 ZZ (g ' h jT.
M € 7L g ' \ T, all these characters are 1 and we get the order of
2 Z ( g / h ) T / 2 Z ( g , h ) T n ffi(g,h)
^
Thus
the
gum
reduces
to:
exp(iri tr(tMfiM) + 2-rri trfSf-Z))
I M€E
(g
'
h)
.T
= LHS.
QFD
The reader will notice that the proof of Riemann's theta relation is much shorter than the statement and its rearrangement into its various forms!
In fact, as often happens in such a case,
if we generalize it even further, the proof will become really simple and transparent.
The natural setting to which these ideas lead us
is that of theta functions associated to quadratic forms. It is in this setting that all the multiplicative properties of theta functions are best studied.
To start, suppose we decide to rewrite a product
of h theta functions as one series.
^ T\d(z i=l
V*
1
fQ)
=
What happens is this:
h
/LA exp( £ (TTitn.rfi.+27Ti ^ . . z . ) ) ^,...,^9 ±4i i i i i
.
In terms of N = (nlf—,nh)
be the
Z = (z l f -.- f z 2 )
"
'
gxh matrix with columns n. J.
.
217
This is T$(z.#rt = L-JrK<3Mn) expdri tr(tN.fl-N)+2Tri t r ( V z ) ) . i=l * r£Z '
A natural generalization of this is:
(6.2) where
$Q(Z,ft) =
2-4
h)
exp(7ri tr (tN-ft-N-Q)+2Tri tr^N-Z))
Q is a positive definite rational h/h matrix and the variable
Z lies in £C "' ' (g h complex matrices) , and ft lies in fy\ . ^) may be reduced by the old v9 if we define a map fi i \
> Q® Q >^hg
where ft®Q is the ghxgh matrix given by
(f28Q)
ih+j,kh+A " °i+l,k+lQj*'
Oii^
l<j,*
If we rewrite Z as a column: vec(Z)
ih+j - Zi+l,j
0 < i < g, 1 < j < h
then it is immediate that ^9Q(Z,fi) = $(vec(Z), QQQ)
218
Under the same map, the theta functions with characteristics give us:
- # Q £B] ( Z , a ) =
^-r* ^expOrri tr(t(NfA)fi-(N+A)-Q) + 2-rri tr( t (NfA) (Z+B))) tfcZ^/h) A,B €
Q(g'h).
Setting Z = 0, one may think of the functions
n I
> $Ql$
(o,n)
as being a natural basis of the vector space of all functions
£Q[f](ft) =
I
,
. .f(N).exp(iri tr^NftNQ))
N € Q(g/h;
a^'h)).
f
Thus all our previous ideas generalize to this setting.
In these
terms, we may generalize (6.1) as follows: Theorem (6.3):
i) i£
The functions
where
definite
^ Ql (z 1# fi)-
A9° 2 (Z 2 ,^)
Z = (Z, ,Z ) .
if_ Q 1 = ^ - Q - T
(RT'Q)
satisfy:
Q = (§jfe) . then ^ Q (z,^) =
ii)
-$ (Z,ft)
hxh
where
T
€
Q ( h , h ) , Q'/Q
both positive
rational symmetric matrices
_$Q'(Z-T,ft) = d " 1-
2*
exp(Tri tr(tA^AQf2tA-(Z+B)))-19Q(Z+ilA(>fB,fi)
219
where K
==2;^^).tT/4(g,h)#tT
n z (g,h)
K 2 -•^ h >.T- 1 A ( ^ h ) .T- 1 n« ( «' h ) d In particular, if
=
#K 2 . Q = Q 1 = I h , (R a ' Q )
reduces to ( R T ) . The
proof is exactly the same as that of (6.1).
A clearer way to
state (6.3), perhaps, is via the functions ~$
Q
[f](Q).
Note that
to prove (6.3) for all Z, it certainly suffices to prove it for Z = ftAQ+B, A,B e Q g '
, hence
(R T ' Q ) reduces to proving
corresponding identity
«&Q> for all
^ ^ ' ^ . ^ ^ Z ^ ^ ^ i A,B
tr W,^:](Z*,
but with Z = 0.
