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Tata Lectures on Theta III
David Mumford With Madhav Nori and Peter Norman
Reprint of the 1991 Edition Birkhauser Boston • Basel • Berlin
David Mumford Brown University Division of Applied Mathematics Providence, RI 02912 U.S.A.
Madhav Nori The University of Chicago Department of Mathematics Chicago, IL 60637 U.S.A.
Peter Norman University of Massachusetts Department of Mathematics and Statistics Amherst, MA 01003 U.S.A.
Originally published as Volume 97 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, 55-02 (primary); 11F27, 14K05,14K10, 22D10 (secondary) Library of Congress Control Number: 2006936982 ISBN-10: 0-8176-4570-5 ISBN-13: 978-0-8176-4570-0
e-ISBN-10: 0-8176-4579-9 e-ISBN-13: 978-0-8176-4579-3
Printed on acid-free paper. ©2007 Birkhauser Boston BirkMuser All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhauser Boston, c/o Springer Science-f-Business Media LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 9 8 7 6 5 4 3 2 1 www. birkhauser. com
(IBT)
David Mumford with
Madhav Nori and
Peter Norman
Tata Lectures on Theta III
1991
Birkhauser Boston • Basel • Berlin
David Mumford Department of Mathematics Harvard University Cambridge, MA 02138 U.S.A.
Peter Norman Department of Mathematics University of Massachusetts Amherst, MA 01002 U.S.A.
Madhav Nori Department of Mathematics The University of Chicago Chicago, IL 60637 U.S.A.
Printed on acid-free paper. © Birkhauser Boston 1991 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Birkhauser Boston for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $0.00 per copy, plus $0.20 per page is paid direcdy to CCC, 21 Congress Street, Salem, MA 01970, U.S.A. Special requests should be addressed directly to Birkhauser Boston, 675 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. 3556-4/91 $0.00+ .20 ISBN 0-8176-3440-1 ISBN 3-7643-3440-1 Camera-ready text prepared in TeX by the author. Printed and bound by Edwards Brothers, Inc., Ann Arbor, Michigan. Printed in the U.S.A. 987654321
CONTENTS
Contents
v
Preface
vii
§1 Heisenberg groups in general
1
§2 The real Heisenberg groups
15
§3 Finite Heisenberg groups and sections of line bundles on abelian varieties 34 §4 Adelic Heisenberg groups and towers of abelian varieties . . 47 §5 Algebraic theta functions 71 Appendix I: i9s as a morphism 87 Appendix II: Relating all Heisenberg representations . . . 89 §6 Theta functions with quadratic forms 94 §7 Riemann's theta relation
118
§8 The metaplectic group and the full functional equation of i9
133
§9 Theta functions in spherical harmonics Viewpoint I: Differentiating analytic thetas Viewpoint II: Representation theory Viewpoint III: The algebraic version §10 The homogeneous coordinate ring of an abelian variety
. .
147 159 170 183
Preface The present volume contains the final chapter of this work on theta functions. Like the other chapters, it originated with the senior author's lectures at the Tata Institute of Fundamental Research during November 1978March 1979. Excellent notes on these lectures were made by M. Nori, but, due to the shortage of time, not all the projected topics were covered. In the next few years, the ideas in this chapter were developed in various directions while the other parts of these lectures were published. However, this final chapter was not completed until a collaboration with Peter Norman beginning in 1988 infused new life into the project. At the same time, the interest of string oriented theoretical physicists in theta functions gave extra impetus to completing these notes. We are pleased that this joint eflFort has now made it possible to publish this volume. The idea behind this chapter was to bring together and clarify the interrelations between three ways of viewing theta functions: a) as classical holomorphic functions in the vector z and/or the period matrix T, b) as matrix coefficients of a representation of the Heisenberg and/or metaplectic groups, c) as sections of line bundles on abelian varieties and/or the moduli space of abelian varieties. Although equivalent on a deep level, superficially these three points of view look totally different and require quite different vocabularies. A more specific motivation was that the purely algebraic theory of theta functions, which comes from (c), has not been very widely understood. This approach originated the senior author's three part paper in Inventiones Math., in 1966-67, On the Equations defining Abelian Varieties, This paper is not easy to read, however, and with the exception of a few papers by Kempf, Barsotti, Igusa, Moret-Baily and Norman, the ideas in it have not been developed very far. For this reason, one goal of these lectures was to give a reasonably simple explicit treatment of the algebraic definition of theta functions, valid over any ground field (or base scheme). In the last few sections many open questions are raised: we hope this will make it clear how little is known beyond the foundations and will stimulate further work in the subject. Cambridge February, 1991
Vll
1. Heisenberg groups in general The abstract approach to the theory of theta functions is intimately bound up with a certain class of non-abelian groups, called Heisenberg groups. We begin by developing the representation theory of this class of groups. We consider locally compact groups G which lie in a central extension: 1 —y €\
—> G —y K —^ 0
i.e., CJ = {z G C I \z\ — 1} is a normal subgroup of G, in the center of G, and G/C\ is an abelian locally compact group K (which we write additively: hence the notation 1 —^ ... —> 0 above). We assume that G admits a continuous section over A', so that we can describe G as G = CJ X K
(as a set).
Then the group law on G is given by: (A, x) • (/i, y) z=z {Xfitpix, y), x-\-y) where xl^'.Kx K —^ C* is a 2-cocycle: 4){x, y) • \l){x + i/, z) = ipix, y + z)xl){y, z). Next, if we choose any elements x,t/ G A', let x,y G G lie over them and form X y x'^y"^.
This lies in CJ and is independent of the liftings x, y, so
we may define: e : A X A —^ CJ by e{x,y)
= X y x""^^"^
2
TATA LECTURES ON THETA III
It is easy to verify t h a t e{x -f x', y) = e{x, y) • e{x', y) e{x,y-^y') e{x,x)
= e{x,y)
= 1, e{x,y)
-e{x,y') =
e{y,x)~^
xp{y,x) e is a skew-multiplicative pairing. Let K be the character group of K (its "Pontrjagin d u a l " ) . Define —^
(p:K
K
by
or ghg~^ = (p{'g)(h) • h,
D E F I N I T I O N 1.1.
(V^f, h e G, "g - irg, li -
G is a Heisenberg
group if (p is an
irh).
isomorphism.
Given such a G, we will want to consider closed subgroups H C K such t h a t equivalently: a) CHXH = 1 cind H is maxima l with this property. b) (f restricts to an isomorphism between H C K and (K/H) c) H = H^,
where H-^ = {x e K \ e{x,y)
(The equivalence is easy noting t h a t (p~^{K/H) 1, X G H^,
C K•
= 1, all y G H}, = H^ and t h a t if e | / / x i / =
then the group H^ which is the closure of H -\-l - x in K also
satisfies e\H'xH' = !•) W h e n CHXH = 1, we say H is isotropic; when jy^ is m a x i m a l with this property, we say H is maximal
isotropic.
Note also t h a t
the following two properties of a closed subgroup H C K are equivalent: a)
H is isotropic.
b)
TT : G —>> K splits over i / , i.e., 3 homomorphism a : H t h a t TT o a = 1H-
—> G such
HEISENBERG GROUPS IN GENERAL
3
Here b) = > a) by the definition of e. To see a) = > b), let H be isotropic and consider ^"^(11).
Then ^"^(11) is commutative, and taking duals we
have: 0 ^— I <— ir-^H)
^— H <— 0.
Lift 1 G Z to an element ^ € ir~^{H). Then C is a character of 7r~^{H) such that C(A) = A, all A E C^. Thus TT restricts to an isomorphism from Ker(C) to H and its inverse here is the section a. In terms of co-cycles, splitting TT over H in K amounts to giving (T(X) = (a(a^), ar),
"ix £ H
such that (T{X + y) = (T(X) • <7(y), or
a{x + y) = il){x,y), a{x) ' a{y)
\fx,yeH
"ix.yeH.
After these preliminaries, we are now ready to state the Main Theorem about the representations of such groups: THEOREM
1.2. (Stone^ Von Neumann, Mackey), Let G be a Heisenberg
group. Then i) G has a unique irreducible unitary representation U :G —y
Aut{Ho)
such that Ux = X ' id., all A G CJ. ii) For all maximal isotropic subgroups H C K and splittings {a{x)^x) of TT over H, this representation may be realized by ( measurable functions f : K -^ C such that a) f{x + h) = a(h)'^rP{h, x)"^ f{x),
no = < I b) [ \f{x)\^dx < +00 I JK/H
Wh e H
a(x)
4
TATA LECTURES ON THETA III
^(A,y)/(^) = ^ • H^y y)' / ( ^ + y)' We will write this Ho as L^KI/H). Hi) AH representations (UjH) such that Ux = ^' id, all A G C*, are isomorphic toHo ^Hij
G acting trivially on H\.
This theorem was first proven independently by Stone and von Neumann in the case K — R^". The general case is due to Mackey. We give the proof first for the case where A' is finite^ where all the steps are completely elementary. Then we indicate the modifications necessary to deal with the general case, but following exactly the same method. We shall use as a reference for the general case the treatment in V.S. Varadarajan, Geometry of Quantum Mechanics^ vol. II, and simply isolate the steps where analytic lemmas are needed. This does not seem to distort the situation. Assume then that (U^H) is any unitary representation of G, that K is finite, that H C K is maximal isotropic and that a(x) = {a(x),x)
is a
splitting of TT over H. S t e p I. Decompose 7i under the action of the abelian group o-{H):
where 7i( = {a E 7i \ Ua{x)0> = C(^) •«> a" ^ ^ H), Another way of writing this is to classify the C's according to dim W^. Let H^z= {(^eH
|dimW^ = n},
(n = 0 , 1 , 2 , . . .;oo)
and let /C„ be a standard n-dimensional Hilbert space. Then oo
n=0
HEISENBERG GROUPS IN GENERAL
5
where L'^(HnylCn) = { space of/C„-valued functions fni^) on Hn} and the above isomorphism carries ?7<7(ar) for x ^ H to the map
S t e p II. 7r~^(/f), which is CJ x i / via the section cr, is a normal subgroup of G. But if Tr{g) = y^h = (A, x), then ghg-^ = (A • e(y, ar), x) = (A • <^(y)(x), x). Therefore if Uh acts on a vector a 6 W by the character (^ £ H, then Ug^g-i acts on a by the character ^^(2/) -f C • Ughg-ia = (p{y){x) 'Uh{a)
= M2/)+C)W-«. Thus dimW^ = dim7i(p(y)4.^.
Since <^ is surjective, this proves that dim W^ is independent of C; hence
for some fixed n. So for f E H, y G H, x £ H,
S t e p I I I . Now let us write down how the rest of G must act on 7i in terms of a 1-cocycle. We use the simple LEMMA.
Automorphisms
V of L'^{H^ICn) that commute with the action
of H are given by:
{vf){0 = AiO{f{0)
6
TATA LECTURES ON THETA III
where MO : ICn-^ JCn is a set of unitary maps depending on CTo apply the lemma, take x E K and let (f) denotes the composition of (p with the restriction from K to H. Consider the map / i—^ Vf, where
= (t^(i,.)/)(c - ^(^))-
{vfm Then for
a\\y£H,
V U(^aiy),y)fiO = f^(l,x)f^(a(y),y)/ (C - H^^)) = eix,y)U^a(y),y)U(l,x)f{C
-
H^))
= e(x,y)a(j/)(C - ^(x))(j/)i/(i,,)/(C - <^(x)) =
eix,y)cciy)iC-H^))iy)-VfiO
= ay) • c^ivwrn =
U(a(y).y)VfiO-
Therefore
f^(i,.)/(0 = ^x(C)[/(C + .^W)] where Ax{C) are unitary isomorphisms of /C„. Step IV. Now define a map
where
{ =
space of maps g : K -^ JCn such that g{z + h) = a{h)-'^{h,x)-'g{x), L\K/IH)®Kn.
Let Wf{x) = [/(i,.)/(e) = A,ie)[fi
{x))].
^h € H
HEISENBERG GROUPS IN GENERAL
7
(The 2"^ and 3^^ expressions are equal by Step III). By Step III, we see that
/ \\wfix)fdx = / wfiH'^m'dx = i wmfdc JK/H
JK/H
JH
So 14^ is a unitary map. Moreover, iov h £ H
Wf{x-\-h) = U^,^,^n)f{e) = [U(xp(h,x)-\h) -U(i,x)f]{e) = V(/^, x)-'
.
aih)-'U^,^h),hmi,.)f){e)
= xP{h,x)-^ 'a{h)-^ hence Wf G L'^{K//H;ICn).
'Wf{x)
Thirdly
=
=
U{xPix,y),x-\-y)f{e)
ij{x,y)Wf{x-j-y)
which is the rule given in the theorem by which G is to act on L^{K//H)<S>fCn (generalized to arbitrary n).
Finally, as / can be given
arbitrary values in K^n for each (^ £ H^ it follows that Ax{e)f((p(x))
has
arbitrary values in JCn for x ranging over a set of coset representatives in K mod H. Thus W is surjective. Putting this together, I^ is a unitary isomorphism of the G-representations 7i and L'^{K//H)(S>^n' Thus ifTi is irreducible, n = 1, and every irreducible representation U such that Ux = X ' id. is isomorphic to L'^{K//H).
This proves (i) and (ii).
Moreover every non-irreducible one is a direct sum of L'^(K//H) with itself n times for some n. QED for finite K We now outline the modifications necessary to deal with general n. The generalization of Step I is the classification theorem for arbitrary unitary representations of locally compact abelian groups. This is called
8
TATA LECTURES ON THETA III
the theory of spectral multiplicity, and may be found, e.g., in P. Halmos, "Introduction to Hilbert space and Spectral Multiplicity". The result is that any unitary representation (U^Ti) of an abelian H is of the form: oo
n= l
where H is the disjoint union of Borel sets Hn, ^n is a measure supported on Hn and L'^{{Hn^fJ'n)]^n) is the Hilbert space of measurable functions / : Hn -^ fCn with norm
and H acts by
UhfiO = C{h) • /(C). In this decomposition, the measure class of //„, (i.e., the set of subsets S C H of measure zero) is uniquely determined. To generalize Step II, the argument given above shows that for every T) G Hj if we modify the representation (Uh^^H) on H by multiplying by the character r;(/i), we get a unitarily equivalent representation. In terms of the decomposition via the measures /i„, this means that we translate all the measures fin on H by rj. Therefore by Step I, the measures ^„ must have the property: All translates of //„ by C '—^ ( + r], rj E H, a,Te in the same measure class as /i„. We now cite the well-known lemma: LEMMA 1.3.
If H is an abelian locally compact group, there is a unique
measure class which is translation invariantj and it contains a unique measure which is also translation-invariant:
the Haar measure.
(See V.S. Varadarajan, op. cit.. Lemma 8.12, p. 19). Thus all //„ may be assumed to be multiples of Haar measure. Since the /i„'s also have disjoint
HEISENBERG GROUPS IN GENERAL
9
support, only one of them can be non-zero, i.e., for some n:
To generalize Step III we need to know that any unitary isomorphism of L^{H^)Cn) commuting with H is given by
{vm) = A{0(m) where A{C) :fCn — ^ /Cn
are unitary maps depending measurably on C; and that if Vi/ is a measurable family of such V's, it is given as above with At{C) : JCn — ^ JCn
depending measurably on t and ^. This is the content of Lemmas 9.4 and 9.5 in V.S. Varadarajan, op. cit., pp. 63-66. It is when we reach Step IV that we are in trouble. The definition of W does not make sense because we have to evaluate U(^i^x)f oi Ax(Q at specific points, and we are dealing with measurable functions, which are only well-defined modulo functions supported on a set of measure zero. This is the key point which Mackey was able to solve. His idea was that Ax{C) is essentially a co-cycle and as such it can be made continuous, mod a coboundary, and then it has good values everywhere. To be precise, let
rp{y,x) Then the fact that for all x,y E K\ ^(l,ar)^(l,y)/ = '^{^^ y)U(i,a:+y)f
10
TATA LECTURES ON THETA III
tells us that {B^^y{z)f){ip{z
+ X + t/)) = V^(z, X + y)-\A^^y{(p{z)f)){(i>{z =
-\-x-h y))
xP{z,x-j-y)-'U^,^,^y^f{(f>{z))
= rP{z, X + y)-'iPix,
y)-'U^i,,)U^i,y)f{(l>{z))
A^{(p{z)) o Ay{(f>{z 4- x))f((z -\-x-\-y)) x)f){4>{z + X + 2/))
= (B^{z) oBy{z^
i.e., for all x^y £ K, then for all z E K except in a set of measure zero (depending on x^y): B:,+y{z) =
B4z)oBy{z-\-x),
Then Mackey's fundamental lemma in this quite special case says that there is a measurable function A : K —^ Aut{ICn) such that for all x G K, then for all y G K except in a set of measure zero (depending on x): B:,{y)=A{y)-'oAix
+ y),
We may normalize A by requiring A{e) = id. So Bx{e) — A{x^. Our old definition of W can now be rewritten:
(Wf){x) = A4e)f{
i;{e,x)B,{e)f{^{x))
=
ilj{e,x)A{x)f{(p{x)).
This formula does not involve taking a function defined up to a set of measure zero and substituting a particular value for one of its variables. It is now straightforward to verify that this W is a unitary isomorphism of H with L'^(K//H)<S)^nj although, since we do not have the formula t/(i a;)/(e) for Wf{x)y the proofs are more roundabout than in the original Step IV. We give the proof of Mackey's lemma in order to convey the style of this type of argument. We write "W" to mean "for almost all".
HEISENBERG GROUPS IN GENERAL a) Let C{x, y) = By^x{x) = A{^)~^ ^^{y)-
11
Then in terms of C the cocycle
condition on B gives W(x, y, z) e K\
C{x, z) = C(x, y) o C(y, z).
It follows that there is some particular value yo of y such that W(x, z) G A'', C(^, z) = C{x, yo) o C{yo, z). Write
so W(x, z) G K\
C{x, z) = C2{x) o Ci{z).
b) Define -B^(y) = ^i(y) o Bx{y) o Ci(x -f y)"^. Then 5 ' satisfies the same hypothesis as J5, and moreover W(x, t/) G A'^
= Ci{y) o C2{y) o Ci(x + y) o Ci(x + y)"^ = C^iy),
say.
c) Use the identities that B satisfies, and it follows that W ( x , y , ^ ) G A ' ^ Cz{z) ^
B',^y{z)
= BUz)oB'^iz
+ x)
= C3{z) o Caiz + x) i.e., Vix e K, Czix) = id; hence
Vi{x, y) e K^, B'^{y) = id.
d) Finally fix any x G A'. Then W(y,z) e A'^, the 3 formulae: BUz) = Bi^^iz)oB'^{z
+
xr'
BUyi') = id. B'y{x + z) = id. all hold. Thus W z G A', 5;(z) = id., hence Wz G A' 5x(z) = C i ( z ) - i o C i ( a ; + z) as asserted.
12
TATA LECTURES ON THETA III We finish this section by making three remarks to amplify the idea of
the Heisenberg representation. PROPOSITION
1.4.
Given a Heisenberg group l ^ d - ^ G ^ K
^0,
let H C K be a compact isotropic subgroup of K, and a(h) = (Qf(/i),/i) a splitting ofG over H. Then 1 ^ q ^ 7r-\H^)/aH
-^ H^/H
-^ 0
is a Heisenberg group, and if Ho is the Heisenberg representation of G, {HQY^ PROOF:
is the Heisenberg representation ofGn = T^~^{H^)/aH. It is easy to see that Gjj is Heisenberg (without assuming compact-
ness of H). To check the assertion about representations, let H C Hi C H^ be a maximal isotropic subgroup of K and extend a to ai : Hi —^ G: Then L?{K//Hi) L^{H^//Hi) Via e, H and K/H^
= the Heisenberg repres. of G = the Heisenberg repres. of G//.
are dual abelian locally compact groups; since H is
compact, K/H^
is discrete, i.e., H^ is open in K. Therefore we may iden-
tify L^{H-^//Hi)
with the subspace of L^(/\//i7i) of functions supported
on the open and closed subspace H^.