When Z = 0, it is simply the explicit form , in terms of standard bases of (R
nat }
g(Q g ' ^
Q ,
) of the formula:
[ f ](^) = $Q[f](fl)
where
f • (N) = f (N^T) .
At this point, the proof reduces to the totally obvious calculation: •>9Q[f](fi)
=
I , , , f (N)-exp( t N^NQ) N€0Tg'n; _1
: j . O . N .i : T Ln « , . T I , , , f (N)-exp( Tf H N.^-N. T"- Q .T N€Qlg'n;
I i , x f ( M . S ) •exp(tM-fi.M.Q') M€Q(g'h)
_ 1±
^ )
where M = N-V" 1
220
(6.3) has the important Corollary: Corollary (6.4).
i)
For all
Q
and
f, §Q [f](Q) is a modular
form in ft of weight h/2 and some level. ii)
For each Q
> ^9 Q tf]
f I is a map w : &(CD "' )
>{v. sp. of modular forms of wt. h/2, any level}
called the Weil mapping associated to Q, and the image depends only on isomorphism type of the rational quadratic form
x.Q.x
iri
h variables. iii)
;\,), then Under multiplication of modular forms , if Q = (i0*1 O Image(w_) = Image (w ,)• Image (w„) .
Proof;
(iii) is a restatement of (6.3.i), and the 2 — half 0 . Now any Q can be diagonalized of (ii) is a restatement of (RT ':*) over Q, and the product
f
-i/f? of modular forms of wt. n..,n2 is a
modular form of wt. n,+n2.
So (i) follows from the fact that
$ [f ] (IQ) is a modular form of weight 1/2 for all I >_ 1.
QED
The 2 fundamental problems in the analysis of theta functions as functions of
Q are the description of the image and kernel of
w . For example, one might ask whether Ker(w ) =
span of the differences f-f, where f • (N) = f (N^T) , T e orthogonal gp. for Q ©vev- (Si
221 O so that (RT '?*) gives the full kernel? I don't know if this is true nat or not. Also, one might ask whether Im(w ) (or ImCWj )) contains all "cusp" forms if h is bigger than some simple function of g. As functions of z for fixed ft, however, we saw in §1 how to produce from v bases for the vector spaces
R*
of quasi-periodic
functions in z of each weight. With Riemann's theta relation, we can go further and work out explicitly the multiplication table of v ft the ring I R0 in terms of these bases. To do this, we apply I < R cn Q )
*
W i t h
/n1+n2
=
\
1
0
0
\
nji^ni+iij) J
~ ( n]L +n 2 ) [l
-1 )
(RT 'Ov ) works out to part (i) of the following Proposition: Proposition (6.4) i)
For all
n
- j / n 2 >. *»
5 [ 0 J^r^-^Lo J(z2'n:ft) 2
n,d+a+b
§\
d€Z g /(n 1 +n 2 )Z g
ii)
For a l l f|n)
be the basis of
n > 1, (z)
Rfi n
let
L
ni+R2
I (V Z 2' ( V n 2 ) f i ) -
J
0
L
Z
L
Z
J n 1 n 2 3+n 2 a-ni£ 1(n_z.-n_ z 0 , n . r u (n,+nJn_)
-^[%n](nzfnQ)
Of §1. Then: —
222 _ a
b
Proof;
^
3€x9/(n1+n2)!zg L
"** it
0
1 2 1 2l
J
(ii) follows from (i) by setting z.. = n..z, z
These identities in the case n..|n
n^a+b
= n z.
have been applied by
Koizumi (Math. Annalen,^, 1979, p.U7) . However the case which has been applied most is when n, = Y\ 2 •
In
this case, using the simple
identity (6.5)
-\9 [ n 1 (nz, n2Q) = 9 N
^ Z-J
eez^/nzg
L
oJ
(z,n)
l
oJ
(6.4.i) reduces to the very simple classical: a/n-i
[
^rb/n-i (zrnn).^ Q (z ,nfl) =
2n 4- ^F lcz.+z,, 2nfi).^[2 2nl(zXrz_, 2nQ) 2 d(EZg/2Zg L 0 J ! 2 I o J
We give 2 applications of this.