Writing out the quasi-periodicity
formula with respect to Hi, it is easy to see that / G L'^{K//Hi) invariant if and only if it is supported on H^. PROPOSITION
1.5.
2=1,2
til en l^d^GiX
(T{H)-
Q.E.D.
Given 2 Heisenberg groups:
1-^Cl-^Gi-^Ki-^O,
is
G2/{(A, A-i) I A € q } -^ A^i x A'2 ^ 0
HEISENBERG GROUPS IN GENERAL
13
is a Heisenberg groupy and its Heisenberg representation is
The proof is straightforward. The third remark is this. If G is any Heisenberg group, we get a "universal" representation {U^Hu) where U\ = \ - id. by taking the space of all functions
such that /(A
/
JGIC\
\f?dg < oo.
Call this Z.?js((j). It is in fact a left and a right G-module via
Equivalently, via the set-theoretic section a oi G over A', this space is just L^(A') with G X G acting by:
I claim: PROPOSITION
PROOF:
1.6.
There is a G x G-equivariant unitary
isomorphism
Let G = G X G / { ( A , A) | A G C J } . This group acts on L'^{K). But
it is also a Heisenberg group in a sequence 1 -^ CJ -^ G -> A X A -^ 0, and Kj embedded diagonally in K x K, is a maximal isotropic subgroup. Working out L^{K x K//K)^
we find that the representation of G x G on
14
TATA LECTURES ON THETA III
L^{K) is just the Ileisenberg representation of G. On the other hand, if G° is G with its center identified with C^ in the opposite way, then G^G^
xG/{(A,A-^) I A G C * }
so by Proposition 1.5, this representation is isomorphic to
But HGO = H*G, hence Prop. 1.6 follows.
QED
2. The real Heisenberg groups We want to specialize the theory of §1 to the case of Heisenberg groups 1 -> CJ -^ Heis{V) -> 1/ - . 0, V a real vector space. We construct explicit realizations for this Heisenberg representation and use these, given an element T of the Siegel upper-half space, to get special elements fri^i
of the representation space. The theta
function appears as a matrix coefficient for the representation using / r and ej.
The commutator e :V x V -^ C^ can be written in the form
where A : V xV —^Ris a, non-degenerate, R-bilinear skew-symmetric form on V. From now on we will write e(z) for e^^*^. We shall choose V to be R2^,
and A to be: A{x,y)
where x = I
),
y= \
= *xi -2/2 - * ^ 2 -yi
), and Xi and pi are o-rowed column vectors.
Choosing a suitable splitting, the group Heis(V)
may be described as:
r set of pairs (A, x), A G C^.x G R^^, with group law Hets(2g, R) = | ^^^ ^^ ^^^^ ^^ ^ ^^^ ^^^^^^ ^^^^^^ ^ ^ ^^ In the notation of §1, xp(x^y) = e(y4(x,y)/2) = e(|,?/). The most obvious kind of maximal isotropic subspace in F is a real sub-vector space W C V which is a maximal isotropic for A. We shall use the notation
•., = {('„.)} CR"
Taking W2 as the group H in Theorem 1.2, we obtain Tio = functions / : R^^ -^ C such that
16
TATA LECTURES ON THETA III
Uix,y)f(x)
= A eiCxi
. 2/2 ~ '^2 • yi)/2)f{x
+ y).
These functions / are obviously determined by their restrictions to Wi, so setting (f{xi)
^Q ])» we find:
= f{i
R e a l i z a t i o n I:
Hi = functions v?: R^ —> C such t h a t
/ |<^(a:i)| dxi < GO
U(l^y,)(p{Xl) = (p{Xl + ? / l )
= e(*xi •2/2)<^(a^i) hence ^(A,yi,y2)V^(^l) = ^ ^ C ^ l • 2/2 + V l • 2/2/2) <^(xi +
yi).
Here we have the well-known Heisenberg representation, i.e., the irreducible group of unitary m a p s of iy^(R^) consisting of translations and multiplication by characters. Another m a x i m a l isotropic subgroup of F is a lattice L such t h a t A is integral on L x L, and L is maximal with this property, i.e., if you express A on L X L by a m a t r i x , then det A =^ 1. Take L to be Z^^, the s t a n d a r d lattice in R^^, and split Heis{2g^H)
over L by
(T(n) = ( e . ( n / 2 ) , n ) , where e J
Vnel''',
^ M = e(2 • (^xi • 2^2))- Then e,(n/2)G
{±1}
if
n G Z^^
and (T(n) • cr(m) = ( e * ( n / 2 ) e * ( m / 2 ) ^ ( n , m ) , n -h m) = (e((^ni • ^2 + *mi • m 2 ) / 2 -f (*ni • 1712 - *n2 • m i ) / 2 ) , n + m ) = ( e ( ' ( n i 4- m i ) ( n 2 -h m2)/2) • e ( - ^ n 2 • m i ) , n -h m) . . n 4- m . . , . = ( e * ( — - — ) , n -h m) = (7(n -f- m)
THE REAL HEISENBERG GROUPS if n, m G T?^. Notice that e* is a quadratic form on \l}^/J?^
17 with values
in { i l } and (T is a section because
The main theorem in the previous section gives Realization II:
functions / : R^^ —y C such that
U(x,y)f{x)
The space Wj will be called
= X-e{-,y)f{x
+ y).
L^iR'^sf/j^gy
The IIeis{2g, R)-equivariant unitary maps between these 2 realizations are easily written down:
/ 6 L^iR^)
corresponds to
/* £
L'^(R'^y/I^^)
iff /* ( r ) = E
-/"(^i + " H * " • ^2 + '^1 • ^2/2)
We leave this to the reader to check. There is a 3rd realization which is very important and which is based, not on a maximal isotropic subgroup of A', but on a maximal isotropic subalgebra of Lie(A')c. Before explaining this, it is convenient to make explicit a little linear algebra underlying complexifying the vector space V.
18
TATA LECTURES ON THETA III
PROPOSITION
2.1. Given a real vector space V of dimension 2g and a non-
degenerate alternating form A : V x V —^ R, then the following data on V are all equivalent: 1) A complex structure JonV
such that A{Jx, Jy) = A{x^ y), all x,y E
V and A{Jx, x) >0, allx eV,x^
0,
2) A complex structure J on V and a positive definite Hermitian form H such that ImH
= A,
3) A g-dimensional complex subspace P C Vc such that Ac{x^y) all x,yePy
iAc{x,x)
= 0,
< 0, aii x G P, x ^ 0.
The set of all these we call the Siegel space S)v associated to V (and A). If, furthermore, we choose a basis e,^^\ ep^ of V in which A{e\^\ef-^) , e^. ^) = 0, A(e\
A{e\
=
,e^- ^) = Sij and let x\ ^,x\ ^ be corresponding coor-
dinates, then a point of i^v" is given hy a g x g complex symmetric matrix T, with Im T positive definite, i.e., a point T E S^g, the Siegel upper half space. The connections between these data are: a) Given J, H{x, y) = A{Jx, y) -f iA{x, y), P = locus of points ix — Jx. b) Given P , the complex structure J comes from the isomorphism
V ^
Vc —^
Vc/P
and T is defined by the property: e,^^^ - DTfye^-^^ G P. c) Given T, H is defined by i y ( e p \ e f ^ ) = {Im T)Jj\
P is the span of
e^ ^ — TiTijCj , and the complex structure J comes from requiring that x^i = I^TijXj ^ -\- x\ ^ are complex coordinates. Finally, the symplectic group Sp{V,A) acts on the space i3v by
y{J) = jJj-\
yeSp{V,A).
THE REAL HEISENBERG GROUPS
19
This action is transitive, and the stabilizer of J is the unitary group U{Vj H), hence f)v=Sp{V,A)/U{V,H). In terms of a basis, the action is given by T—yiDT-C){-BT
+ A)-\
if 7 =
A C
B D
(The usual formula, except for the automorphism
ic I) - i-'s f ) of Sp{2g,R).)
(For more details see Tata Lectures on Theta, L)
We now chose i/, J, and T as in the Proposition, so that we have a complex structure on R^^, and even a definite isomorphism
with the complex coordinates X = Txi +X2. From Chapter 2, we recall that the formulae =
t?(x, T) d{x,T)
t^U + ^2, T)
n2 G Z^
= e ( * n i -x + ^ni • T • ni/2)i^(x-f T n i , T )
ni G Z^
describe the invariance of t^(x, T), and that these extend to actions of R^^ on the space of holomorphic functions on C^:
U^i,y,)fix) =^BCyi • x + 'yi -T • yi/2)f(x
+ T yi).
Combining these, we find that we have a representation of the Heisenberg group Heis{2g,R),
provided we set Ux = \~^ • Id., hence
U(x,y)f{x) = A-ie('yi -x + 'yi- y/2)f{x
+ y).
20
TATA LECTURES ON THETA III
This called the Fock representation of Heis(2gjR).
This A~^ looks like an
accident but it is actually rather basic as we shall see. We make this into a unitary representation by setting
= / i/u)i2 g - 2 . ' x . . / m T . x .
.^^^^^2
It is easily checked that the t/(A,y)'s are unitary on the Hilbert space nl{C^,T)
of holomorphic / ' s for which ||/|| < -foo.
We shall show that the Fock representation is irreducible. It then follows from Theorem 1.2 that it is canonically the dual of the other two representations, and also, because it is a Hilbert space, hence is conjugate-linear to its dual, there is a Heis{2g, R)-equivariant conjugate-linear isomorphism of the Fock with the other representations. To prove this irreducibility, a modified version of the Fock representation is more convenient: namely one in which the action of Wi and W2 are symmetrical. In previous chapters, we have used the trick of modifying the periodicity of i? by multiplying by e^(^\ Q quadratic. We do the same here: define W?(C^,T) = space of holomorphic functions f{x) on C^ such that
where H{x^x) = *x • (ImT)"^
• x as in Prop. 2.1.
Define a unitary isomorphism
nl{a,T)^'Hlia,T) by
Then it is an elementary, though tedious, calculation that the group action on K^ takes the form:
THE REAL HEISENBERG GROUPS
21
The deeper analysis of real Heisenberg groups depends on the fact that they are Lie groups, hence have Lie algebras. We use the Lie algebra structure to construct for each choice of T E i3 a unique element / r of7i
.
'—^
This limit will exist for a dense set 5 of x's in W and S Ux : S —^ 7i will be a skew-adjoint, but not bounded operator. One defines Hoo CH to he the set of all x G W such that 8Ux^O'"o8
Uxr, {x)
is defined for all X i , . . . , Xn G Lie G. It is a theorem that Tioo is dense in H. We do not want to discuss the general theory here, but only want to illustrate what it says in our example. Here Lie G has a basis:
such that exp(ExrU) = (l.(f'))
exp£:rp)5,) = (1,(^(2))) exp(/C) = (e'"',0). Then
[Au Aj] = [Bi, Bj] = [C, Ai] = [C, Bi] = 0 [Ai,Bj] = 6ijC.
In realization I: 6 t^A.(/)(x) = lun
^ 2'KitX, f(
S UB.{f){x) = lim 6 Ucif){x)
^ y
= \ _ f(
—ix)
\
^^ ^ = 2 « x , . fix)
= lim/'"'-^^^^^"-^^^^ = 2^i
fix).
TATA LECTURES ON THETA III
22
6 UA^ are called the momentum operators, 6 UB, the position operators, set of functions with "L^-derivatives g ^ " of all orders with < -foe. By Sobolev's lemma, Tioo is the set of C^-functions
f{x) with ax«
allA^,a,
eO(\\x\\-^),
also known as the Schwartz space 5(R^). We do not work out 6U for realization II, but note that, by our formula for the isomorphism between these, the space Woo will be the space of C^ quasi-periodic functions. In the Fock representation, in its symmetrical form, abbreviating Im T~^ by 5, we find: ^
UAJ{X)
= lim^[e-^'^^^(^^*)-('^/2)^(^^»)^^^(*^*)/(x + tTeO - f{x)]
= "^ E ^i • ^^^^k ' ^(^) + Yl dx_j^i j.k
S
UBJ{X)
= lim^[e-'*^*'^'-'/2'('^-)^ (''•)/(x + <e.) t—0 t
= -x^5.fcXfc./(x) +
/(x)]
dx.
The complex span of these operators is the same as that of the operators:
/
I—> df/^^i,
called the annihilation operators
/
I—^ Xj/,
called the creation operators.
and
In fact: WT = {Span of 6UA, - Y, ^ i ' ^^B, all i} = {Span of / h—. WY = {Span of SUA, ~ J2 ^*i ' ^^^. ^'^ '^ = {^P^^ of / . - ^
xj}
df
^ }
dx-
and these are conjugate abelian complex subalgebras of Lie{G) 0 C. These subalgebras have a remarkable property:
THE REAL HEISENBERG GROUPS THEOREM
2.2. FixT
23
eS)g.
a) In the Heisenberg representation Ti of Heis{2g^ R), there is a element fx, unique up to a scalar, such that S Uxifr)
is defined and equal to
0,allXeWT. b) Ji3L2(R5)^y-^^e^.-^r.Tx c) In L 2 ( R 2 ^ / / Z 2 ^ ) ^ /T^e^^^'^i-^i^feT). d) In 7^?(C^, T), there is a unique /T killed by Wf and it is /y = 1, the Vacuum state". Hence 7i\ is irreducible and in the conjugate linear isomorphism with L'^(W), 1 corresponds to e^^ ^^^. P R O O F : In
7ii
^"'•f-ii-, 6UBJ
^2mxif.
So a function / annihilated by WT is one satisfying
The only solution to these equations is f{x) — e'^* ^r TX l}{^9
Going over to
jjj}9^ by the formula given above, the function becomes
^
^
n — ^'^iCxiTxi
+ *xiX2) V ^
^Tri*nTn+2wi*n(Txi-\-X2)
n
Finally, in W?, / is killed by W^r if and only if df/dx^ constant.
= 0, all z, i.e., / is a Q.E.D.
This theorem says that in the big Hilbert space Tiu we have a canonical way of singling out a finite-dimensional submanifold of the "most elementary vectors", i.e., the / T ' S . And in the quasi-periodic representation, we get exactly the theta functions, multiplied by a simple exponential factor
24
TATA LECTURES ON THETA III
which puts their periodicity in the simplest form. The Fock representation suggests a whole filtration of 7i defined by 7iff,(C^,T) Z) Vn = (vector space of polynomials in x of degree < n). Then Vn = Span of {{6U:,, o •
\ Xi G
•-SU^.J/T
Lie{Heis{2g,R))},
In physics, Ki is called the space where there are < n particles present. In order to complete our construction of theta as a matrix coefficient we now single out a second element eg. Following the idea of distributions, we can enlarge 7{ canonically by defining Ti-oo = {Space of conjugate linear continuous maps £ : T^oo -^ C} where 7i C ^ - o o by mapping x £71 to ix '- "Hoo —^ C, 4(2/) =
{x,y).
(The inner product {x,y) is taken to be linear in x, conjugate-linear in y. Here continuous means that there is a finite set of conditions \\SUxi o • • o ^^am^ll < ^ on X G Woo implying |^(x)| < 1.) In our case, it can be shown that: In realization I, W-00 = Space of tempered distributions on R^. In realization II, W-00 = Space of all distributions / on R^^ such that / ( x + n) = e . ( | ) e ( x , ^ ) / ( x )
allxeZ^^.
In the Fock representation 7i^: Tioo = Space of holomorphic functions / ( x ) such that /(x) = 0(ei^(^'^)/||x|r),
all n
THE REAL HEISENBERG GROUPS
25
7{_oQ = Space of holomorphic functions f(x) such that fix) = 0 ( | | ^ | | " . e^^(^'^)),
some n.
We will not use these facts though. Our reason for introducing 7i-oo is to prove the following theorem. THEOREM
2.3. Let cr(l^^) C Heis{2g,Z)
be the set of elements:
Tl
{(^*(2)'^)'
a) In the distribution completion W_oo of Tij there is a element e j , unique up to sca/ars, which is invariant under Ug for all g G b) In L^(R^),
€z = ^nei9Sn,
(6a = the delta function at a).
ez = E„€Z= » ^*(t)^"-
c) In L2(R2.//Z2.),
ei =,?(£, r ) .
d)In7lUC0,T), PROOF:
(T(1?^).
Consider an arbitrary (no continuity assumed) conjugate-linear
map
{ such that i{U^€^(^),m)f)
C^ - functions / on R^^ s.t. "I } -^
C
= ^/j all / . I claim i(f) = a - / ( 0 ) ,
some a.
In fact, f/(c.(^).m)/(^) = e * ( y ) e ( - , m)/(x -f m) = e(x,m)/(x) so ^([1 - e(x, m)]/(x)) = 0 , Let €i,..
all m G Z^^
.,€2g be the unit vectors in Z^^ and for some 6 cover R^^ by Uo = Bs+l^',
5 , = {x I ||a;|| < ^}
Ui = {x\e{x,ei):)tl},
l
26
TATA LECTURES ON THETA III
Let {pi} be a partition of unity by C°^ functions on R^^/Z^^ such that supp{pi) C image Ui. Take any / G S{R'^3//t^^) such that /(O) = 0. If 5 is small enough, then by Taylor's theorem, we can write
Then po/ = ^9i,
where supp gi C
JB<5,
and xi — 0 = ^ 9i{^) — 0. Extend
^f,- to a quasi-periodic function in S{R'^^//l?^).
Then
t= l
Thus ^ is of the form g i—y »5'(0). This particular linear functional is continuous by Sobolev's lemma again . We may write g \—y g(0) as an inner product between quasi-periodic functions:
^(0)=
/
^
-
X]
eAn(x)dx
and this proves (c) and (a) together, (b) follows from the transformation formula between Realizations I and II: In fact, for all f{xi) corresponds to
E L^(R^), /
THE REAL HEISENBERG GROUPS
27
and
(if F C R^^ is a fundamental domain containing the lattice point mi, 7712) = / * ( m i , m 2 ) e ' ' * ""^""^
n£is
As for (d), the transformation formula for i? says that t? is invariant under the required Ug^s,
Q.E.D.
With Theorems 2.2 and 2.3, everything can be fitted together very elegantly. First, the essential relation between the theta function and the Heisenberg representation is given by: COROLLARY
2.4.
FixT £ S^g, let Ti be the Heisenberg representation of
Heis(2g, R), let fr G Woo be killed by WT, and let ex G U-o^ be fixed by (T{T?^).
Then ifx_ = Txi -\- X2j we have: (t/(i,.)/T,e2)=c.e'^*"--^i^(^,T)
for some c G C*. PROOF:
We compute in L^iR'^^//!^^). Then ej = ci J ^ e.(f )^„,
/T = C 2 e " ' ^ ' M £ , T ) , so {t^(l,x)/T,ei} =Cif/(i_^)/T(0)
^cifrix) =
ciC2e"''-^^ix,T). Q.E.D.
28
TATA LECTURES ON THETA III
Recall that in Ch. 2, §5 we defined i?^
X2
(T) = e""''^' (^^1+^2) . ^(j^xi
+ X2, T ) .
In terms of theta-functions with characteristic, Xi X2
iT) =
^2
(0;T).
We will see that in many ways ^ " is the most natural and most important variant oft?. In particular, Cor. 2.4 says that 'd" is just a matrix coefficient of the representation Til Moreover, by looking at matrix coefficients, we construct the quasi-periodic and Fock realizations ofTi directly, i.e.. COROLLARY
2.5.
I The space of functions on R^^ ^ XI—^(f>f{x) = (t/(i,^)/,ez), some f G Tioo (The space ofC^ - functions g{x), x eW^\ I such that V . I g{x + n) = e,(f )e(x, ^)g{x), all nel^' J Hence the space on the right is contained in L^(R^^//Z^^).
Let
L^(R^^//Z^^) be acted on as in Realization II; then the map
is equivariant: ^UgU) = PROOF:
^gi^f)'
Calculating the space on the left in the realization L^(R^^//Z^^),
we see that
so the space on the left is L^{R^3//I'^3)oo, i.e., the C ^ functions in L^(R^^//Z^^). Now we check the assertion concerning the group actions:
THE REAL HEISENBERG GROUPS
29
= (^(Ae(f,y),a:+y)/,e2) X
QED Likewise the Fock representation^ is just the action of Heis on a space of matrix coefficients.
COROLLARY
2.6.
The space of functions on C^ some f G '^-oo ( The space of holomorphic < functions g{x)^x^ G C^ such that ^ • [ gU) e 0(||^||"ei^(^'^)), some n The action of Heis{2g^R)
e~^^^-'-\
on the space on the right given in the Fock
representation is just
AH J, 1 _
^(A.-y)[e^
PROOF:
AH
V'/l^e^^-Vc/^,.,)/.
We calculate the space on the left in the Fock representation 7^?
to which 71 is conjugate linear isomorphic. Here / T corresponds to the function 1, so if / corresponds to a holomorphic function / ( x )
^We state this assuming the identification oiTi-oo in the Fock representation asserted above. If this is not assumed, one can limit the corollary to the subspaces where / G W on the left and g G Ti'i on the right.
30
TATA LECTURES ON THETA III
= j hy) •
Uir,r)fTiy)e-'"^l'S)dy
= e-T«(£.£) f f{y' -x)e^''(yJ'^^ .e-'^^y-'-yJU^ if j / = y -\- X. But by the Mean Value Theorem, there is a consant c such that
for all holomorphic functions g. Therefore
We calculate easily:
= e+^^(^'^).A-ie(|,3/){C/(i,._,)/T,/)
= f/(A,-y)/>) = t/(A,-,)(e^^-V/)(x).
QED
THE REAL HEISENBERG GROUPS
31
We may summarize the situation in the important diagram:
7i*0n
"essentially" all fens. on R'^^jHeis x Heis acting
Space spanned by fens.
& VVr-right-annihilator
\ ^ (r{l?^) — left-invariants
Space spanned by fens. (f/l,x/T,/),
Space spanned by fens. {Ui,xf,e.z),
Fock space
fen-oo Quasi-periodic space L2(R25//Z2^)
H
(o
^^ the unique fen.
32
TATA LECTURES ON THETA III
To explain this, we define the left and right actions of Heis{2g,R)
on the
space of functions on R^^ by
We urge the reader to check that these are commuting actions of Heis{2g^ R) and that Ulty)[{Ui,.)f,9)]
=
{Uii,.)f,U(,,y^g)
t/(t;;[(f^(i.x)/,ff>] = (f/(i,x) • U^,,y)f,9). But as we saw in Proposition 2.3, L^(R^^) is the Heisenberg representation of Heis X H€is/{(X,X)\,X
G C^}, and is isomorphic to H* ^H.
Thus, if
we replaced the space spanned by the matrix coefficients
by the space of L^-convergent combinations V_^a»;{f^(i,a7)/«)/j)j
y j l^i'iP < oo,
/,• E '^ an orthonormal basis,
we would get L^(R^^). As it stands, the space in the upper right is essentially all functions on R^^, in a function space that we do not make precise. What the above diagram does, in essence, is to give a representation theoretic proof that d^ is the only function on R^^ which is a) quasi-periodic for Z^^, and b) multiplied by a simple factor, holomorphic.
THE REAL HEISENBERG GROUPS
33
We can describe the situation by a picture:
The Fock spaces for various T
Mysterious subset of the functions d-[x]{T) for some T
L2(R25//Z2«)
3.
F i n i t e H e i s e n b e r g g r o u p s a n d s e c t i o n s of l i n e b u n d l e s
on
abelian varieties
We have seen in §2 t h a t the t h e t a function occurs as a m a t r i x coefficient of the real Heisenberg representation. In Ch.2, §1, we used a finite set of translates of the t h e t a function to embed various complex tori XT,L in projective space. We now want to relate these two aspects of the theory of t h e t a functions and show how a finite version of the Heisenberg representation theorem occurs naturally in each of these projective embeddings. We first recall some fundamental definitions. A complex line bundle on a complex analytic space consists of (A)
a complex analytic space L and a morphism p : L -^ X such t h a t , for each point x G X,
p~^{x)
is endowed with the structure of a
one-dimensional complex vector space, and (B)
each point of X has a neighbourhood U and an isomorphism (pu • C X U -^ p~^{U)
of analytic spaces such t h a t p{(pu{^i ^)) = ^ and for
each X £ U^ the m a p from C to p~^{x)
given by 2: K-^ <^(/(Z,X) is a
linear transformation. T h e trivial line bundle on X is simply L = C x X , and p : L - ^ X is t h e projection: p(z, x) = x for z E C,x £
X.
A section of a line bundle p : L —> X is a holomorphic m a p s : X —^ L such t h a t ps{x) = x for al\ x £ X. T h e collection of all sections of L forms a vector space denoted by r ( X , L ) . Note t h a t a section of the trivial line bundle is necessarily of the t y p e s(x)
= (/i(x),x) where /i is a holomorphic function on X.
T h u s , in this
case, r ( X , L) is j u s t the vector space of all holomorphic functions on
X.
An a u t o m o r p h i s m of a line bundle L on X is a pair (<;6, ip) where (p and
FINITE HEISENBERG GROUPS
35
ip are analytic isomorphisms of X and L respectively such that the diagram L
-^
> L
P
P
commutes, and in addition, the restriction of V' from p~^{x) to p~^{(f>{x)) is complex linear. The collection of automorphisms of L, to be denoted by Aut L, forms a group under the obvious composition law (<^i, V'l) o (<^2) V'2) =^ {i ^2^'^i^ ip^)' The group Aut L acts on the vector space r ( X , L) by f^(,^,t/^)5 = xj) o s o ^^,
We shall now define an action of Heis(2g, R) on the trivial line bundle on C^ (equivalently, a homomorphism from Heis{2g^ R) to the group of automorphisms of the trivial line bundle on C^) such that the corresponding action of Heis{2g,R)
on the sections, that is, on entire functions
on C^, is identical to the Fock representation ?i|(C^,T) if one ignores the square-integrability hypothesis. Fix T E S^g and hence fix an isomorphism R2^ - ^ C^ via ^ = Tt/i + 2/2 as in §2. For any h = (A, y) G Heis{2g, R) where 2/ = (2/1,^2) put for all z G C^, and
(f)h(z) =z z — y
iph{(^,z) = {aX~^exp 7ri*yi{2z — y), z — y),
for all a G C, 2r G C^.
Then h 1-^ ((^/i, V'/i) is the action we are looking for. If s{z) = {f{z)i z) is a section of the trivial line bundle on C^, then '^hS(l>l^{z) = xl^hsiz + y) =
^h{f{z-\-y),z-\-y)
~ {/{z -\-y)X~^exp = {f{z -\-y)X~^exp which is the Fock representation ?i^(C^,T).
iri^yi{2z-]-2y-y),z) 7ri^yi{2z-\-y),z),
36
TATA LECTURES ON THETA III Now for any lattice L C Z^^, we defined, in §1, a complex torus XT,L =
C^/L. We can now define a basic line bundle L on each of these complex tori. Recall that there is a section a : Z^^ —> Heis{2g, R) defined by cr(n) = (e*(n/2),n) where e*(n/2) = e(^(*ni • 712)) if n = (^^1,^2). Via cr, Z^^ acts on the trivial line bundle on O : n G Z^^ acts by the pair {a{n)y'^a{n))^ and of course, <j>a{n) is simply translation by —n. The action of Z^^, and therefore of any sub-lattice L, on C x C^ and C^ is free and discontinuous. Thus the quotient of C X C^ by L is a complex manifold and we get a commutative diagram C X C^
C^
> L = C X C^/L
>
XT,L
=
C3/L
where the horizontal arrows are universal covering maps and the vertical maps give line bundles on C^ and XT^LDEFINITION.
The basic line bundle on
C^/crL constructed DEFINITION.
XT,L
is the line bundle L = C x
above.
For a line bundle L on a complex torus X,
G{L) = {(<^,V') ^ ^^^ L \ (j) is a translation}, K{L) = {a £ X\ if(j){x) = X + a, then there exists a pair ((^, tp) G ^(L)}. Thus ^(L) comes equipped with the exact sequence: 1 -^
C* —> C7(L) - ^ A'(L) - ^
0.
In fact the kernel of w is identified with the group of all holomorphic nowhere zero functions defined on all of X, which by compactness of X and the maximum modulus principle is just C*.
FINITE HEISENBERG GROUPS
37
Let N{
There is an action
Under the action of cr(jL), x is identified
with (T{t){x) for all i e L, Is Vn(a^) identified with Vn (cr(^)(a?))? Yes, since V'n<''(^)(^) = <^(^)^n(^) for some m G L by normality. This shows that ^„ descends to a map xl)^ : L —)- L. Also, if n G L^ the induced action ot
—>• R^^ be the usual projection. Since 7r~^(L) ncr(L) has
exactly one element, any element which under conjugation normalizes cr(L) actually centralizes it. Thus if ar G N{a{L))^ then 7r(x) G L-^- This gives a homomorphism between exact sequences: 1—^
q
—^
—^
N{(TL)/(TL
L^/L
—^0
Because of our conventions, the reader will see that the vertical arrows here CJ — • C* and L^/L
—y K{L) (between subgroups of XT,L) are both
the reverse of the identity. From the general results of §1, we see that 1 —yC[
—y N{(TL)/(TL
—y L^jL
—> 0
is a Heisenberg group built from the jiniie abelian group L^/L.
Let us
define, for any field fc, an algebraic Heisenberg group over Ar as a group G plus an exact sequence: 1 —^ r —^ G-^K
—y 0
with K finite abelian and fc* = center of G. Then we have: PROPOSITION
3.1. L^/L ^ K{Vj, hence N{aL)/aL ^ g{L) and g{L) is
an algebraic Heisenberg group over C. PROOF:
The homomorphism from N{aL)/aL
to ^(L) already showed that
L-^/L is contained in A'(L). If a G C^ is such that its image m X'j' j^ is in
38
TATA LECTURES ON THETA III
/\ (L), then by applying an elementary lifting argument there is an automorphism {Ta, V') of the trivial line bundle on C^ (where Ta = translation by a) which normalizes the crL-action, and therefore centralizes it. Now putting ^ ( a , z) = {ah{z), z -\- a) where h is nowhere zero, the above condition can be read off as h(z — a)~^h(z — a — y) — exp 27ri*T/ia for all y G i^ and z £ C^. Putting h(z) = exp 27ri/(z), one finds that f{z — y) — f{z) is a constant function for all y E L. It follows that all the partial derivatives of / are invariant under translation by L and are therefore constant because XT,L = C^/L is compact, implying that f{z) = A + g{z) where A E C and ^f is a linear form on C^. Substituting for h in terms of g in the last equation now gives g{y) = ^yia
(modi)
for all 2/ G L. If a = 6 = Tbi -f 62, then 9{y) - ^ypi = *yia=
= A{y,b)
*y6i
{mod Z)
'yi{Tbi-\-b2)-\Tyi-hy2)bi
(modi).
Now L generates C^ as a real vector space, which implies that the complex linear form g{y) — ^ybi takes only real values and is therefore identically zero. This implies that A{L, 6) C Z, or equivalently, that b £ L^. Therefore QED
K{L) = L^/L. Next we consider the sections r{XT,Lj^)
of the basic line bundle L
over XT^L' We easily see that r(Y
t \ c:^ f Sp^c^ of cr(L) — invariant sections of the 1
I A T , L , •-; - I ^^—^j bundle C X C^ on C^
J
^ f Space of entire functions f{z) on C^ invariant by 1 "" \ (T{L) in the Fock representation W^(C^,T) J '
FINITE HEISENBERG GROUPS
39
In fact these invariant functions must belong to the space 7^^(C^,T)_oo, as one checks as follows: a) Define the norm AT : C X C^ — . R+
by
b) a small calculation shows that N(a,z)
= N(xph{(^,z)),
all h G Heis{2g, R),
where il)h{(^,z) = {aX'^exp
7ri^yi{2z-y),z
-y).
c) hence if { ( / ( l ) , z)} is a (T(L)-invariant section, N{f(z_),z) is a function of z^ G C^/L, hence is bounded, d) hence /(l)GO(e'^*"^^^^-^^). Using the characterization oiHfp^-oo before Theorem 2.3 and the map from 7i^ to 7/,^ we see that
v{XT.LX)^{nl{a,T).^r'^, where L = C x C^/L is as constructed above. In fact, r(XT,L5 L) is the irreducible Heisenberg representation of ^(L). To prove this, we next define a whole set of elements e[, ] in the distributional completion W-oo of the real Heisenberg representation space 7i. For all a,6G Q^, set 5 ^ e 2 ' ^ ' " ( " - « ) X - a W in realization I
^^l^^
n6Z3
=
Y^ n,m^T^
e'^*'("-^)('^+^)-6n-a,m-6W
in realization II
40
TATA LECTURES ON THETA III
These are easily seen to correspond by the proof used in Theorem 2.3. Or else, their equality follows by noting that ez = e[ ] is the unique a{I?^)invariant vector of that Theorem and that
Moreover, in the conjugate-linear isomorphism with W^, e[A corresponds to the unique cr(Z^^)-invariant element, namely i?(x), hence e [ , ] corresponds to
-- y ^ g7r.-[*(n+a)T(n+a)-|-2*(n+a)(£+6)]
the theta function with characteristic of Ch. 2, §1. We now prove PROPOSITION
3.2.
Let {ai,bi) be coset representatives for L^ jZ^^.
Then
identifying r(XT,L,L) with its image in 7^^(C«,T)_oo, the i?["*] form a basis ofT{XT,Lj L), and T{XT,LJ tation
ofg{L).
PROOF:
group
L ) is the irreducible Heisenberg represen-
We show that
T{XT^L,^)
NOW
N{(TL)I(TL.
is irreducible under the action of the
any A G CJ C N{aL)/aL
acts on this vector
space by multiplication by A~^, and by §1 it follows that
T{XT,L,^)
=
fC ® fC .. .^ fC where /C is the unique irreducible representation such that A G CJ acts by A"*^. To show that r(XT,L,L) = JC it suffices to prove that the subspace of r(XT,L) L) fixed by crJ?^/(TL has dimension one. Now this subspace of
T{XT^L^^)
is identified canonically with the subspace of
r ( C X C^, C^) fixed by crt^^ For ni G Z^ x {0} and ns G {0} x Z^ we have by the formulas in §2. ^o{ni)f{z)
= f(z -h Tni)exp{7ri*niTni
Ua{n,)f{z)
= f{z
+ n2).
-f 27rini^z)
FINITE IIEISENBERG GROUPS
41
And any holomorphic function on C^ invariant under these two sets of operations is a scalar times 'd(z,T). If Cj = (ci», 6t) form a system of representatives for L^/l?^^
then
i
and therefore the functions [/(i^c^^C-^'^) span a A/'((7Z/)/crL-invariant subspace of r(XT,i,jL). Because r(XT,L,L) is irreducible and its dimension is [ M :Z2^], it follows that
form a basis of
T{XT,L
, L) .
QED
Thus we have arrived again at the situation of Ch. 2, §1: to embed XT^L in projective space, we take the basis ^[,*] for the global sections of the basic line bundle. Now we use: LEMMA
3.3. If L C nL-^, then there is a line bundle M on
XT,L
such that
M®" ^ L. PROOF:
Note that if e^*) is a basis of L, then L is C x C^ modulo the
automorphisms generated by (a, z) K-. (a . e"'^'."(2£-^-i"),^ - e(')),
e(') G L.
Define M to be C x C^ modulo the automorphisms generated by (a, z) K-. (ae"'^*"(2£-Te<")/n^ ^ _ ,(.))^
^(0 ^ i .
Using the fact that ~e^*) G L-'-, i.e., A{e^^\e^^'^) G nZ, one checks easily that these maps commute, hence define the needed M.
QED
Now a theorem of Lefschetz (cf. D. Mumford, Abelian Varieties, p. 29) says that if n > 2, the sections of M^" define a holomorphic map T,L : XT,L —
P"-',
V = [I^^ : L]
42
TATA LECTURES ON THETA III
and if n > 3,
T,L is an embedding. This gives us a projective version
of the Heisenberg representation: the action of ^(L) on
T{XT,L,^)
induces
an action of the finite abelian group A (L) on P^~i which is irreducible, (i.e., there is no linear subspace P^~^ (- pi/-i ^vhi^Ji [^ mapped to itself by K{L)) and which makes
(J>T,L
A'(Z/)-equivariant, i.e., if a G K{L) induces
Pa :P''-^-^P^-Sthen T,L{x + a) = Pa{T,L{^))'
This group action leads in low dimensional cases to very beautiful explicit descriptions o f / m (f)T,L' In chapter I, we studied the case g = \^ L = 2-1?. The cctses 5f = l , L = 2Z-|-Z and ^ = l , L = 3Z-f-Z are the well-known representation of elliptic curves as double covers of P^ ramified in 4 points ± a, ±a""^ and as cubic curves XQ -f Xf -f -^f "+" AX0X1X2 = 0, respectively. The case g = 2, L = 2T? + T? is the representation of principally polarized a 2-dimensional abelian surface as a double cover of a "Kummer" quartic surface with 16 nodes. The case g — 2, L = 42 -{-Z^ leads to a beautiful class of octic surfaces in P^; the case g = 2, L = 5Z + Z^ leads to an interesting story in P"* (cf. Horrocks L Mumford, Topology, vol. 12, 1973). Much of the above theory concerns only the algebraic varieties obtained when a complex torus is embedded in projective space. This part of the theory is really a branch of algebraic geometry and has nothing to do with analysis. We want next to sketch this variant of the Heisenberg circle of ideas. We begin with some basic definitions: DEFINITION.
Let k be a field. An abelian variety defined over k is a pro-
jective variety X defined over k with a morphism f : X x X ^^ X and a k-rational point 0 E X which makes X into a group. For every k-rational point a, let Ta : X —^ X be given by Ta{x) — f{x, a). Facts about abelian varieties that we shall freely use are: LEMMA 3.4. -l:X
-^X.
(A) X is a commutative group, whose inverse is a morphism
FINITE HEISENBERG GROUPS (B) The set of all points
x of X defined over the algebraic closure k of k
such that nx = 0 for some n > 1 is dense in X, (C) Xn={xe
X(k)\nx
43
= 0} ^ (l/niy^,
ifn
and is not divisible
by char k.
W h e n X = XT,L these facts are obvious. For the general case, see D. Varieties^ §§4,6. T h e fundamental definitions related to
Mumford, Abelian
line bundles in the algebraic case are: D E F I N I T I O N . A line bundle L on a variety X defined over k is a p :L ^^ X and isomorphisms UQ of X such
morphism
(pa : Ua 'X f\^ —^ P ~ ^ ( ^ a ) on an open
that
{
X A^ ^ f/« fi C/^ X A^
X f\^) : UaDUp
is given by (x^t) \—>• (x, '^aj3{x)i) where tpap and tp'p are regular on the open set
functions
UaC\Up.
A section of L on an open subset U ofX
is a morphism
that ps = iu where iu is the inclusion morphism of sections
cover
of I. defined on U is a k-vector
s : U -^L
ofU in X.
space denoted
sheaf of sections of L is the sheaf on X given by attaching
The
such
collection
by T(U,L.).
The
to every open set
U in X the abelian group T{U, L ) . This sheaf is a locally free sheaf of rank one and this sets up a one-to-one
correspondence
on X and locally free sheaves of rank one on DEFINITION.
The tensor product
between
line bundles
X.
Li (g)L2 of two line bundles Li and L2 on
X is a line bundle on X such that its sheaf of sections is the tensor of the sheaves of sections
of Li and L2. Equivalently
there is a
product morphism
L i X x L2 - ^ L i 0 L2 making
the fibre of Li (g) L2 over each k-valued
of X into the tensor product
over k of the fibres of Li a n d L2. The
biindiesL(8)L,L(g)L(8)L,... are denoted by L*^^, L ® ^ , . . . . The line L is ample if for some n > 1, L®" is generated sections
define an embedding
L
of X in
by its sections,
and
point line
bundle these
P^.