In the first, we assume
n is even and n >_ 2 and, following §1, embed the torus into
Eg/L
3P " , N = n g , by:
-(•••-, ^ n ] < n ? , n a , . . . ) a£Z9/nz9 ' Then (6.6) gives us a simple set of quadratic equations in IP which vanish on the image.
Let
n = 2m.
Substitute a+me for a,
6-me for 6 in (6.6), multiply by exp(7ri e.J) e €2 g /2Z g .
This gives:
N-l
and sum over
223
Tp
a+e
2-1 a c p d r i ^ - i ^ ^ ^ z ^ ) - ^ ] e€Z g /2Z g
-^
b
e
1 1
"
5
]^!^
3 + a+b
,
3 t a-b
Setting z. = z. = nz, and writing
n) fl (z) = ^ r n ] ( n z , n f i ) , a
<e,l>
= expCiri^-t)
0
we find the i d e n t i t i e s
... i
X,. 1
gezg/2zg
<e,t>
r(n)n
( nn) f - l ^,( z.) . f_ A ^(z) a+me
2
e€X^/22;g
b+me
c+me
d+me
whenever a+6
E c+3 mod n2Zg ? € Zg/2Zg
where the constants are given by:
*i = + £ 1
g
e€ZS/2Z
c-a+ne
2
±
-Z%/22gg eezy/2z
a-b+ne
(o)
he homogeneous homoge This means that if the coordinates in labelled
X-)-, a € Z g / n Z g , then the tori
N-l IP are
E g / L 0 in IP®'1
satisfies
224
(6.7)
A.,7 <e,?> X-*-, +.Xr>- + = 1^ ' a+me b+me
for a l l
X0Y <e,?> X+ ->.X^, -> 2L ' c+me d+me
a+S = c + 3 , ? e z P / 2 Z g - ( A . . , X2 a s a b o v e ) .
p r o v e n i n Mumford, I n v . M a t h . , v o l . 1,
1966
In f a c t ,
it
is
> PP- 341-349, t h a t these
q u a d r a t i c e q u a t i o n s a r e a complete s e t of e q u a t i o n s f o r t h e image of t h e t o r u s . * k
k
As a second application, in (6.6) take n = 2 , z . = z
( ?
= 2z
and consider the bases
fi2,k>(«) = ^ [ a / 2 k ] (2 k 2 / 2^) of
R , . In terms of these bases, we get a very simple and 2K beautiful multiplication table
(6.8)
fi^.fX2")^ I fl^.f^W). a b g +] g c a J,des /f z
H3* c-dab
This implies the identity (6.9.a)
f|2
)
(0) .fi2
}
(O) = _>_ I
on the modular forms
f-i2
)
v+1
„^
'(O).^ 2
(o) = $ [a£2 ](0,2kfi).
'(0)
In fact,
any solution of these identities plus the further identities;
(6.9.b)
2 f| >(0) = _. Ig a
S€Z /2K+^g . S=2S mod 2 k + 1 Z g
4D 2
'(0)
(a special case of (6.5)) *More precisely, the ideal they generate equals the full ideal of (Cg/L in sufficiently large degrees.
225
f|2
(6.9c)
comes from some
}
(O) = f[| ) (O)
Q or a "limit" of ft's. This is proven in
Mumford, Inv. Math., 3, l^t'7, £(0-I I
, where a complete
description of the "limiting" values of the given.
f->
(0) is also
In terms of inverse limits, as in Ch. I, §17, we can
restate the result as:
V
K« 2*
CUspS
J
(6.10) Projffi f - ' - f ^ J /
L
'
lall
/
k>1
identities 6.9.a,6.9.b,6.9
with
tgp/2w /
4 2 ) (0)
re laced
P
Y
* k,S There is another interpretation of the "data" {f-V the identities 6.9.a,6.9.b,6.9.c that
(0) }
which is quite beautiful. -^/o2k
01
d2 2kv>(0) = ^ [ ra/2 ^ J(0,2 2k «) ?