D E F I N I T I O N . For a line bundle L on an abelian variety X defined over an
44
TATA LECTURES ON THETA III
algebraically closed field Ar, g(L) = {(0, rp) e Aui \.\
for some a e X}
/<(L)=:{aGX|T;(L)-L}.
Clearly there is an exact sequence
1 —^ Jfe* - ^ G{L) —^ K{V.) —> 0.
The main results of the algebraic theory parallel those for locally compact Heisenberg groups and assert the existence of a canonical Heisenberg representation: PROPOSITION 3.5. Consider all k-vector spaces (finite-dimensional or not) equipped with an action of an algebraic Heisenberg group G such that X E k* C G acts by multiplication by A. There is one among these, denoted by Tij with dim 7i = y/#Kj
which is irreducible, and furthermore,
such representation is a direct sum of copies ofTi. Heisenberg
any
Ti will be called the
representation.
PROPOSITION
3.6.
If L is an ample line bundle on an abelian variety X
defined over an algebraically closed field k, with first chern class ci(L), then (A)
dim T{X, L) = •^(ci(L)^) > 0, and this integer is called deg L. If the characteristic of k does not divide deg L, then
(B)
G{\-) is an algebraic Heisenberg group; /\ (L) has (deg L)^ elements, and
(C)
T{X,L)
is the Heisenberg representation
ofg(L).
There is still another variant of the Heisenberg theory to cover the case
FINITE HEISENBERG GROUPS
45
char(k) \ degL. Using the language of schemes, we define group schemes £(L) = group scheme whose R — valued points are the pairs L Xspec k Spec R
—y
L Xspec k Spec R
Xspec k Spec R
—>
X Xspec k Spec R
^
where is translation by an R — valued point of X and tl) is an isomorphism of line bundles; the bijection between points of £(L) and pairs {(fjip) is functorial in R.
K{t.) = the sub-group scheme of X whose R — valued points a (for any local ring R) are those such that if (f) is translation by a, then <^*(L X Spec ifc Spec jR) = L XspecfcSpec i^. If char k \ deg L and k is algebraically closed, then ^ ( L ) will be discrete and finite, and the Ar-valued points of ^(L),
K_(L) will be the ordinary
groups ^(L), K{L) defined above. If char k
f deg L, but k is not al-
gebraically closed, the "Scheme" structure just amounts to an action of Gal(ik/Jfe) on g(L Xjfc I ) and K{L Xklc). But if char k\ deg L,
£ ( L ) may
have nilpotent elements in its structure sheaf. Next, we define a Heisenberg group scheme to be a group scheme G plus an exact sequence
where ^ is a finite abelian group scheme and Gm is the center of G. The main results generalize to: PROPOSITION
3.7. Consider all representations p : G —y GL{n) such that
the center Gm Sicts by p{X) = A • / „ . There is one among these, denoted
46
TATA LECTURES ON THETA III
by Tif with dim 7i = yjorder K_ (here order K_ = dimfc T(OK)), irreducible.
which is
And furthermore any such representation is a direct sum of
copies ofH.H PROPOSITION
will be called the Heisenberg representation of G. 3.8. If L is an ample line bundle on an abelian variety X
defined over fc, then ^(L) is a Heisenberg group scheme, ^ ( L ) has (deg L)^ elements and r(A'^, L) is the Heisenberg representation ofQ_{\-). For proofs, see D. Mumford, Abelian Varieties §16, §23 and T. Sekiguchi, J. Math. Soc. Japan, 29 (1977), p. 709.
4. Adelic Heisenberg groups and towers of abelian varieties The algebraic theory of Heisenberg groups of §3 seems like merely a faint residue of the real Heisenberg representation in the context of projective varieties and might suggest that only a small part of the theta function could be reconstructed from abelian varieties. This is not correct, however, and by using adelic methods, we will see that a purely algebraic theory of adelic Heisenberg groups and adelic theta functions can be given which is quite parallel to the analytic theory. We will explain how this goes in §5. In this section, we will merely define the adelic Heisenberg group ^(L) associated to any line bundle L on any abelian variety X and give a systematic exposition of its basic properties. We begin with some basic definitions: DEFINITION 4 . 1 . Let A he an abelian group. The groups Vp{A), V{A), VP{A) andTp{A),
T{A), Tp(A) are:
i) Vp{A) = the group of all sequences (ao,ai,a2 5 • • •) where ai G A such that: pai^i = ai and p^ao = 0 for some N > 0 ii) V{A) = the group of all sequences (ai,a2,a3,...) where ai E A such that: mamn = «n and Nai — 0 for some N > I in) V^(A) is the same as V(A) except that the a^ are defined only for i not divisible by p and there is an N, not divisible by p, such that Nai = 0. JV; Tp(A),T{A)
and TP{A) are the subgroups ofVp{A),V{A)
and VP{A)
given by ao = 0, ai = 0 and ai = 0, respectively. Note that VJ,(Q/Z) is the field of p-adic numbers Qp;Tp(Q/Z) is the subring of p-adic integers Z^; V^(Q/Z) is the ring of finite adeles A/; T(Q/Z) is the subring of integral finite adeles, Z, a completion of Z; \^^(Q/Z) is the ring of finite adeles without the p-factor, which we write A ^ , and T^(Q/Z)
48
TATA LECTURES ON THETA III
is the subring of integral finite adeles without the p-factor, which we write
LEMMA 4.2.
T{A) = Jl^^i^) t
^^^ TP{A) = YITI{A) i^p
where the £ range
through all primes J and V{A) and VP{A) are the ''restricted"
direct prod-
ucts: V(A) = Uai) e Y[Vi{A)
I all but finitely many oct belong to Ti{A)\
I
V^{A) = Uai) e Yl M^)
I ^ii ^"^ finitely many at belong to
Ti{A)\.
There are exact sequences, 0 ^ T{A) -^ V{A) - . Ator -^ 0 O^T,{A)^V,{A)-^A{p^)^0 0-.TP{A)-^VP{A)-^Al,-^0 where Ator is the torsion subgroup of A, A{p^) is the subgroup of A of points of order p^ for some N, A^^^, is the prime-to-p torsion subgroup, and the homomorphism from V{A)
to
Ator is given by (ai, a2, «3, • •) —^ «!•
We omit the proof which depends only on the fact that any element a of finite order in any abelian group A has a unique decomposition a = /^^cit where each ai is annihilated by some power of i. DEFINITION
4.3. Let G be any commutative group scheme defined over a
field k. In practice we shall put G = an abelian variety or G = G^ (the multiplicative group scheme of the field k) only. Let G{k) = all points of G defined over the algebraic closure kofk.
We put
Vp{G) = Vp{G(k)),
ifp ^ char, k,
Tp{G) = Tp{G(k)),
ifp ^ char, k,
V{G) = V{G(k))
and T{G) = T{G{k))
ifchar.k
= 0,
and T{G) = TP{G(k))
if char. k = p.
and finally V{G) = VP{G(k))
ADELIC HEISENBERG GROUPS
49
Actually, these groups remain unchanged if k is replaced by ksep the separable closure of k, and all of them are acted upon naturally by the galois group LEMMA
Gal(kgep/k).
4.4. For an abelian variety X of dimension g defined over k of
characteristic 9^ p, the pair {Vp{X), Tp{X)) is isomorphic to (Qp^^lp^) and the pair (V;,(G^),Tp(G^)) to (Qp,Zp). Also the pairs {V{X),T{X)) {V{Gm)iT{Gm))
and
Sire isomorphic to (A^^,Z^^) and {FKf.Z) when the char-
acteristic ofk is 0 and to
((AJ^))^^,
(Z(^))2^) and {f\^f\i(P^)
when the
characteristic is p. PROOF:
The second statement follows from the first after an application
of Lemma 4.2. Let X(p^) = all points of X defined over k annihilated by some power of p. By definition V^(^) is the inverse limit of the chain of arrows:
There is an isomorphism X{p^) = (Qp/Zp)^^. Once this isomorphism has been fixed, Vp{X) is the inverse limit of: >
(Qp/Zp)2^
- ^
P' Q^^
(Qp/Zp)^^
- ^
1
p\
^ (Qp/p'^P?" —
i^/pip?"
(Qp/Zp)^^
—
(Qp/Zp)^^
which by the commutative diagram above and the completeness of Q^^ in the p-adic topology is just Q^^. It is clear that Tp{X), given by the first member of the chain = 0, gets identified to Z^^. The Lemma holds for G ^ because G ^ ( p ~ ) ^ Qp/Zp holds too.
Q.E.D.
On various occasions below we will fix an isomorphism e:A//Z-^I*^^ (or f\f/2^^^ such a map.
- ^ k^^j. in char, p) and we will always use bold-face e for
50
TATA LECTURES ON THETA III
DEFINITION
4.5. If E is a line bundle on X and f :Y —^ X a morphism,
then the hne bundle f*EonY
is the fibre product:
rE =-.Y
xx E
i
*•
E
i.
/^ X
Y
If g = (<^, tp) G Aut E and (f)^ : Y -^ Y is an isomorphism so that <j) o f = f o (j)' then (<^',V'') ^ Aut f*E is the lift of g covering <j)' where ij^^y^i) = i\y),rP(^)) for all{yj)
eY
XxE-
The most important group associated to a line bundle L on an abelian variety is: DEFINITION
4.6. Assume first that characteristic k = 0, Then g{L) = all
sequences of pairs {xnj<j>n) where: A.
The Xn G X{k) deBne a member of V{X),
i.e., mxmn = ^n and
Nxi = 0 for some N > 1. B.
The(i> n are defined for all n such that Xn G A'(n*L).
^'
(^iCmn J ^mn) is the lift of (Tx^j n) covering Tx^^ as in Definition 4.5. with E = nmj^L, Y = X^ / = mx-
D.
The group law in G{L) is given by iXn,n)o{yn,i^n)
= (Xn -\-yn,
n ^ tpn)
In G{l-j if char. A: ^ 0, then (x„, <^„) are to be defined only for n not divisible by p, and 3N such that p )[N and Nxi = 0. The power of ^(L) is due to this: For any element a: = (xi, • • •) G ^ ( - ^ ) , there is an N such that for all n divisible by AT, x„ G K{n*L). have PROPOSITION 4.7.
The group G{L) has an exact sequence:
1 ^^ F
—^ g{L) - ^ v{x)
-^
0
Indeed we
ADELIC HEISENBERG GROUPS
51
where TT takes {xnyn) to the sequence (a?„) defining an element
ofV{X)
and the inclusion of k in ^(L) is given by a i—^ (x„,<^„) where Xn = 0 and (j)n = a(n*lL) for all n. Further, k is contained in the centre ofQ{\J). P R O O F : TO
see that
TT
smallest integer so that
is surjective, let {xn] € y{X) and let A^ be the NXQ G A ' ( L ) ,
so
TNXQV-
= T^va^jvL 0 L~^
0
L~^
is trivial. Now
(by the theorem of the square)
= (trivial bundle). QED COROLLARY
4.8. Jf <^ : Li ^ L2 is an isomorphism of line bundles on X,
then the induced isomorphism ofG{Li)
and G{L2) is independent of (p.
check the Corollary, it suffices to show that any isomorphism
P R O O F : TO
<^ : L = L induces the identity isomorphism from ^(L) to itself. And this is so because any such (p is ai_ where a £ k and therefore the corresponding automorphism of ^(L) is conjugation by a G ^ C ^(L) which is the identity.
QED
The next proposition describes the functorial nature of ^(L). PROPOSITION
4.9. If f :Y -^ X is a homomorphism of abelian varieties
and L is a line bundle on X, there is a commutative 1 — . fc* — . |l 1—y
G{rVi
-^
|j(/,L)
k* —>
g{L)
diagram: V{Y)
—>0
|F(/) -^
v{x)
—.0.
if Li and L2 are line bundles on X and H = {gu92) e d{Li) X g{L2)\ 7r((/i) = 7r(^2)} and A = {(ai,a2) els* xlc* C H\ aia2 = 1}, then there is a diagram: 1
k*
H/A
V{X)
\h
1
e(Li(8)L2)
V{X)
0
52
TATA LECTURES ON THETA III
PROOF:
Let {ynjn) be a sequence in £?(/*L). Choose an A^ such that
A^'^t/Tv ~ 0; this implies that <J)N is defined. Then /{yw) € A'(A'^L) allowing one to choose a {Tj(^yj^)^P) G Aut A'xL. By modifying /? by a unique element of k one may assume that the lift of (T}(y^), /?) to Aut f*N^L
is
(Ty^,<^iv)- Let {f{yn)yn) be the unique sequence in ^(L) so that 7v = /?. Put j{fX){yn,n) = {f{yn),n)' This gives j ( / , L ) in a well-defined manner, and gives j ( / , L)(a) = a for a G A^ ; that TT o j ( / , L) = ^ ( / ) o TT is immediate. We shall content ourselves with defining /i' : /f —^ Q{\®^2) such that /i'(^) = 0. Let u = {xn,(i>n) G ^(Li) and v = {xn.tpn) G ^(Ls). Then (Ti:„,n) and (Tx^jipn) are in Ai// '^x'-i and ylw^ ^x'-2 respectively; they induce {Tx^j(j>n 0 ^n) an automorphism of Li 0 L2. Now put h'{u^v) = {Txn^^n ^ i^n)' This satisfies h\A) = 0 and gives a factoring of h':
h' : H -^
HlA^g{U^U) QED
The homomorphism TT : ^(L) —> V{X) has a section o^ over the subgroup T{X) C V{X): DEFINITION
4.10. If x e X„, let {Tj:,an{x)) G Aut n^^L be the lift of
(IA-, 1L) G Aut L covering r^. For x G T(X), w/iere x = (a^i, ^2, 2^3,...), leta^{x)
eG{L) be the sequence of pairs (x„, cr„(x„)). Alternately,
(T^{X)
is the unique element of g{L) with 7r(cr^(x)) = x and <j)i — 1L {wheie n))' LEMMA
4.11. The map (Tfi defines an action of X^ on ?^^L so that
p{crn(x)£) = p(£) + X. The quotient of n^L by this action is precisely the line bundle L on X. The functoriality of a^ is given by:
ADELIC HEISENBERG GROUPS LEMMA
4.12.
53
With notation as in Proposition 4.9, we have
j(fXW^^{x) for all X e T{Y) and h{a^'{x),
= a'^{V{f){x))
(T'-2(X))
= a^'^^^{x)
for all x G T{X).
Lemmas 4.11 and 4.12 follow by just plain checking. PROPOSITION
4.13. The three subgroups ofQ{L) defined below are iden-
tical: (A) the normaliser of
a^T{X),
(B) the subgroup of{xn,(t>n) with xi G ^"^(L), and (C) the elements g ofQ(L) which have a representative of the type {xn^n) with
is isomorphic to the
subgroup of elements g of ^(L) with 7r{g) a torsion element of K(L). PROOF:
Step I : We prove ACB,
where g = {xn,<j>n)' Hue showing
i.e., if ^f G N{a^T{X)),
(T^T{X),
then gug-^u-'^
then xi G A^L)
e k* n
(T^T{X)
= 1
g commutes with u. Assume that (pm is defined. Reading off"
the m^^ element of the sequence for gug''^u'^^ one sees that <j>m(^m{y) = o'm{y)m for all y G Xm- By Lemma 4.11, because (f)m : ^ x ' - —^ " ^ x ^ commutes with the (Tm-action of Xm, there exists ^ : L —)• L and a commutative diagram
L
-^
L.
Clearly {Tmxm ^ V') = {Tx^, V') € Aut L implying that xi G A"(L). Step I I : Next prove B C C. If g = (a:„,0„) has xi G K(L), then <;!>i iiIS defined and so Xn G K{n*L) for all n. Step III : If ^ = ixmj
This is so because both go^{y) and (T^{y)g have
their m^^ coordinate as the lift o{{Tx^,(f>i) covering Tx^+y^.
54
TATA LECTURES ON THETA III This shows that the three groups are equal. Now define h : Ar((7'-T(X)) - . e(L)
K{xi, i), (x2, h ) . • •.) = (^x,, i) e g{L). By definition the kernel is just a^T{X)
and the image is all (T^^^, i) G
^(L) with xi a torsion point of A'(L). In particular, if L is ample, GiL) ^
N{a''T(X))/
PROPOSITION
4.14.
X. For x^y £ y{X)
Let X be an abehan variety and L a line bundle on choose x, y E ^(L) with 7r(x) = x and 7r(y) = y. Put
e^{x, y) ='x y 'x''^y~^, A)
then
e'-(x, y) is independent of the choices ofx and y and is an alternating form with values in the group of all the roots of unity in k .
B) e^{x,y) =
lforx,yeT{X).
C) LetA'-{x,y)
= {e^{x,y),e''{^,y),e'-(f,y),,.)inV{Gm).
Then
A"- : V(X) X V{X) — . V{Gm) is an alternating f\f-bilinear
forni on V{X), and if
0 ^ T{Gm) -
V{Gm) ^ k l , ^ 0
is the exact sequence of Lemma 4.2, then e{A*-{x,y)) = e^{x,y)
for aUx,ye
V{X).
D) IfL is an ample line bundle on X, then define TiX)^
= {ye
ViX)\e*-{x, y) = 1 for all x G T{X)}
= {ye
V{X)\A'-ix,y)
e Z for allxe
T{X)}.
Then [r(X)-^ : T(X)] = (deg L)2 in characteristic 0, or the primeto-p factor of (deg L)^ in characteristic p. ( See Proposition 3.4 for definition of deg L.)
ADELIC HEISENBERG GROUPS E) In particular, if deg L = 1, then T{X) =
55
T(X)-^.
F) It f \Y -^ X is a homomorphism of abelian varieties, then A'-iVif)x,
Vif)y)
=
Af''-ix,y)
and e'-{Vif)x,Vif)y)
= e^'*-{x,y)
and finally, G) A*-'®*-'{x,y) = A*-^ix,y) +
A'--ix,y)
and
The group-law on V{Gm) is written + rather than • because it is an A/-module. We give only some of the proofs. A) e^{x,y)^
= e^{mx,my)
— 1 if TTI is chosen so that mx and my both
belong to T{X). B) The existence of the group homomorphism cr*- : T{X) -^ ^(L) shows that e(ar, t/) = [the commutator of o^{x) and
A ' - ( X , X)
is additive in x. Also e'-(x,^) - e ' - ( ^ , ^ ) " ' = 0
= 0. We omit the rest of C).
56
TATA LECTURES ON THETA III
D) Letting Xf^^ be the prime-top torsion subgroup, the sequence
0 -^ T(X) - V{X) ^ XI, -^ 0 from Lemma 4.2 allows us to identify T{X)-^ IT{X) with a subgroup of Xf^^. It follows from Proposition 4.13 that this subgroup is precisely Xf^^n/ir(L). By Proposition 4.4, K(\-) has (deg L)^ elements and is therefore contained in XiorF) This follows from Proposition 4.9. Put j = j{f,L).
Choose x,y E
G(/*L) so that 7r(x) = x and 7r{y) = y. Then: e^ '^{x,y) =
xyx~^y~^
=
j{xyx~^y~^)
=
e^{7rj{x),7rj(y))
=
e'-{V{f)x,V{f)y)
Now A^ is defined using e*- so the same formula holds for it, and G follows from Proposition 4.9 too in a similar manner.
QED
LEMMA 4.15. Let K be the reduced connected component of/^(L), where L is a line bundle on an abelian variety X. A) The null-space ofe^ is V{K), and e^ gives a non-degenerate pairing on
V{X)IV{K). B) Also, for any prime p ^ char, k, the null-space of e^ on Vp{X) is Vp{K). C) IfK{L.) is finite and an isomorphism A/ = y{Gm) is fixed, then there is an isomorphism V{X) = f\f such that A^ becomes the form^xi'y2 — ^X2'yi for all (a^i, 3:2), (2/1,2/2) G A^^, and T{X) becomes a subgroup of finite index in 1?^ (replace A/ by FvP in characteristic p > 0). PROOF
A: The pairing e : K{\-) x A (L) —>• Gm is trivial when restricted
to K X K(L) because K is complete, connected and reduced and e^(0, x) = 1. The same argument shows that e"^*- is trivial on A' x K(n*L) D K x Xn
ADELIC HEISENBERG GROUPS
57
for all n > 1 because K C /ir(n*L), and consequently, e^(x,y) = 1 for all X G V{K), y G V{X),
Denoting the induced pairing on V{X)/V{K)
and the image of T{X) in V{X)/V{K)
by T, we have T^/T
=
by e*K{V)/K
by 4.13, where T^ = {xe
y) = 1 for all y G T } .
V{X)/V{K)\e^{x,
Thus [T-L : T] is finite, showing that e"- is non-degenerate. B) follows from A) by looking only at Tp{X), Vp{X). To prove B) implies C), we cite the normal form for non-degenerate skew-symmetric forms on each Vp{X), defining Vp{X) - ^ Q^^ separately for each p such that Tp{X) goes over to Z^^ except for finitely many p. All these isomorphisms together give us the isomorphism of V{X) with A / . We leave the details to the reader.
QED
We add the last bit of structure to ^(L). Assume L is isomorphic to i*L where i is the inverse map. This induces an automorphism of ^(L) which allows us to define a section r to the map TT : ^(L) -^ ^ ( ^ ) - The section r is the key to an algebraic definition of ^. PROPOSITION 4.16.
Let Atd^X
be the group of all automorphisms f of
X such that /*L = L. Then there is an action of the group Aut^X
PROOF:
on
Any / G Aut X induces a i ( / , L) as in Proposition 4.9: GirL)
-^
\jiLL) G{L)
V{X) \f
-^
V{X),
By functoriality j(/,L)oj(5f,/*L) = j ( / o y , L ) . Now / G Aut^{X),
so there
exists A : /*L = L; further the isomorphism induced by A : G{f*L) -^ G{L) is independent of the choice of A (Cor. 4.8). Thus j{f^ L) can be regarded as an automorphism of ^(L) when / G Aut^X; furthermore if / , ^ G Aut^X,
TATA LECTURES ON THETA III
58
then j(f,L)J{9X)
By Proposition 4.9, j ( / , L) acts on the
= j{f ^ gX)-
exact sequence for C?(L) as follows: 1~^
r
—^
^(L)
1
-^
V{X)
—.0
y{f)
KfX)
V{X)
1
0.
QED DEFINITION
4.17. A line bundle L on X is symmetric if ( - l x ) * L = L, or
equivalently if — lx € Aut^X.
The involution j ( — l x , L ) on G{L) will be
denoted by f'-, so that there is an exact sequence: V( X) -1
1—>
r
-^
g{L) —
ViX)
0.
This allows us to make the important DEFINITION.
Let x e V{X) and choose y e §(L) such that 27r(y) = x.
Define r(x) G ^(L) to be REMARK 1: REMARK
2:
W{T{X)) T(X)
=
yi^{y)-^. x.
is independent of the choice of y. To show this note
y can only be replaced hy a - y with a E k* and ay(i{ay))~^
== y{iy)~^
because i(pcy) — oii{y). We shall abbreviate (T^, Z^, etc., to cr, ?, etc. when there is no possibility of confusion. PROPOSITION
4.18.
Let L be a symmetric line bundle on an abelian va-
riety X. A.
i o a(x) = cr(-x) for x G T{X).
B.
Any element ofQ{L) can be written uniquely as A • T{X) with X E k
and X G ^ ( ^ ) ; the multiplication X Xr{x) • fiT{y) = Xfie^(-,y)T{x
table ofQ(L) is given in this set-up by 1 + y) = A/ie(-A*-(x, y))T{x + y).
The involution i is given by i(XT(x)) =
XT(—X).
ADELIC HEISENBERG GROUPS C.
If we define e\: by
(T^{X)
quadratic form on ^T{X)
= e!r(f )r(x), for all x G T{X),
y)eUx)-'eUy)-'=e'-{x,y)\
Lemma 4.12 says j(/,L)cr-^ "-(x) = a^Vf{x)
A.
Ji-lx
then e\^ is a
with values in the group ±1; e^ satisfies:
e'^{x + PROOF:
59
,L)
which gives for
= a*-{-x),
which in our notation reads as i^a^(x) = a^(—x). B. We verify the multiplication formula: assume 27r(p) = x and 27r(q) = y, then XT{x)fiT(y) =
Xfipi{py^qi{q)~^
= A/ie'-(7r(p2(p)~^),
Tr{q))qpi{p)-^i{q)-^
Also i(Ar(x)) = ii^piip)"^) C.
a^{2x) = a^{xy
= Xi{p)p~^ = Xp~^i{p) =
= e^Cf )V(x)2 = e\:{^yT{2x)
o-^(2x) for all x G T{X) because T{2X) = a^{x)ia^{x)-^
(T^{X)
XT{—X).
by B. But T(2X) =
is a lift of x so that by definition
= a^{x)a^{x)
=
(T^{X)\
This shows e|[r(|)^ = 1 for all x G T{X). It only remains to check that
We have
60
TATA LECTURES ON THETA III
by the multiplication formula in B. On the other hand, by the definition of ei" we have (7 (a: + 2/) = e^ {—^)r{x
-h y). QED
LEMMA
4.19.
Any non-degenerate quadratic form in 2g variables over 9
9
Z/2Z is equivalent to either Y^a,-6,- or a\ -h ai^i -\-h\-\PROOF:
y^ajhi. 1=2
»=1
Let q be the quadratic form and A{x^ y) = qix-hy)'-q{x)
— q{y) the
associated alternating form. If ^ > 1 and x 9^ 0 there is certainly a non-zero y different from x such that A{x^ 2/) = 0 because {i/|yl(x, y) = 0} has 2^^~^ elements. For such a pair at least one of the values q{x), q(y), q(x + y) is zero. So let vi ^ 0 be such that ^'(^^i) = 0 and choose V2 such that q(v2) = 0 and A(vi,V2)
= 1, and take the orthogonal complement and
proceed by induction until one has chosen a basis so that 9-1
«=i
The only possibilities for the last bit in ag and hg are 1)
2)
a ] - ^ a g h g ^ h ]
aghg
3)
a g { a g - ^ h g )
4)
{ag
4-
hg)hg
,
and 2), 3) and 4) are all equivalent. It only remains to show that the two quadratic forms in the statement of the lemma are inequivalent. Simply count the number of zeroes. The first hcis 2^^"^ -h 2^"^ and the second has 2 2 ^ - 1 - 2 ^ - 1 zeros.
QED
From this, it is now easy to show PROPOSITION
4.20.
Let L be a symmetric ample line bundle of degree
one on an abelian variety X. {V{Gm),T{Gm)) with (V{X),T{X))
Fix an isomorphism, once and for all, of
with (A/,Z). Then there is an isomorphism
of{f\f,P^)
so that A^ and and e^ are now given by A^{ui,Vj)
= Sij
ADELIC HEISENBERG GROUPS
61
and A^{uiyUj) = A^(vi^Vj) = 0, and
i
i
where (wi, W2, * * *, w^, i;i, t;2, • • •, i^^) is a basis. Of course this means that the Ui and Vi form a Z-basis for T{X)
and an A/-basis for V{X).
The
L 's for which the first normal form holds are called "even symmetric" and those for which the second holds are called "odd
symmetric".
One uses Proposition 4.14E, Proposition 4.18C, Lemma 4.19 and works a little to get to the proof of this fact. If the characteristic is p, replace f\f and Z by Rp^Sp. We remark here that this problem of the two choices for e* occurred earlier in §2. We had L, a maximal isotropic lattice in V, and found that the collection (±1,^) with £ E L formed a subgroup of Heis(2g, R). Calling this subgroup G for the moment, the exact sequence
0-^{±l}-^G^L^
1
has its splittings in one-to-one correspondence with quadratic forms:
£ i—>•
((~iyW,^)isasplittingifandonlyif/(^i4-^2)-M)-/(^2) = ^(4,^2) (mod 2). And under the symplectic group of the lattice L these quadratic forms break up into two orbits (Lemma 4.18). Only in the case ^ = 1 is there a canonical choice: a^ -h a6 -f P has the property that any quadratic form is equivalent to this one. This corresponds to the fact that there is a canonical choice in an algebraic equivalence class of line bundles of degree 1 on an elliptic curve, but not so when g > 1. The subgroups of V{X) can be used to classify isogenies. DEFINITION 4.21. A "good" isogeny f : X —^ Y of abelian varieties is a homomorphism which is surjective and has finite kernel whose order is prime to char. k. A rational "good" isogeny or a Q-isogeny from X toY is a triple
62
TATA LECTURES ON THETA III
(Z, / i , /2) where Z is an abelian variety and fi.Z—^X are good isogenies.
and j ^ '- Z —^Y
In what follows, we shall omit the adjective good,
but always make the ^^prime-to-p" assumption.
Two Q-isogenies (Z,
fi.S^)
and (W, gi,g2) are equivalent if there is an abelian variety R and isogenies a : R -^ Z and b : R —^ W so that fioa
= gi o h for i — I and 2. A line
bundle L on X and a line bundle L' on X' are related by a Q-isogeny if there is a Q-isogeny (Z,/i,/2)
from X to X' such that fiL = /2L'.
Many of the formal properties of isogenies carry over to Q-isogenies. Since an isogeny / : X —y Y induces an isomorphism V{f) : V(X) V{Y),
a Q-isogeny a = (Z,fi,f2)
—y
from X to y induces an isomorphism
V{a) : V{X) —^ V{Y) defined by V{a) = ^ ( / s ) o V^(/i)-^ Equivalent Q-isogenies induce the same map. LEMMA
4.22.
Equivalence classes of Q-isogenies from X to Y for a fixed
X are in 1-1 correspondence with compact open subgroups ofV{X),
hence-
forth referred to as lattices. P R O O F : We can describe the correspondence between isogenies and finite subgroups of X in a slightly different way. An isogeny induces: 0 —^ T(X) —^ V(X) — . Xtor —^ 0 0 —.
T{Y)
—
V{Y)
—.
Ytor
—> 0
as in Lemma 4.2. It follows that ker / can now be identified with the group V{f)-^T(Y)/T{X). L{a) = V{a)'~^T{Y).
Associate to a Q-isogeny a from X to Y the lattice Note that L{a) depends only on the equivalence
class of a. Conversely, given a lattice L C y(X) m T{X) C L. Fut Z = X and fi = mx. contains T(Z) and M/T(Z)
there is some m > 1 for which Then V{fi)-^L
gets identified to a finite subgroup H of Z.
Now let f2 : Z -^ Y be the quotient map of Z by H. V{f2)-^T(Y) V{a)-'T{Y).
= M C V{Z)
= M and therefore L = V{fi)M
It follows that
= V{fi)V{f2)-^T{Y)
= QED
ADELIC HEISENBERG GROUPS
63
Let / : X —> Y be an isogeny and let L, M be ample line bundles on X and Y respectively so that /*(M) = L. We can recover M from ker(/) plus a splitting of the natural map ^(L) —>• K{L.) over the subgroup ker(/) C K{Ly.
g(L) —
KiL)
^-.^
T
ker(/).
Equivalently, to give (F, M) we need to give a lattice L with T{X) C L C V{X) and an extension of a : T{X) —^ G{\-) to L. This procedure generalizes to rational isogenies. Let a = (Z, / i , /2) be a rational isogeny from Xi to X2, and let Li and L2 be line bundles related by a, so, / i Li = /2L2. We write a*L2 = Li. Z h /
\
/2
Xi
>X2 a
Since j(/,-,Lj) : G{f^V.i) —> Qi\-i) ^tre both isomorphisms we can define an isomorphism j{oc) : ^(Li) —^ G{^2) by j{a) = i(/2,L2) o j ( / i , L i ) - ^ Let L(a) = V{a)-'\TX2)
and define a{a) : L{a) —^ ^'(Li) by a{a)x =
j(a)""^(cr'-2(F(a)x)) for all x G L{a). Conversely if L is a lattice in V(Xi) and ai : L —)- ^(Li) agrees with a^^ on LnT{Xi),
then we can construct
a rational isogeny a : Xi —> X2 and a line bundle, L2, on X2 so that «*(L2) = Li. PROPOSITION
4.23.
A. If Li and L2 are both symmetric line bundles
related by a, then i*-^ • j{a) = j(a) - i^^ and i^^a(a)x X€ B.
L{a). If Li is assumed to be symmetric, then L2 is symmetric if and only if
i^^aiix) C.
= a{a){—x) for all
= (TL{-x)yx
G L.
Choose a G Xn and ^ = ( 6 , 6 , ' - J
G ^ ( ^ ) ^ith 6 = «•
Let
E = T*L. Then n ^ L = ?^x^' ^-^-^ ^ — ( ^ ' ^ x , ^ x ) is a Q-isogeny from X
64
TATA LECTURES ON THETA III
to itself by which L and E are related. The pair {L(a)j ^{^)) associated to the triple ( X , E , a ) , is then: L{a) = T(X) given by a{a){x) = C^^i^)^
and a{a) : T{X) —^ ^(L) is
= ^^{C x)a^{x)
where ^ is some lift of(^ to
a(L). D.
is symmetric and2a = 0, then E is symmetric
In the notation ofCifL
ande^(x) = c!r(x + Oe!r(0PROOF:
i^^j{a)
A. follows quickly from the definitions. = j{a)i^^
For example, showing
reduces to the case of a plain isogeny. B. is left to the
reader. C. beX
For g E Xny let
{T^(f>g}. The diagram n L — • -u n L —^ TgT^n L —>• TgU V. shows that (^,6)-i-(,^,5).(^,6)
is the descent data needed to construct T^L from n*L = T^n*L. To finish it is only necessary to formulate this in terms of the big group G(L). D.
If 2a = 0, then the line bundle E = T^L is symmetric because TE = i*T*L = T1J*L ^ T ; L .
Define j : ^(L) —>• ^(E) by fitting together the various
jm:g{m*L)^gim*E) for m divisible by 2. For cp G ^(L), by the construction of j , j{(fi) G ^(E) over the fiber x E X "is" (p{x). Consequently joi^((p) is (p{—x) on the fiber over X E X^ but by the same kind of reasoning i^ o j((f) "is" (p{—x) over
ADELIC HEISENBERG GROUPS X E X. Hence i^oj
= joi^.
This implies r^ = jor^:
65 Let 2y = x E
y{X),
and y G ^(L) such that 7r(y) = t/. Then T^{x) = j(y)(i^ oj{y))-'
= j(y){j o
i^{y)y'
= j{y o i^iVT^) = i o ^^ WApply j " ^ to the equation
(T^{X)
= T^{x)ef{x/2)
and we get
=
T*-ix)e*:{l)eHtx),
= r'-(x)e:r(|+Oe!:(0
(by 4.18C).
Now by comparing the RHS of the first and last equations we get D. QED COROLLARY
4.24.
If L is a symmetric line bundle of degree one on X,
there is a point a E X2 such that T*L is even symmetric. PROOF:
By 4.23D, it suffices to prove that if ^ is a non-degenerate quad-
ratic form in 2g variables with Z/2-coefficients, then there is a t/ E (Z/2)^^ such that Qy defined by qy{x) =• q{x -\- y) — q{y) and q are non-isomorphic quadratic forms. But if q has s zeros and qy has t zeros, it is clear that s = t if q{y) = 0 and s + t = 2'^^ if q{y) = 1. But s ^ ^. 2^^, by Lemma 4.19, showing that any y with q{y) = 1 will do.
QED
The following will be of use in §6. LEMMA
4.25. If e^^ = e*-^ and ejr^ = ejr^ for symmetric line bundles Li
and L2 on XJ then Li = L2 (assuming char k ^ 2). PROOF:
If
L
=
LI
0 LJ^, then e*- = 1 and ejr = 1. By 4.15 we have
K{\-) = X and the group-scheme G{L) with involution i sits in the exact sequence: 1 —»• Gm —^ ^(L)—^X —> 0. Mimicking the construction of r, define h : g{L) -> C?(L) by h{x) = X'i{x)-K
Then h{Xx) = h{x) for A G G^
66
TATA LECTURES ON THETA III
and X E ^(L) showing that /i = <^ • TT and w - <j) = 2x where <j) : X —^ ^(L) is a morphism. Now (f) is actually a homomorphism: rp{x,y) = <j){x -\- y){x)~^<j){y)~^ defines a morphism from X x X to Gm such that ^^(0,0) = 1, and therefore V' = 1 (because X x X is projective and Gm is affine). Also for any x G X2, {x) = ellrC^) where ^ = ( x , . . . ) € ^T{X).
This shows that (f) is trivial
on X2 so that <^ = 2r where r : X -^ ^(L) is a homomorphism so that WT = Ix'
The morphism from L(0) x X to L given by {a,x) -^ T{x)a is
an isomorphism and this finishes the proof that L is the trivial line bundle. QED A construction which makes the definition of ^(L) and the theorems that followed more transparent is: DEFINITION 4.26. Consider the so called ''tower'' of all isogenies f :Y —^ X (degrees prime-top if characteristic k = p):
\
\
y yi
/ Y2
X More precisely J we have a partially ordered set whose elements are isogenies f :Y -^ X and f >• g means there exists a homomorphism h with f = goh. Cofinal in this tower are the isogenies nx : X —^ X under the ordering ux >" THx ifTn\n. DEFINITION 4.27. Let X be the inverse limit of the tower on X so there are maps fn'.X-^Xso
that nxfmn = fmX=
Put f\ = f.
lim y.
fY-^X
The limit X has the structure of a proper k-scheme.
Put
E = /*L. A,
Then d{L) is the subgroup of ^(E) which sits above V{X) C K{E) C geometric points of X.
ADELIC HEISENBERG GROUPS
67
B. X is the quotient ofX by T{X) and L is the quotient of E by
(T{T{X)).
It may seem that X is a rather abstract and "unreal" sort of object. This is not so: to make it explicit, let {£/«} be an open affine cover of X, and consider the inverse and direct systems:
r{nm} iU^,Ox)
{nm)^\Ua)
C
X
m*x
mx
rinx'Ua,Ox)
n-^(C/«)
C
X \nx
"x V{Ua,Ox)
Ua
Let Ra = limT{n^^UaiOx)-
C
X,
Then Spec Ra is a scheme over Ua, and
n
X is formed as the union of the affine opens Spec Ra^ In characteristic p, a more comprehensive theory will be obtained if we replace X by the inverse limit of all coverings nx : X
-^X
including n = p*. If p|n, nx is inseparable, and kev(nx) must be considered as a group scheme. We can construct V(X) as a formal scheme, viz. the direct limit of schemes /"^(Xn)
C X for all n. Likewise, we get a formal
group scheme ^(L) extending V{X),
An immediate advantage is that for
L ample, e*" is non-degenerate on p-torsion too, and that in 4.22 we can treat all isogenies. Our primary interest is in characteristic 0 however and we will not discuss this further here. We now tie together the algebraic approach of this section with the analytic approach of §3. Now we shall put k = C,X = XT,L as in §3 where T £S)g and L C Z^^, and L the basic line bundle defined on X. For x € Q^^,^ = Txi -f X2, the sequence ^ x (or rather its image in XT^L) gives an element of V{X),
Thus,
there is a homomorphism from Q^^ to V{X)^ inducing an isomorphism Q2i? (8) A/ = A^^ ^ V{X) so that T{X) is the closure of the image of L.
68
TATA LECTURES ON THETA III Recall that L was defined, using a particular action of Heis{2g,R) on
the trivial line bundle on C^ in which (A, y) = (A, 2/1,2/2) acts on C x C^ by {a,z)
I—^ {QX^^(py{z),z - y) = (aX-'^exp{Tri\yi)
' {2z - y)),z - y).