rt
+ Lk_o a \
ex
P(lri
*«•«•">
while ,92k+l. fi ' (0) =
J
Equivalently, we may define 2 measures
expUi
u,v
on
n-2ft.n)
Q^
by:
.
and
Note
226
y(U) =
1 i
exp(iri
neure; £ ] g
n*ft-n) 1
(6.11)
v(U) =
for all open sets
I , expUi n€Un» [ | ] g U c Qg.
n-2ft-n)
Then
U (2k«| + \ ) = f f
> (0)
6.9.b is subsumed under the fact that y and v are measures, and 6.9.c says they are even measures: y(-U) =
y(U);
V(-U) = v(U).
A little calculation will show that 6.9.a says that y and v are linked as follows: £ : ffig xffig
let
2
(6.12)
> 0)2 * Q?
2
C(x,y)
= (x+y,
be
x-y).
Then
Thus
(6.13)
(6.10)
can be r e s t a t e d \k/v V* '
r
, 2k *
as:
U certain! S cusps CUSPS f pairs y,v
~
of even measures on
*• s a i t s f y i n g 6.12 mod s c a l a r s ( S e e Mumford,l«v. rtert..^^ p . | l 4
}.
Qg \ J
227
§7.
Theta functions with harmonic coefficients. Starting with the functional equation for 7(z ; fi), we have seen
in
§5 that we can define a large space of modular forms of weight 1/2
by -#[f]Uft> = ^fCnJexpUi^n-fl-n),
The functions -\9ffHfcft)
f € «Qg) .
are all linear combinations with elementary
exponential factors of the functions
a,£ € Q g
-#(Qa+£, £ft) ,
obtained by restricting ask:
z
to a point of finite order mod L .
are there other ways of getting modular forms from
We
S(z,Q)7
In fact, another way is by differentiating
-Q- with respect to
and then setting
z = fta+b. To illustrate
z = 0, or more generally
z
this, look again at the one-variable case:
j~d
[£](zfT)l
=
I
= 27ri °
~ exp(TTi(n+a)2T + 2iTi(n+a) (z+b))|
£ n»exp(iTin T+2-ninb) . n €S+a
If we differentiate the functional equation . 2 TTiyz
(where (a p € SL(2,Z) is in a small enough congruence subgroup), with respect to z, and set z = 0, we see that
228
-or- <°- ?££> = i.e., &*[?J/
c(/7^) 3 / 2 - - ^ I O , T ) (
is a modular form of weight 3/2.
But if we
differentiate the functional equation twice, we see, for instance, that ^
[
b 2]~ - ( 0 , T )
=
-4ir"2
3z
is not a modular form.
Jv n 2e x p (7rin Z2 T+27rinb) n GZ+a
But persevere!
A longer calculation will
show you that ^ - ( 0 , T ) •-£!£,HO,T) -
| •
-y-E-to.T) •
3—<0,T)
is again a modular form, now of weight 4. functional equation introduces a factor
The point is that the 2 Xz e and we need to form
combinations of the z-derivatives at z = 0 which are invariant under substitutions
$(z) -I > e
z
-«$(z) . Written out, this last
modular form is ,~ . x 4 (2TTI) •
v
2.
2 2 TW \ ffi(n 4m) T 2^Tinb4mb, P(n,m)e . e
n€Z4-a mez+a' 2 2 or (in terms of theta series for the quadratic form X +Y ) :
•^•^>-S 1 , [S£'.]«"i'-2>'^
z z
r 2=0
229
where 13
P(n,m) = Y11 " n m
2
Note that P is a spherical harmonic polynomial.
For several
variables, the situation is of course more complicated, as we have g partials 3/3z..
In fact, the natural thing to expect to
find are vector-valued modular forms. Here is what happens: Definition (7.1):
Let
T: GL(g,(c)
>GL(N,(E) be an N-dimensional
polynomial representation of GL(g,(E) or a 2-valued representation given by
T(A) =
representation.