This is the Fock action. The bundle L is the quotient of C x C^ by the maximal isotropic subgroup of Heis{2g^ R) consisting of {(e.(f),n)|n€Z2^} . To connect the algebraic to the analytic constructions first note that we can replace the tower of coverings of X X
X
y n
\m
X with a single universal covering space C^. Then a £ Q^^ corresponds to the sequence {a, ...=,...)
in V(X).
Secondly we relate ^(L) to
Heis{2g,R).
Let 6 € Q^^, so 6 corresponds to a sequence ( . . . , ^ , . . . ) e V(X).
Assume
Nb G Z^^, and N\n. Consider n*L as the quotient of C x C ^ by the pullback of multiplication by n of the relations defining L. Then (A, =) G Heis{2g, R) descends to an action on n*L, hence (A, ^ ) G K{n*L). This defines a map Heis{2g,R)
DHeis{2g, Q) -^ g{L) (A,6)K-.(...,(A,6/n),...),
and gives the diagram: 1—y q —<• Heis{2g,R)
CI
Heis{2g,q)
mverse
ci REMARK:
-^
R2»
—.0
Q 2fl mverse
^(L)
V{X) = f\f
— . 0.
This diagram of groups respects the various actions on C x C^
and n*L. Indeed part A of the proposition below follows immediately.
ADELIC HEISENBERG GROUPS PROPOSITION
69
4.28. A. For x,y e V{x) = A^^, we have
B. The line bundle L is symmetric and the subgroup Heis{2g,Q)
of
G{\-.) is stable under the involution i. The element (A,z) I—^ {\ exp{7ri^yi{2z-y),z-y)
= {X(py(z), z - y)
of Heis{2g^Q) has its image in G{L) equal to r(y) whenever y G Q^^. PROOF:
For
T G
i5, let L be the basic line bundle on X = C^/(TZ^ 0 Z ^ ) ;
the bundle i*L is constructed as the quotient of C x C^ by the pullback via i of the action defining L: (a,z)
—^
(avp_n(-^)e•(-|)-^2:-n) i
Since y?-n(—2:)e*(~-^)~^ =
same. We compute the map r : Q^^ -^ Heis{2g^ Q). Let y G Q^^, and let g be an element of Heis(2g,Q,) that projects to y/2; then, by definition, T(y) = g • 2''(^)~^. First we compute i^{g). By definition i^ is the composition
Since z*L = L, the first map is the identity. If v? G ^(L), then j(2, L)(V') = V' means f*<^ = x/^. Since 2*02* is the identity, j{i, L) is just i*. Ug = (A, y/2) G Heis{2g,Q,), then i*"(flf) is the map C X C^ —^ C X C^ (a, z) I—^ ( a A - V y / 2 ( - ^ ) , 2r + t//2); in terms of Heis{2g^ Q),
*'^(5^) = (M/2(-^)"V-y/2W,-y/2).
70
TATA LECTURES ON THETA III
Plugging in the explicit formula for (p gives ^{g) = (A,—1//2). From this we see that
riy)=9-i'-i9r'
= il,y). QED
5. Algebraic t h e t a functions Our aim is to define a t^-function algebraically. Let L be an ample symmetric line bundle on an abelian variety X, and let s E r(L) (if the characteristic is p, we assume p / deg L). We will associate to every s a function ??, on V{X).
DEFINITION
We begin by studying the basic representation of
5.1. f(L) = limr(X,n*L), where g{L) acts on f(L) as fol-
lows: for g = {xn, n) € ^(L), Ug(s) = <^„ o 5 o T^Xn) which is independent of the n chosen. (In the above limitj the n^s are assumed prime to char(k) ifchar{k)
> 0.)
According to 4.15 and 4.18, ^(L) has the normal form k x A / or k X Rp^ {char = 0 or char > 0) with group law (A, x)' (^, y) = {X^e{-Cxi PROPOSITION
5.2.
• 2/2 - ^^2 • t/i), x + y).
Consider k-vector spaces W with an action U of the
group Heis{2g^ FKj) such that Ux = Xlw
for A G ^ , and for each w £W
there is an m such that U^(^rnx)'^ = ^ ^or all x G Z^^. We caii these continuous
representations.
Among all such representations there is a unique irreducible one which we call the Heisenberg representation Ti, and any such representation is a direct sum of copies of 71. 7i has the following models: A.
7i can be realized as the space of all locally constant k-valued functions g on Ay with compact support and the action ofHeis(2g,
A/) is given
by: ^i\i,yi,y2)9{x) B.
= Ae(*y2 • x -f ^^yi • y2)g{x + yi).
H can be realized as the space of k-valued functions f on f\f
with
compact support and quasi-periodicity: f{xi-\-ni,X2
+ n2) = (-l)^'^i'^2e(~(*xi • 712 ~ *ar2 • ni)) •/(xi,ar2),
72
TATA LECTURES ON THETA III for all (ni, 712) in 2^^, and the action of Heis(2g, A/) is given by ^(X,yuy2)fi^i^^^)
= ^ 2 ^ ^ ! •2/2-*ar2 •yi))f{xi
+ 1/1, 2^2 + 2/2).
The map between these 2 realizations is given by f{xi,X2)
= N'^
g{xi + n)e(*n •X2 + -*xi • X2),
Y^ n6(l»/iVZ^)
(A^ large enough so that g is constant on cosets of NZ^ and X2 G j[Z.^) 9(^1) =
/ ( ^ i , ^ 2 ) e ( - - ^ x i .X2).
^ X2GAJ/Z5
P R O O F : Let a : 1?^ —y Heis{2g, A/) be the section
(T{XU X2) = Let Gm = N{a{mZ^^))/a{mZ^^).
{{-iy'^'-^^,xuX2) This is a finite Heisenberg group.
For the representations V of Heis{2g, Ay) under consideration, it is true that V = ^m>iym where Vm = {v £ V\Ua(mx)'^ — v, x £ Z?^}. Given a representation V of Gm -> by the Stone-Von Neumann theorem it is isomorphic to V ^^^ '^^Hm
where 7^ m is the unique irreducible representation of
Gm. Each Vm is a representation of Gm and hence Vm — ^m 0 Vi canonically where Tim is the irreducible representation of Gm with the following model: all functions / on N{a(mZ'^^) such that f{Xa{my)
• x) = A/(x)
for all X e N{(T{mZ?^), t/ G Z^^, A G A:*. Extending these functions by zero outside
N{(T{WJ?^)^
%m is realized cis a space of Ar-valued functions on
Heis{2g^ Ay) and Tim C y-mn for ^H ^ and n. Call H the increasing union of the Tim • The canonical isomorphisms Vm =• Tim 0 Vi for all m lead to an isomorphism V = Ti ^ Vi: equivalently, V^ is a direct sum of copies of Ti. That Ti is irreducible follows from the fact that Tii := {f E Ti\U^(^x)f — f for all X G Z^^} is one-dimensional. Thus ifO^VcTi,
then F^C^"") ^ 0 and
consequently V^^^ ^^ = Ti\ which generates W as a Heis(2g, The rest of the proof we leave to the reader.
A/)-module. QED
ALGEBRAIC THETA FUNCTIONS
73
To apply this theorem, we prove: PROPOSITION
PROOF:
r ( L ) is the irreducible Heisenberg representation of
5.3.
When deg L = 1, this follows from observing that the subspace
of r ( X , L) fixed by aT{X) is just T(X^ L) which is one-dimensional. If deg L > 1 and H is a, maximal isotropic subgroup of K{L.) then there is a line bundle L' on X/H
= Y such that the pull-back of L' is
isomorphic to L. As we have seen in the last section ^(L) = ^(L'), and clearly r ( X , L) = r ( y , L') so that we are reduced to the previous case. QED Next the realizations constructed for the Heisenberg representation in Proposition 5.2 can be adapted to irreducible representation of ^(L) as follows: (B) becomes C^(V(X)//T{X)),
the space of A;-valued functions / on
V{X), with compact support that are quasi-periodic for T(X): f(x + t) = e.(i)eix,i)fix)
yteT
with 6{\-) acting by U'^%)f{x)
=
X.e(l,x)f{x-y).
The superscript o denotes compact support. To give a version of A, we fix V{X) = V^i 0 V2, a decomposition into subspaces Vi,F2 maximal isotropic for e. Then the analogue of (A) is C^(Vi), the space of Ar-valued locally constant functions / on Vi with compact support, with ^(L) acting by Uxf = A/, Ur(y)f{x) = fix + y)
for
Ur(y)fix) = e(y, x)fix)
for y € ^2-
yeVi
74
TATA LECTURES ON THETA III
COROLLARY
5.4. There are Q(L)-linear isomorphisms, unique up to a con-
stant:
f(L)-C,"(y(X)//T(X))
Next we are going to interpret members of T(X^ L) as functions on V(X).
The following is the main definition of this book:
DEFINITION 5.5.
Fix an isomorphism e : L(0) — • k. Let x G V^X)
and
assume T{X) = (x„,<^„). For each s G r ( X , L), define a k-valued function on V{Xl i?„ by
This is defined of all sufficiently divisible n and it is independent of the n chosen. Thus ??,(x) is defined by the chain of maps s{x„) € (n*L)(x„) = (r;^n*L)(0) t^^^ n*L(0) - ^ k There are 2 ways to interpret this definition, both quite important. One is that via the section r of ^(L) over V{X), we can trivialize the pull-back of the bundle L to V{X). To be precise, consider p*L V{X)
—^
L
.
X.
^
Then ^(L) acts as a group of automorphisms of p*L and we define an isomorphism Ai X ViX) -^ p*L by requiring firstly that for all x G y{X)^ T{X) should be identity map between the fibres over 0 and over x, i.e., r(x)[$(A,0)] = $(A,x),
ALGEBRAIC THETA FUNCTIONS
75
and requiring secondly that over 0 , ^ is €~^ . By this trivialization, each section of p*L over V{X) is just a function fs{x) via (/,(a:),x) = $ - i s ( x ) . It then follows that
six) = mMx),x)) = r(x)$((/,(x).0)) =
or
T{x)e-\f,ix))
/ , ( x ) = e(r(x)-i(«(x))) = «?,(x).
Thus i^s is the function corresponding to the section s. This definition can be beefed-up to show that when X is a family of abelian varieties parametrized by a scheme S (an abelian scheme over 5), then V(X)
over S becomes a direct limit of schemes Tn{X) over S and t?,
becomes a family of morphisms
TniX)
I We shall work this out in an appendix. For the present, we note only one important consequence which follows without any fancy apparatus: PROPOSITION
5.6.
If X^L and s e T{X,L)
are all defined over the field
kj then ds{(Tx) = (Tds{x) for all
Note that every a G Gal{k/k) acts as automorphism of ^(L) and
that a{Tx) = i?5.
T{(TX).
The formula is then immediate from the definition of QED
76
TATA LECTURES ON THETA III The second way of interpreting t?, is as a matrix coefficient of the
irreducible Heisenberg representation r(L) of G{L). In fact, evaluation at 0 defines linear functionals
4:r(X,n^L)—L(0)^fc, and passing to the limit, a linear functional
io : f(L) —^ k. Then for all s G r ( L ) ,
is a matrix coefficient of this representation. I claim this is essentially 'ds{x). In fact, say T(X) is represented by ((j)niXn) and s by 5„ G r ( X , n*L). Then {Ur(r)S){y)
= n{s{y " a^n)),
hence
Since T{—X) is represented by (<^^^, —Sn)^ it follows that
This interpretation makes it clear that 'Oug{s) csin be computed from ds for all g G ^(L). In fact: LEMMA 5.7.
du^,^^^^(s){x) = A • e{y, | ) t ? , ( x ~ y).
PROOF:
= 4(C/,,(^,).(,..)(.)) X
= Ae(2/,-)£o(C/,(j,_,)(5)) X
QED
ALGEBRAIC THETA FUNCTIONS
77
Thus one of the functions t?, determines the rest. We express this as follows: as above let Ck{V(X))
be the space of all locally constant ib-valued
function on V{X), and let ^(L) act on Ck{V{X))
by the right action, so
U'>:fi)fi'^) = ^e(l,x)fix-y). Then the map
f(L) ^
Ck(ViX))
is linear with respect to these actions of ^(L). DEFINITION
5.8.
When deg L = I, let 0 ^^ s e r ( X , L ) and put i9^(x) =
T^six), a k-valued function on V{X). We shall see below that if k = C, X = Xxj^g,
then the 'd^' above
coincides with the i?^ of §3 on Q^^. From Lemma 5.7 and the invariance of 5 G r ( X , L) under crT(X)j we have LEMMA 5.9. PROOF:
e ( | , x ) e , ( | ) ^ « ( x ) = i?^(x + y) forallxeViX),
yeT{X).
Using that e*(|)r(y)5 = 5, we have t^«(x -f t/) = e{T{x H- y)'^s{x
+ y))
- e{T{x)-^T{y)-^e{^,x)s{x
-[- y))
= e(|,x)e.(|)^«(x) QED Let us see what happens now if we investigate the space of all matrix coefficients of the representation of ^(L) on r(L), as we did in §2 for the real Heisenberg representation. We start with some linear algebra revolving around the actions of ^(L) on Cjk, the space of locally constant A:-valued functions on y{X).
There are two actions:
78
TATA LECTURES ON THETA III
One checks that rrleft _. rrright _ rrright r/left ^AT(y) ° ^Ar(y) - ^Ar(y) ° ^Ar(y)-
Let Ck(V//T)
be the space of functions
{/€Ctif/,7;;/=/ , vyenx)} = { / G C*| / ( x + y) = e . ( | ) e ( | , x ) / ( x ) , Vy € T(X)}, and recall that C^{V//T) is the space of functions on V that have compact support and satisfy / ( ^ + y) = e * ( | ) e ( x , | ) / ( x ) ,
VyenX),
and that C^(V//T) is an irreducible right ^(L) module. LEMMA.
There is a perfect pairing C,{V//T)
X
Cl{VllT)—.k
respecting the action ofG(L) up to a change of sign in argument:
{iu]:Ji_,))-'f, 9) = {/, <•;>). PROOF:
Define
(/, 9) =
E
x€V{X)/T(X)
/(^)»(^)-
This is well-defined: iff € TCX), then ff(x + t)f{x +t) = g{x)e,{^)eix,
| ) / ( a : ) e . ( ^ ) e ( ^ , x)
= 9ix)f{x). We check the assertion regarding group actions:
=
E
A e ( | , x ) / ( x + 2/Mx).
79
ALGEBRAIC THETA FUNCTIONS On the other hand E
/(^)(C^Axt»W=
x^V/T
E
/WAe(f,xM^-y)
x^V/T
= X I / ( ^ + y)Ae(|, X + i/)5'(ar). QED In the case of real Heisenberg representations we found that a choice of r G i3 gives two subspaces of the space of functions on R^^, one a right Heis{2g,R)
module, the other a left Heis{2g,R)
module; furthermore the
functions in one space enjoy the quasi-periodicity of theta while the functions in the other space are analytic with respect to the complex structure associated with T. We have the adelic analogue of this. We assume deg L = 1 and 0 :^ SQ £ r ( X , L). Then we get the diagram of spaces:
{
( space of functions / W = EA(C/r(-,)«.)> < ii € Hom(f (L), k) Si € r(L)
space of functions C
all s e f (L)
}=V
( space of functions
II
all linear functionals £ : r(L) -* k
{^«(X)}
> =Vx.
f(L)* PROPOSITION
5.10.
A) V = the space C'^,{V{X)) of all f € CkiV{X)) we have U'^fj'^^f = f,allyeNB) Vi = the space Ck{V{X)llT{X)) all y G T{X),
such that for some N
T(X). of all f e Ck such that J/^j-f^V = / ,
i.e.,
/(=^ + y) = e . ( | M | , x ) / ( x ) ,
ally&T{X),
C) There is an isomorphism of ^(L) x ^(L)-mocfuies
x
eV{X).
80
TATA LECTURES ON THETA III
such that for some 61,62: d^ corresponds to 61 <S> 62 r(L) corresponds to7i<S> 62 Vi corresponds to 61 <^7i*. Here the action t/"^^* of G{\-) on V restricts to the given action of G{\-) on r ( L ) and the action U^^^^ ofG(L) restricts on Vi to the dual action ofQ{L) on r(L)*. So if f G Ck satisfies the quasi-periodicity condition in B then it corresponds to an element of the form ei 0 /i for some h G W*. PROOF:
We begin with (B). Since r(L) is an irreducible representation of
^(L) we know there is an isomorphism V : C°iV{X)//TiX)) Note that e^(^)ST(x){^)
^
f (L).
is the unique cr(T(X))-invariant function in
C^(V{X))//T{X)
(here Ss{x) is the function with value 1 if x G 5, 0 if
X ^ S). Therefore
IP(6^{^)6T(X)(^))
= ^o- Using the ^(L)-equivariance of
7p, it follows that for all y G V{X)
X
= ^r(-y)V'(e*(-)5T(X)W) =
Ur(-y)So.
Now we identify Ck{V//T) with the dual of C^{V//T).
Using our description
of ip we calculate the ^(L) isomorphism V,' : f (L)' ^
C,{ViX)//T{X).
l{rP*{£) =g, then ^(f^r(-y)«o) = ( e * ( ^ — ) e ( —,x)6T(A:)-y(a;), =
9{-y)-
ALGEBRAIC THETA FUNCTIONS
81
Thus the matrix coefficient f'{UT{-y)So) (except for reversing the sign in y) is the element g of Ck{V{x)//T{x),
and since ip* is an isomorphism, this
proves (B). To prove (A) and (C), note that the matrix coefficient map s (^£ I—y the function f{x) = i{Ur(-x)s) is always a map «i: f (L) ® f (L)' —
Ck{V{X))
and with our conventions it will carry the action of ^ x ^ on the left hand side to the action of ^ x ^ by f/^^s^t x 17^^^^ on the right hand side. We verify this:
u:'S]\s®£) = {(Ur^y)s)®e). This maps to the function
—X
=
ei—,y)f{x-y)
On the other hand which maps to
= / ( ^ + 2/)e(|,x)
The action of Q on r(L) is continuous in the sense of 5.2: Ww G r(L), 3m s.t. Ua{mx)'^ = '"^j all X e T{X).
Thus under C/^'6^\ the image of (f) is
continuous, i.e., Im
TATA LECTURES ON THETA III
82
in the C/^sht.^ction to 7^0[
n^Ck{V/
Thus <j) is an isomorphism if its restriction to the ^(T(X))-invariant
subspace is an isomorphism. But this restriction is I/J*. This proves (A) and (C).
QED
We can describe the situation by a picture as in §3:
images of r(L) under ^ for various abelian varieties irred. by W'^^^ action of Heis
Mysterious subset of those fens, which occur as ??^ for some abelian variety over k some symplectic isom.
(V{X),T(X)^{f\y,p3)
'possible' functions d^ Dual of irreducible Heisenberg module
ALGEBRAIC THETA FUNCTIONS
83
Note that one feature that is missing, compared to §3, is the characterization of the image of r(L) as the holomorphic functions for various abelian varieties. An obvious question is to try to characterize those functions which arise as d^ in some more algebraic way. We will try to do this in the next section and in §11. Another way of phrasing this problem is to use realization A of the Heisenberg representation. Choose V{X) — Vi 0 ^2We get
f(L) S Cl{V{X)//T{X))
^ Cl{V,)
Note that the dual of Cj^(Vi) is the space Mk{Vi) of all finitely additive ^-valued measures ^ on the Boolean algebra of all compact open subsets of Vi, hence
r(L)* ^ Ck{v{x)iiT{x)) ^ Mkiy,) Thus 4 G f(L)* (or d"^ G Ck{V//T))
defines, up to scalars, a measure /i.
Can we characterize those measures fi which arise from abelian varieties? Let us go back to the classical case: Let T £ S), X = XT,L and L the basic line bundle and let's see what F, G, 'dgj'O^ and fi are. The vector space L(0) is canonically identified with the fibre of the trivial line bundle C x C^ at 0 and thus there is a natural e : L(0) =. C. We pick up the situation as in Proposition 4.28 and the remarks preceding it, as well as §2, big diagram. PROPOSITION
5.11. A.
r{X,L)
is the linear space spanned by 'da,b(z,T)
where (a, 6) G Q^^, and in fact the da,h{z,T) for {a,b) G Q^^/Z^^ form a basis. B.
The natural action of Heis{2g,Q) on the span of the days
coincides
with the action ofO{L) on F(X, L) restricted to J/eis(2^,Q). C.
For any x G Q^^,5 G F(X,L) (s is identified with an entire function
on C^), 'dg{x) = exp 7ri^xi{Txi -\- X2) • s{Txi -f ^^2). D.
In particular J when L = Z^^, deg L = 1 and x G Q^^ , i?^(x) =z d{Txi + X2,T)exp 7ri^xi{Txi + ^2)
(defined as
d''Q]{T))
84
TATA LECTURES ON THETA III whenever x E Q^^.
E.
From Proposition 4.28, A? x 0 and 0 x A^ are maximal isotropic sub-
spaces off\y
= V{X).
Call them Vi and V2 respectively. Then the finitely
additive measure /i on Vi (now identified with M x 0) corresponding to io ^^*
is given (up to scalars) by n{U) =
^^P 7ri*xTx
2_\
for all compact open C/ C A^.
x£UnQ3
Thus fi is countably additive when restricted to compact subsets of Ay (i.e., a Radon measure on A? j and is totally singular, being supported exactly on the countable subset Q^ C A?. PROOF:
A follows from Proposition 3.2.
Parts B, C and D follow without any tedium in our set-up. In particular, B is contained in Proposition 4.27. The specific formulae in C and D rely on Proposition 4.28B. When x G Q^^ C V(X), T{X) is the transformation (A, z) I—)• (A exp 7ri*xi(—^-h 2z), z — x). Therefore
T(X)~^
takes {s(x)jx)
to {s{x) exp
TTZ^XI^,
0) showing that
'dsi^) — s{x) exp TTz^xix for all X G Q^. D follows by putting s{x^) = 'd(x_,T). We prove E. Let H be the space of locally constant functions on A? and xp : r(L) —^ H the unique equivariant map. The evaluation at 0 map: r(L) —^ k induces via 'tp a measure dfi on A^ so that f tP{t)dfi = t(0)
tef{L).
Since deg L =: 1,0 ^ 5 G r ( L ) corresponds, under ^, to (5, the characteristic function of Z^. Since ^^c/;^^. )t(0) = A 'dt(—y) we have
ALGEBRAIC THETA FUNCTIONS
85
Recall that t?«[^^](T) = ^ e ( A « n r n + 'n(Tari+X2) + |'a;i(ra;i + a;2)). To complete the proof we must verify that for the measure fi in E that
' ' " [ ° ] ( r ) = f iUi,o,-rJ)dfi
= i
eCx2•y)d^l. QED
This is straightforward.
When A: = C we may also relate the real and adelic Heisenberg representations. We do this in an appendix to this section. Proposition 5.11 plus Proposition 5.4 have the following corollary: COROLLARY
5.12.
Let T £ S^g and consider the abelian varieties
XTJ^Q,
i.e.,
and the basic line bundle L on it. Also, for aii a, 6 G Q^ and let x(^j ^)m = Image in XT of —(Ta 4- b). m Then suppose that (a, 6) G ^Z^^, k C C is a subfield such that XT ^nd L can be defined over k and x{a, b)2n Is rational over k. Then
,?«[^](r)/r[°](T)efc. PROOF:
The Galois group Aut{C/k)
acts on
XT,
hence on V{X), which is
isomorphic to A^^. This representation does not preserve the "lattice" Q^^ in f\y.
However, if a: G y{X)
over ky then crx — x G T{X), (a, 6) G Q^^ C V{X),
has its image XQ £ X mod T{X) rational all a G Aut{C/k).
the hypotheses of the Corollary imply that for all
a G Aut{C/k): a{a,b) = (a, 6) + 6 some 6 G 2nT{X).
Therefore for the point
TATA LECTURES ON THETA III
86
Now using the equality of the analytic •d"' with the algebraic d", we get
d-[l]{T) J
Vn(o,o)); _^Xa,6)) t?«((0,0))
:fMi
e.(f)e(|,(a,6))-,?-(a,6) »?«(0,0)
»9"(0,0)
^-[J](r) QED It is this result that distinguishes d" from the other functions which differ from it by an elementary exponential factor and that motivates the definition of d".
This is noi true for d{Ta + h,T); it is almost true for
»?a,6(0,T) since this differs from t?"[, ](T) only by a root of unity, but still not true. It is unfortunate to add another notation for an almost identical function, but this corollary is the ultimate reason.
ALGEBRAIC THETA FUNCTIONS
87
Appendix I: dg as a morphism We shall now define the 'ds scheme-theoretically. This has three substantial advantages: A. Statements like Proposition 5.6 follow trivially. B. It proves that when an abelian variety in characteristic 0 has good reduction mod p, the values of its theta function are integers which reduce mod p to the char p theta-function. C. There is no need to avoid p-torsion when working over a field of characteristic p. Let j : X —)- S be an abelian 5-scheme (see D. Mumford and J. Fogarty, Geometric Invariant Theory, Ch. 6) and let L be a relatively ample line bundle on X such that (—lx)*L = L with a fixed isomorphism e : 0*L = Os where 0 : S —^ X is the zero section. S t e p I. The construction of/i — f : X —> X and /„ : X —^ X such that '^xfmn = fm ^^ ^^ Definition 4.26 goes through without a hitch. Define Tn{X) = fn^i^)
— /"^(^n)
which is a closed subgroup scheme of X.
Each Tn{X) is a closed subgroup scheme of Tnm{^)'
For each k and n,
multiplication by k from Tjfc„(X) to itself maps Tkn(X) isomorphically onto the subscheme Tn{X). In particular we have inverses
We define V{X) to be the directed system of closed immersions: T{X)=:Ti{X) ^ Tn{X) ^ TnmiX) ^ .,.
Step II.
S(n*^L) makes sense as a group scheme over S with the exact
sequence: l—^Gm—
G{n*xL) ^
A'(n^L) — 0.
Put Gn = TT~^{Xn^), and Tfi = j*n*xL a locally free coherent sheaf on 5. There is a natural action of Gfi on Fyj (when S is aflfiine, Yn is given as a co-module over the co-algebra which is the coordinate ring of G„).
TATA LECTURES ON THETA III S t e p I I I . Now fn induces a map from T„2(X) onto X„2. Put Gni}-) — We get the following commutative diagrams of exact sequences:
1
We can then define ^(L) as the directed system of inclusions:
Finally, put r ( X , L ) — lim Tn = {j o / ) * / * L . We have a homomor—•
n
phism from ^n(L) to Gn and an action of G„ on F^, hence we get an action ofan(L)onf(X,L). S t e p I V . Given a global section of F(X, L), we shall construct a compatible system of morphisms from ^n(L) to A^ which we shall call a morphism from ^(L) to A^. One only has to check that the old formula e(Ug-is)(0) for g G G„, s G F„ makes scheme-theoretic sense. S t e p V. The construction of r.
The symmetry of L gives as usual a
compatible system of involutions i on ^n(L). The morphism x —)• xi(x)~^ from ^n(L) to itself clearly factors through a morphism h : Tn{X) —>• ^n(L) such that nh is multiplication by two. Put T = h o ^. Combining this with Step IV, each s G F(5, F(X, L)) gives a sequence of morphisms i?, : T„(X) —y A^.
ALGEBRAIC THETA FUNCTIONS
89
Appendix II: Relating all Heisenberg representations First consider the abstract Heisenberg representations H = the irreducible Hilbert space representation of Heis{2g, R) with Ux = A, VA G C* Woo C 7i C W-oo) the C^- vectors in Ti and its dual, 7{f = the irreducible continuous representation of Heis{2g, A/) with Ux = A,VAGC*
(cf. 5.2; here Heis{2g, A/) is C* x A / with group law as in 5.1 ff; (A, x) • (^, y) = (At/ e(*a?i • t/2 - ^^2 • 2/i), 2^ + 2/)
e(a + 6 ) . r e 2 ' ^ ^ • ^ a G Q , 6 G Z ) . PROPOSITION
5.13.
There is a unique Heis{2g,Q)-Iinear
embedding
(j) : W / — ) ^ W_oo
the image of which is the linear span of the vectors e[, ], a, 6 G Q^. PROOF:
The image of such a map must contain a vector fixed by
and e[ ] is the only such vector in Ti-oo-
(T{1}^)
Thus the image of (p is the
linear span of the e[, ]'s. But Heis{2g^Q) acts continuously on the span of the e[, ]'s with respect to the topology defined by the subgroups cr(mZ^^), and Heis{2g^ fkj) is the completion of Heis(2gyQ) for this topology. So Heis{2g, A/) acts continuously on this span and by 5.2 it is isomorphic to QED
Hf.
This abstract result allows us to pass between the realization of these representations via the full adeles as follows:
A^B^C
TATA LECTURES ON THETA III
90 where
span of the funs. A= <
fa,b{^)
= (f^r(r)a,6)
I X G R^^ span of the funs.
B [ span of the funs. C=
<
fa,h{x)
= iUr(x)Cl,
aeni^.benj [xef\y
b)
i) Note that the functions in the middle space are quasi-periodic w.r.t.
fa,b(^
+ y) = (^T(aroc+yoc)«. X
=
=
^Tixj+yj)b) 1 {e{^,yoo)'Ur(y)Ur(x^)Cl,e{-A{xf,yf))'Ur(^y)Ur(xj)b)
e\^A{x,y))^U,,{x)
where e' : A —> €{ is the map e'(Aoo, Ay) = e^'^*'^~e(—A/) and A(x,y)
=
ii) The map ri is given by restricting a function on A^^ to the infinite factor, a map which gives an isomorphism
Fens, / o n A 2 ^ , C ~ in real"] variable, locally constant in finite-adele variable, and quasi-periodic w.r.t. Q^^ 3
C ~ fncs. / o n R 2 ^ , quasi-periodic w.r.t. N • 1?^ some A^, i.e. for all k G Nt^3
ALGEBRAIC THETA FUNCTIONS
91
iii) The map r^ is given by restricting a function on A^^ to the finite-adele variables and substitutes —x for x, giving an injeciive map: Fens, / o n A 2 ^ , C ° ^ in real variable, locally constant in finite-adele variable, and L quasi-periodic w.r.t. Q^^
Loc. constant fens. / on A^^, quasi-periodic w.r.t. A^ • Z^^ some AT, i.e.
f{x-^k) =
.for all k e Nt^3
II
B iv) The group Heis(2g,
e{^,x)f{x),
C
A) of pairs (A,x), group law
(A, x){fi, y) =z {Xiie'{-A{x,
y)), x-{-y)
acts on the middle space by combining the action of Heis{2g^ R) on a, Heis(2g,
A/) on 6, and the middle space is its irreducible Heisenberg
representation (more precisely, the subspace of "Schwartz functions" in it). v) The Heisenberg actions match up (with various sign changes) as follows: left Heis{2g, Q) on A ^ - ^ Heis{2g, Q) C Heis{2g, A/) action on B right Heis(2g,
R) on A ^ - ^ Heis{2g^ R) action on B
right Heis(2g, A/) on C ^ - ^ action on B left Heis{2g, Ay) on C ^ - ^ Heis{2g, Q) C Heis{2g, R) action on B The situation is summarized in the diagram below:
TATA LECTURES ON THETA III
92
Fock repres. Holo. fens, on C^;/ci5(2y,R) acting on left J Udense
span of «>J.,(i),
r Fens, on R'' Hei8(2g,H)^ aeting
U dense rC^'fens. on R'^ fixed by left aetion of (l.nZ") Un Hei8{2g, Q) acts on left //ei«(2<;,R) acts on right extending to Hei8{2g,f\) action
u c • i^^^d)
Quasi — periodic reprea. C~ fens, on R^*, such that /(x + a) = e.(f)e(x,f)/(x) all a e l^' Hei8(2g,H) acting on right
"]
93
ALGEBRAIC T H E T A F U N C T I O N S
Schwartz fens, on A ' ' {C°^ in real variable, loc. cnst. in f>-adic variable) left quasi-periodic for Q^' : / ( i + a) = e ' ( / J ( f . a ) ) / ( x ) , a € Q » » //ei»(2},fl) acts t^(A,v)/W = Ae'(/i(f,»))/(x + j,)
Fens, on flj', loc. cnst. / / e i » ( 2 j , A / ) ' acting re»o(-l)
U Schwartz fens, on A^' left quasi-periodic wrt Q^' right quasi-periodic wrt 2 ' ' Hei8(2g,R) acts
a U
rc«o(-I)
Fens, on A,', s t. /(ir-f-a) = e . ( f ) e ( f , x ) / ( x ) all a e P' L//ci«(2y,R) acts on left
6. Theta functions with quadratic forms The purpose of this section is to construct generalizations of the theta functions considered so far, which are associated to an auxiliary positive definite rational form. These functions are important for various reasons. In Chapter I, we discussed the application of theta functions to the problem of the representation of integers by quadratic forms and these general theta functions can be used to study the representation of one quadratic form by another. Our interest, however, is in the algebra of theta functions, e.g., the polynomial identities satisfied by them. We have encountered Riemann's theta relation repeatedly in earlier parts of this book, but this is only the simplest in a large clciss of theta relations. These relations are naturally deduced from the more inclusive algebra of the full family of theta functions associated to quadratic forms. We begin by constructing theta functions with quadratic forms over C. In what follows, for any ring R, R(g, h) will denote the g x h matrices over R. DEFINITION 6.1. For a rational, symmetric, positive definite hx h matrix Q and a T G ^g, define d^{Z, T)=
YJ
^^P ^iTrCNTNQ
+
2'NZ)
Nel{9,h)
where Z G C{g^ h).
It is easy to see that the following holds: ^^{Z + TMQ -f A^, T)exp TriTrCMTMQ
+ 2^MZ) = d^{Z, T)
for all M,N £ I(g, h). This suggests that 'd^iZ, T) is just Riemann's theta function for the complex torus C{g,h)/TZ{g,h)Q
fix the isomorphism Z = {Z\,Z2,-•
-f- T{g,h).
if
To see this,
•, Zh) —• ^ — \ ^ | °f ^(s'' '*) ^ i ^ ^ \Zh.
THETAS WITH QUADRATIC FORMS C{gh, 1) = C^^. U Z,W,F^"
95
' e C{g, h) we shall denote the corresponding
elements of C^^ by 2r, it;, / , • • • . Now T* =T<S>Q can be represented by the gh X gh block matrix:
/TQn rp* ^
TQ21 \TQhi
TQn TQ22
• • • TQiH • • • TQ2h
TQh2
• • • TQhH
Once it is checked that (i) W = TZQ implies w = T*z, and (ii)
Tr{*WZ)=*wz,
we can prove our claim: LEMMA
A.
6.2.
-f- TI{g,h)Q)
C{g,h)/{l{g,h)
^ XT*j2gh.
This torus will be called
XQ.
C. The complex tori X^ and
(XTJ^Q)
are isogenous.
P R O O F : A is clear. By (i) and (ii) above, we get
TrCNTNQ
+ 2^NZ) = ^nT^'n + 2^nz
which shows that
^^{Z,T)=
YJ expiri{^nT*n-^2'nz)
= ^{z,T*).
This proves B. The two complex tori in part C are C(^, h) modulo the lattices TI{g,h)Q
+ I{g,h)
and Tl{g,h)
-f l{g,h)
respectively. But both these
lattices generate the same rational vector space (because Q is a rational invertible matrix) showing that the two tori are isogenous. More of the underlying geometry that connects X^ and
QED XTJ^9
and
the basic line bundles they inherit will be explained later. Meanwhile we define the analogues of ^ [ J ] ( z , T ) and ??"[^](T) :
TATA LECTURES ON THETA III
96 DEFINITION
6.3. With Q and T as above, A,B £
Xi,X2eR{9,h), A.
= E;ve2(,,/,) ^^P ^iTr{\N =
J2
+ A)T{N + A)Q + 2'(iV + A){Z + B))
x[t]WexpTriTrCNTNQ+2*NZ)
N€Q(,9,h)
= exp mTrCATAQ
+ 2*A{Z + B)) •d^{Z + TAQ + B, T),
where X[ D ] ( ^ ) = e«P 2TriTr*NB, if N - A £ 1(9, h) = 0
otherwise.
B. Xi + X2)) • ^^(TXiQ + Xz.T) (T) = exp %iTr{'XiiTXiQ X2 Y, expniTrCXiTXiQ + 'XiX2+'NTNQ + 2'NiTXiQ + X2))
=
Neli9,h)
= exp( =
-wiTr(XiX2))-J2expiriTr{*Xi+N)T(Xi+N)Q+2\Xi+N)X2) exp{-iriTr*Xi-X2)-^^
Xi X2
(o,r).
By (i) and (ii) above, we have A'. ,?<3
iZ,T) = ^
iz,T*).
B'. ^c.Q
Xi X2
(T) = exp wi*xi{T*xi + X2) • d'^{T*xi + X2,T*) = t?"
Xi X2
(T*),
the "algebraic" theta function for the torus
X^.
Following the approach in §2 we define Heis(2{g^ /i); R) to be the set of (A,Xi,A'2) with A G Cj and Xi,X2 G R{g,h) with multiplication defined as follows: {\,XuX2)'{^i.YuY2)
= {\^xexp mTr{' XiY2-'yiX2)^
X^^Yu
X2^Y2).
THETAS WITH QUADRATIC FORMS In fact, Heis{2{gyh)]R)
=. Heis(2ghjR),
97
and it acts on the space of all
holomorphic functions on C(^, h) by the formulae: ^(A,0,0)/ = '^~ V U(i,A,o)f{Z) = exp(7ri Tr{'ATAQ
U(x,A,B)fiZ)
= X-^f{Z
+ 2'AZ))f{Z
+ TAQ)
+ TAQ + B)exp{7ri TrCA{{TAQ
-\-B) + 2Z))),
Then d^{Z^ T) considered as a function of Z is, up to a scalar, the only holomorphic function on C(^, h) invariant under the action of the discrete subgroup (lxZ((/,/i)xO).(lxOxZ(^,/i)) oi Heis(2{g^h)\R). ^a,Q
We shall call this subgroup a(I{g,h)'^).
Similarly,
(T) can be realized as a function invariant under o-(I.{g, h^) for
the following action of Heis(2(g, h); R) on the space of all functions defined onR(<7,ft)2: U(^x,AuA,)fiXuX2)
= X-'expiriTrCX2Ai-'XiA2)fiXi+Ai,X2
+ A2).
Denoting the first action by U^ and the second by C/^, and by R the linear transformation from the first space to the second given by R:f{XuX2)
—> exp{7ri Tr'XiiTXiQ
+
X2))f{Xi,X2),
then we see that U^Rf = RUlf REMARK:
for all g G Heis{2{g, h); R).
Since Rd'^iTXiQ
+ X2,T) = d"''^
we can convert »?*?-identites to »?'^'*5-identities.
X2
in
TATA LECTURES ON THETA III
98
The following results contain the generalization of Riemann's theta relation: 6.4
Let Qi and Q2 be rational symmetQi 0 1. ric positive definite (/ii X /ii) and (/12 x hz) matrices. Let Q = I F I R S T FUNDAMENTAL IDENTITY:
Then A . i9«(Zi,Z2;T) = t?«'(Zi,T)t?«»(Z2,T), and B . ^c,Q
Xi Yi
X2 (T) = t?°'
where Z,- € C(fl',/ii) and 6.5
X.-.Y; G
Yi
(T)t? «,
X2 Y2
in
R(fir,/i,).
SECOND FUNDAMENTAL IDENTITY:
Let A e GLhCQ) and Q' = *AQA.
Set P = [Z'' : ylZ'' n Z'']-«. Then
R
S
R
S
=PJ:J:^'^ R
{Z,T).