Tn(A)«Vdet A
where
Then an N-tuple
T n is a polynomial
? = (f ,•••,f ) of holomorphic
functions of ft is called a vector-valued modular form of level T, type T if^ f.((Afi+B)(Cft+D)
1
n )
I T ( O H D ) .13 .f . (fi)
j=l
3
for all 1 £ i £ n, (£ ®)€ r.
Definition (7.2): polynomial
Let
X G CE(g' '
be a matrix variable.
P(X) is called pluri-harmonic if h
E k=l
32p
3X. V 9X. V lk
3k
E
°'
X
1 ^
£9-
A
230
W e shall denote by IH. the vector space of all pluri-harmonic polynomials P which are homogeneous of degree
SL .
Note that if
P(X) is p l u r i - h a r m o n i c , P'(X) = P(A-X-B) is also pluri-harmonic A £ GL(g,(C),
for all
on
B € 0(h,(E).
Thus
GL(g,(E) xO(h,(E)
acts
mv
Definition
(7.3):
For all Q rational p o s . d e f . hxh symmetric, f G 5(Q>(g,h) ^' ' ) ,
P € JH.,
let §P>Q[f](^) =
I . M f(N).p(NVQ)-exp(TTi tifN-fi-N-Q)). V / N€Q^ g , n j
The m a i n result is t h i s : Theorem 7.4: GL(g,(E) , let
Let
V <= m
be a subspace invariant under
{P } be a basis of V and let GL(g,(C) act on V
via the representation
T:
V A - X) =
Ke ( A ) -V x ) P
r P ,Q -J Then for all Q,f, the sequence of functions « $ a [f ] > vector-valued modular form of type
xQdet '
A word about the history of this r e s u l t :
is a
and suitable
r
H e c k e , M a a s s and
others have investigated v a r i o u s types of theta series w i t h harmonic coefficients and proved this Theorem in m a n y c a s e s . Kashiwara-Vergne
(Inv. M a t h . , £ £ (1978)) worked o u t very
completely these results from a representation-theoretic
point
231
of view and also decomposed 3H, as a representation of GL(g,(E) xO(h,(E) . Theorem (7.4) in its full generality was proven independently by Freitag (Math. Annalen, 254 (1980) , pp. 27-51) and T. Oda (Theta series of definite quad, forms, to appear). The approach that we use is based on the ideas of Barsotti (Considerazioni sulle funzioni theta, Symp. Math,, 3^ (1970), p. 247) analyzing theta functions from an algebro-geometric point of view.
We will describe both Kashiwara-Vergne's results
and Barsotti's in more detail in Ch. IV and consider only the purely classical-analytic results in this Chapter. The theorem could be proven using generators for
r, ~
allowing a transformation on Q,f, following the ideas of
and
§5.
However we can also, following Barsotti, draw a proof directly out of the ideas of the examples above, i.e., by differentiating the functional equation for $
(Z,ft) with respect to Z.
To do
this we need first to see clearly why pluri-harmonic polynomials come in. Put an inner product on the polynomial ring
(C [" • * , Z . . , • • • ] ,
1 £ i £ g, 1 < j < h, by
= (P(---, 3/3Z±.,...)Q)(O).
Note that 2 monomials Za,Z^ P
are perpendicular if a ^ B and
is a. positive integer, hence <,) is positive definite Hermitian.
Let
232
71 = (ideal in <E [•••, Z ..,••• ] generated by]
I
h w. . =
J
> z., z.,
.
Then we have: Proposition 7.5.
i)
<E [•••, Z ..,••• ] = 3H ©fl
and
3H, f] are
perpendicular with respect to the above inner product, ii) (g h)
(C '
Let
near 0, and for all 6
P € (C[ ••• ,Z . . - • - ] , define ID
:
>
P by_
6
Then
P
(f)
= ( P ( - - f 3 / 3 Z i j f - - ) f ) (O) .
i s pluri-harmonic
(7.6)
Proof:
iff
6 (f) = 6 (e tr Z ' C ' Z . f ) ,
all symmetric hxhC.
To prove (i), note that
k
ik
jk
2 {(R(K7T>
° S3z*az. > p ) ( 0 ) " °'
13 (R(
llr: ) P ) (0) ID
*
P
eft1.
ik =
3k
°'
'
all R € 77
a11 R
233
To prove (ii) , note that it suffices to take f to be a polynomial N because 6p(f) depends only on f £(0/B , some N, and polynomials map onto
6n has invariance (7.6) < = > ^ - S^ie* P 3Cpq P
«=*
Corollary definite
(7,7):
P
Z C Z
* * 'f) = 0, a l l f,p,q 00
( j ^ ZpkZqk*f)(0)
<=s>
Pb~-)f
<=>
PC?}1
<==>
PEE.
(O) = 0 ,
°'
a11 f P q
' '
f € Yl
QED
I_f P i s p l u r i - h a r m o n i c
hAh s y m m e t r i c m a t r i x ,
all
=
then
and Q i s a r e a l
P* (X) = P ( X . i / § )
positive
satisfies
"I . Proof:
Substitute
Z = W* *Q
m
(7.6).
To prove the Theorem, note first that the span of the $ ' [f] ' s for fixed Q, depends only on the rational equivalence class of Q because ^ p , ' Q , [ f ] = £ P ' Q [f] if
Q' = ^ Q A , f'(N) = f(N- t A), P'(X) = P(X(v^T"" ^A-^Q)).
234
,
"I
Here
A € GL(h,Q) (and note that VQ'
that
P"
r-
t
• A'/Q
is orthogonal so
is again pluri-harmonic). Therefore we can assume
Q.. = £.6.. 13 1 i] Then
and we may also assume
f = f,$••*®f* , 1 h'
2
f. € £ (Q^). 1
(27ri)Ha' If 1(0) = [p*(---^,---)(TT^[fJ(z i ^^))] L a 9z i ; j 1 1 1 J^Q i=1 where
1—
*
•+
P (X) = P (X"/Q), ot
Z. = (z . , — , z . ) . 1
ex
11
gi
On the other hand,
the functional equation for Riemann's theta function tells us that
TT^[fiHt(C^D)"1*zi/Jli(Afi+B) (OHD)" 1 ) [ +det (Cfi+D)]h/2exp(7Ti
for
(£ ^) i n a s u i t a b l e
i d e n t i t y and s e t
T.
t
I
Apply
z. = 0 , a l l i .
z ± . (CftH))" 1 ^^-^ 1 ) T T ^ [ f i ] < V * i f i )
P^ (• • ,3/3 z ± . , • •)
to this
By (7.7) t h e LHS g i v e s us
(+det (Cft+D) ) h / 2 . (P* (• • f^~r a
=
' • )TT^ If, ] Cz. x
ij
x
1
,lfi))\ IZ=0
w h i l e t h e RHS g i v e s us J TUCft+D)' 1 )
a3
ft(P*(..f-~— B
fe
,--)lTx9[f\] (z.,A.fi)) I
ij
1
1
1
|Z=Q
p Q Combining these, we get the function equation for W 0 a ' [f]}.
235
At this point, we have produced a great quantity of new modular forms, even new scalar modular forms.
The most important
outstanding problem is to find identities among them.
The only
non-trivial example is Jacobi's identity (Ch. I, §13) for g = 1, and its generalizations to higher g (Fay, Nachr. der Akad. Gottingen, 1979, N— 5 ; Igusa, On Jacobi's derivative formula, to appear). Even for g = 1, there must be many further identities (e.g., because many modular forms can be represented as theta series in many ways with different P,Q's:
cf. Waldspurger, Inv. Math.,
50 (1978), p. 135). Is there a systematic way of deriving these from Riemann's theta formula? In another direction, one of the most interesting applications of these vector-valued theta modular forms is to construct holomorphic differential forms on the Siegel modular variety
h
/Sp(2g,Z) (more precisely, on a smooth compactified version
of it). This idea is due to Freitag (Math. Ann. 216 (1975), p. 155;
Crelle 296 (1977), p. 162) and has been developed by Anderson (Princeton Ph.D. thesis, 1981) and Stillman (Harvard Ph.D. thesis, 1983).
We refer the reader to their papers for more details.