S
B. ^«,Q'
Xi-'yl-i X2-A
m = ^ E E ^(l«.o)t^(lo,5)'?"''^ ii
,?
5
J]^
X2
(T).
{ZA-\T).
S
D. t?a,Q'
X2
(T)
P Ei? Ese^P'rirr('7?X2>l-i-«5Xi'^-'5P)-t?«'«
X2'A-^
+ S (T),
THETAS WITH QUADRATIC FORMS
99
where R and S are {g x /i)-matrices whose g rows run through a (finite) system of coset representatives for Z(l, /i)*i4/Z(l, /i)*A fl Z(l, /i) and Z(l, h)A''^/I{l,
h)A-^ n Z(l, h).
Formulae C and D will be used in applications for various choices
PROOF:
are immediate consequences of 6.5A and B: replace Z and they are
o{Q,A,Q',
by ZA-^ and
X2
by
XoA-^ 12-'
P R O O F OF 6.5A:
^^'{ZA, T)= =
Y,
exp mTr{*NTN^AQA + 2*NZA)
"^
expmTr{A*NTN*AQ
Y,
expT(iTr{*NTNQ + 2*NZ)
= Y
Y
R
+ 2A^NZ)
expiriTr{*{N + R)T(N + R)Q + 2*{N + R)Z)
Nelig,hyAn2(g,h)
= Y,
^^P ^iTr{*NTNQ + 2'N{Z + TRQ)+
J2
R N^T[g,hYAn2ig,k)
*RTRQ + 2*RZ)
where the R range through a system of coset representatives for I(g,hyA/Z{g,hyAnIig,h),
Y1 Xl>^(^)ea;p %iTri*NTNQ + 2'iV(Z + TRQ)+
= PY
RNei(g.h)x
*RTRQ + 2*RZ)
where the x range through the character group of Z(
h)
and P=
[Zig,h):Z{g,hyAnZig,h)]-'
= [Z(l,/i) : Z(l,/i)M n Z(l,/i)]-« = [Z'' : y l Z ' ' n Z ' ' ] - ' ' . Observing that any such x can be uniquely represented by X{N) - exp 2mTr{*SN)
where 5 € [Z{g,h)A-^
+ Z{g, h)]/Z{g, h), the
above expression boils down to S
PYYI R
S
e^PT^iTr{*NTNQ + 2*N{Z+TRQ + S)+*RTRQ + 2^RZ)
N€2(g,h)
= PYYexpmTr{*RTRQ R
S
+ 2'RZ) • i?<3(Z + TRQ + S,T)
100
TATA LECTURES ON THETA III
= [Z" : AZ''nZ'"]-^53^C/(\_«,o)C^(\.o,5)'^^(^.^)R
S
The S above were chosen as coset representatives for 2{g,h)A~^
-f
Z(^, h)/Z{gj h)] they could equally well have been chosen to be coset representatives for Z{gjh)A~^/l{g,h)A~^n2{g,h).
For such /^ and S, R = M^A
and S = NA~'^ for some M,N e Z{g, h) and therefore exp 27riTr{^RS) = exp 27riTr(^MN)
= 1. Consequently, the t/A ^ QN and Uh Q ^X can be in-
terchanged; also - expniTr{*RTRQ = i?0
+ 2^ R{Z + S))d'^{Z + TRQ + 5, T)
{Z,T).
This proves the second and third formulae in 6.6A. We show that upon applying R to formula (A) we get (B). It is evident that Rd'^ = d"'^. On the other hand, if Z = TXiQ + X2, then Rd'^\ZA,T)
= exp{m Tr 'Xi{TXxQ
+ X-i)^'^'{{TXiQ
+
= exp{T^i Tr{\Xi*A-^)
• {TX{A-^Q'
X^A)).
d^'{TXi
*A-^Q' +
(since ^A'^Q'A'^ »?".«'
X2A
+
X2)A,T)
X2A,T)
= Q) (T). QED
REMARK
1: Note that the summation over R (resp. S) disappears when
A (resp. A~^) is an integral matrix. REMARK
2: In 6.5D, the exp •p:iTr{*RX2A-^ - ^SXi^A - *SR) are roots
of unity when Xi and X2 are rational. REMARK
3: Replacing Z by Z + TMQ' + N in 6.5C and multiplying both
sides by exp mTr{*MTMQ'+ M ^Q' {Z,T) = N
2*M{Z +N)) gives a generalization of 6.5C:
M^A + R {ZA-\T), NA-^ + S where c = [^Z'' : Z'' n AT'']
THETAS WITH QUADRATIC FORMS
o\
(di COROLLARY %.%. IfQ
101
is diagonal, and
=
\ 0
dh/
then
h >=1
and
h
m=n^"
^«,Q
(rf.T)
«=1
This follows immediately from the first fundamental identity. COROLLARY
6.7. Let Q be rational positive definite. Choose B G GL/i(Q)
so that D — ^B.Q.B is diagonal. Replacing the matrices (Q, Q', A) in 6.5.C by{D,Q,B-'),
we get:
^^{Z,T) =
a-^Y^l[d\2]{J2bjiZj^diT) R,Si=l
^ «J
j
where a is some natural number, the di are the diagonal entries of D, and the ri, Si, Zi are the i-th columns of R, S, Z respectively. This shows that the theta functions with arbitrary quadratic forms are easily calculated in terms of the usual theta functions for the period matrices dT. Put Q = 2/2, Q' = /2 and ^ = ^ [ I ^~'=
_ \ ) is integral, and I{g, 2) • 'A/I{g,
(\
A/T{g,2)
_ M in 6.5C; so 2)'A D I{g, 2) = I{g, 2) •
hcis coset representatives (r/, r/) where r] G \l.^ jV.
COROLLARY
Therefore,
6.8.
d{z,,T)d{z,,T)=
Y.
^[J](^i-f^2,2TMj](zi-.2,2T).
This theta relation will play a major role in §7.
TATA LECTURES ON THETA III
102
Next put Q = Q' — U and
^ = 2
/I 1 1 1>^ 1 1 - 1 - 1 1-1 1-1 \ l -1 -1 1/
in 6.5C. Here A = ^A-= A ^ and coset representatives for l{g,A)Al[l{g,A)Am{g,A)\ COROLLARY 6.9
are {T],r),T},r]) where 77 G i Z V Z ^ Thus:
(RIEMANN'S THETA RELATION).
,?(zi)l?(z2)l?(23)t?(24) =
•'^[J](|(^l - ^2 + ^3 - ^4)) • '^[^KK^l -Z2-Z^
+ Z^)).
Finally, choose an orthogonal basis A , • • •, ^/i of Q'* starting with /?i = (1, • • •, 1). Let B be the h x h matrix whose i^^ column is Pi and let di be the square of the length of /?». Put A = B"^, (h 0\
Q' — Ih and Q =
. Then Q' = *A • QA. Apply Remark 3 following 6.5 Vo dh) and specialize Z G C^^''^^ to Z = (z, >2r, • • •, z) with 2: G C^ ; we get: COROLLARY
6.10.
m
n-=i'?r:'|(-.7') = L * J c-^ Efl,5 exp2TriTr(M • «A5) • J?
{0,djT).
ihz,hT)UU^
On the left hand side, we are taking products of the -d rui where X =
XTJ^Q
rii
ef(x,L)
and L is the basic line bundle. The R.H.S. consists
of linear combinations of 1?
(hz.hT)
G T{X,L^).
Thus the above
formula gives an explicit description of the multiplication map 5'*(r(L)) -^ r(L'*) in terms of the natural bases that both vector spaces possess. We now wish to explain how, up to separate scalars for each Q, we can define the functions
THETAS WITH QUADRATIC FORMS
103
purely algebraically and over any ground field. The basic idea is quite straightforward, although its execution is notationally involved. We start with an abelian variety X and an ample degree one symmetric line bundle L on X, to which we can associate the function "d^ on V{X) as in §5. However suppose you consider L" and wish to associate to (X, L") a single theta function. Since L" has degree bigger than one you need to construct an isogeny Z^'*): X —^ y, and a line bundle M on Y such that
(/("))*M^L" where M has degree one. Over C one obtains in this way 'd{zj ^T) for this bigger lattice, hence it defines the new function. Indeed, if h = 1, Q(x) = ^ , then d^{Z,T)
is just d{Z,^T).
Algebraically, the operation
of passing to a bigger lattice is dividing X by a so-called "Gopel group"
HCXn (Xn = n-torsion in X)\ write V{X) = VI 0 F2, K- isotropic for e*-, so that T{X) = Ti 0 ^ 2 , Ti = VinT{X). for such i7, if y = X/H,
Then H = ^ T i / T i . We must verify that
then L" is the pull-back of a degree 1 line bundle
ony. This construction can be extended to products X^ of X. Thus on X^, for example, we have three line bundles to play with: pJL, p^L,
{pi+P2yL
where pi : X^ —> X are the projections and pi -{-p2 : X^ —^ X is addition. If Q is a 2 X 2 integral symmetric matrix, we can form the combination
which turns out to be ample if Q is positive definite.
It is again not
hard, via a splitting F = Vi 0 V2, to define an isogeny f : X'^ -^ Y
104
TATA LECTURES ON THETA III
and a degree 1 bundle M on Y which pulls back to L^^^. This gives us still more theta functions. Over C, one gets Y by dividing C^ 0 C^ not by (TZ^ 0) + (Z^, 0) -h (0, TZ^) -f (0, Z^) but by the bigger lattice of vectors ( T n ( Q - i ) i i , Tn(Q-i)i2), and (Tn(Q-i)2i, Tn{Q-^)22), (Z^,0) -f (0,Z^). This torus is the torus X^~\ theta function is just d^
nel^
plus
so it follows that the new
i^^T).
To work this out in detail, we first do the complex case geometrically and describe the relations between X = Xxj^g
and X^ = X^^j^gh
very
explicitly. Let L and L*^ be the basic line bundles on X and X^ respectively and f^
: X^ -^ X^
the rational isogeny^ defined by f^{z)
= z for all
We have already seen in §4 that there is a canonical isomorphism AJ e AJ ^ V{X) such that any (x,y) G Q^ 0 Q^ corresponds to the sequence -^{Tx -f y) in V(X). to T{X) e^{x,y)
By this isomorphism, Z^ 0 Z^ goes over
and a^ = r"- on the subgroups Z^ 0 0 and 0 0 Z^.^ We have = e{-A{x,y))
= e(-*xi?/2-f *2/12^2) where x = (a?i,X2), y = (t/i,i/2)
and e(a -\- /3) = exp 27ria where a G Q and /? G Z. Under the canonical isomorphism of Ay (2^,/i) with V{X), the pairing becomes
PROPOSITION
6.11. A.
Identify V{X^) with !\j{2g, h) as above; similarly
^This means that f^ is given by an equivalence class of diagrams Y X^
X^
where p and q are isogenies and f^ is the formal combination q op~^. ^Equivalently, e^{x) = 1 if x G | Z ^ 0 0 or x G 0 0 ^Z^. To determine a symmetric line bundle uniquely, we need to specify both e*- and ejr, so for all our line bundles we shall keep track of e*- and ejr.
THETAS WITH QUADRATIC FORMS identify V{X^)
= f\f{2g, h) usingT^Q.
V{f^)
where X,Y B.
105
In these terms
: ViX"") ^ f\j{2g, h) —> 1/(X^) ^ A;(2^, h)
.(?)"(r
ef\j{g,h).
V{f^){t{g,
h)Q®t{g,
h)) = T(X
h)Q e 0) and K(/«)(0 ® Z(<7, ft)) ofTiX'^l).
V{fQ){i{g, C.
wAere L^*^ is the basic bundle on X'', i.e., L^") 2 pi(L) ® . . . (2)p^(L), the Pi are the projections from X'^ to X, and Xi,Yi € f^fig, h). D.
G Q(sf, h) and xi,X2 are the corresponding elements of Q^'',
If Xi,X2
then ^o,Q
Xi X2
] ( n = conse.^-'-''(F(/
(T) = d"
Eeie the first two are analytically defined expressions while the last is algebraically defined (see 6.5). PROOF:
A.
The two isomorphisms ji and J2 of R{2g, h) with C(gj h) cor-
responding to the complex tori X^ and X^ are Ji ( ^ ) ~ ^^ J2(Y) X,Y
= TXQ + Y. e Q{g,h).
This shows that V{fQ)fy^
~^ ^ ^^^
= (^y"')
Because Q is dense in A/, V{f^)(yj
= (^^
for ) for
X,Yef\f{g,h). B follows from A and the remarks preceding the Proposition. We prove C. Under the identification of V(X'^) with f\{2g, h) using e'-''
T®Q,
i{xl)'{y.))=<^'^-'''^'^^^'''^'^^^^-
106
TATA LECTURES ON THETA III
Given this, C is straightforward:
.-(n/',(-;),n/«)(^;))=.-((\r").(^f:)) 6.3 B' says that the first two expressions in D are equal and by 5.1 ID the second is equal to const, i?^'^'^ T M = const, i^^'"-"^ Wif^)
\ l ]
QED We take this Proposition as our cue for defining ( / ^ , X ^ , L ^ ) and d^'^ purely algebraically. For ease of exposition, we shall assume we are dealing with an abelian variety over a field k of characteristic 0. With straightforward modifications and suitable restrictions that various integers are prime-to-p, everything extends to char (A:) = p ^ 0. DEFINITION
6.12. Let k be a field of characteristic zero, X an abelian
variety defined over k, and L an ample, even, symmetric line bundle of degree one on X. A Gopel structure for (X, L) is a pair (Vi, V2) of subspaces ofV{X) A.
isotropic for the pairing e*- such that V{X) = Fi 0 V2 and
a^ = T^ on Ti = Vin T{X) for i = 1 , 2 ; or equivalently eHr (x) = 1 for X e ^Ti, and
B. T{X) = Ti^T2. In what follows, members of y(X)'* will be thought of as row vectors with entries in V{X), and if a Gopel structure is chosen, then members of V{X^) will be thought of as a 2 x ft matrix with top row from Fi, bottom row from V2. PROPOSITION AND DEFINITION 6.13.
Let (VI, F2) be a Gopel structure
for (X, L) and let Q be a rational symmetric hx h positive definite matrix. Then there is a triple (/, y, M) where f is a rational isogeny from X^ to
THETAS WITH QUADRATIC FORMS
107
Y and M is an even symmetric ample line bundle of degree one on Y such that A. e^{V{f)A,V(f)B)
= e^^''\AQ^\B)
for all A,B
G V(X)^,
where
LW=P*(L)0...0P;(L),
B. Vif){TtQeT^) C' i^ifWi'y
=
T{Y),and is a Gopel structure for (Y, M).
^if)^2')
If ( / ' , y ' , M ' ) is another such triple, then there is an isomorphism ^ : y —> y such that f = g o f triple by
(f'^^X^X^)•
IfQ = Q\®Q2,
PROOF:
and g*M =. M'. We shall denote such a
then
Choose a natural number n such that 5 = n^Q^^ is an integral
matrix. Let L,- = pJ'L and L,j = (pi + pj)*L 0 p*L~^ (8)p^L~^. Define H = (^L,- "0»
to compute the pairing e". From 4.14F and 4.14G it follows that e^*(a,b) = e^(a,-, 6,) =
and
e^{ai,bj)e^(aj,bi)
where a = (ai, ^2} • * • > a/^) and b = (6i, 62? * • • > ^/i) with the a^ and bi in V{X).
A repeated application of 4.14G now shows that e"{a,b) =
l[e'-(ai,bif"'[[e'~iai,bjf''e'-iaj,bifi
= e'-'"\aS,b)
i<j
=
e'-"'\n-'aQ-\b).
Consequently V^ and V2 are maximal isotropic spaces for this pairing, and in addition, n~^{TiQ 0 T2) is a maximal isotropic subgroup for this pairing: if ai G Vi^ and 02 G ^2^*5 ^^en e"(ai -f 02, n - i 6 i Q + n-%)
= e*-^'^(nalQ-^ 62)^^ • e'-^'^(na2, 61) = 1
108
TATA LECTURES ON THETA III
for all 61 G Tf ,62 G T^ if and only if naiQ-^ and only if ai G n-^T^Q
G T^ and na^ G T^, i.e., if
and 02 G n-iT2''.
Now the subgroup r"(n-^T/^Q) • T^{n-^T^)
of ^(H) gives rise to an
abelian variety X^, a symmetric line bundle L^ and a rational isogeny g : X^ -^ X^ by 4.22B so that H and L^ are related by the Q-isogeny g (so if ^ is an actual isogeny g*L^ ^ H). In addition j{gX^) takes T^{n-^Tl'Q)
• T^{n-^T^)
isomorphically onto
- ^(H) = ^ ( L ^ )
( T ^ ' ^ ( T X ^ ) . NOW
put
f^ = n^^g^ and it is easily seen that the triple ( / ^ , X ^ , L ^ ) satisfies A, B and C. What is missing, however, is the ampleness of L^ which depends on the fact that Q is positive definite and not just non-degenerate which is all we have used so far. We postpone the proof of this fact. Now let (/, y, M) and ( / ' , y ' , M ' ) be two such triples. Because
Vify^TY
= V{f')-^TX
= T^Q e T^
4.21D implies that there is ^ : y = y such that / = g o f.
In addition,
e^' = e^*^ and e^'(x)
and for x G
\V{f')T^,
= ef^{x)
= 1 for x G \V{f')T^Q
showing that e\:\x) = ei^{x)
for all x G \TY'
4.23 (/*L = L' finishing the proof of the Proposition.
by 4.17C. By QED
We head towards a proof of the Fundamental Identities in the algebraic case; along the way we show that L^ is ample. When A is an integral (h x h) matrix, let XJ^A • X^ -^ X^ be the homomorphism defined by
V'A(^)
= y where yi = 2_]^ji'^j'
I ^ ^ ^ GL;i(Q)
j
choose a non-zero n G Z such that nA is integral and define tpA = ipnA o (f^xf^)'^ which is a Q-isogeny from X^ to itself (see 4.20). PROPOSITION 6.14. Let Q' = ^A- Q - A where A G GLh{Q) and Q is a rational h x h positive definite symmetric matrix. Choose an n > 1 so the
THETAS WITH QUADRATIC FORMS Q-isogenies f^,f^
109
, II^A are represented by morpbisms in:
Then, for suitable rij the pull-backs of L^ and L^
to the top X^ are
isomorphic. Let / ^ o V^^ o {f^)'^
PROOF:
for all a;, 2/ G V(X^)]
this clearly being necessary for the Proposition to be
correct. If x = V{f^)v
=
be the Q-isogeny V'^. We first show that
and y = V{f^)w,
then
e^""'{V{f^'){vA),V{f'^'){wA))
= e^^''\vA '
Q'-\wA)
= e'-^'^t; . yl . Q'-^ . *A, ti;) = e^^'\v • Q - \ ti;) = e^'^ix, y). Choose a non-zero m so that mtp'j^ = ^ is a morphism. The above shows that e^ ^
= e^ ^
. Replacing m by 2m if necessary, we may assume that
e* of both bundles vanish identically, so by 4.23, g*L^' ^ m*L^.
QED
We can now prove the ampleness of L^. It is clear that if two line bundles are related by a rational isogeny both are ample or neither is ample. By the above Proposition we need to show that L^ is ample only for diagonal, integral Q (because any quadratic form can be diagonalised), and by the last statement in Proposition 6.13, it suffices to do so when /i = 1, Q = d > 0. In this case, L^ on X^ and L^ on X are related by a suitable rational isogeny, showing that L^ is ample. Notice that given i?^ and A, with d^ the unique element fixed by a maximal isotropic subgroup of Eeis{2{g x /i). A/), we can construct 'd^ in a purely group theoretic way without reference to the underlying abelian varieties. We now make the algebraic definition:
TATA LECTURES ON THETA III
110
DEFINITION 6.15.
Given any rational symmetric positive definite h x h
matrix Q, abelian variety X, ample, degree 1, even symmetric line bundle L, and Gopel structure for (X, k), define: ^a,Q
^''[V{f^){xQ-hy)]
Here t9^ is the algebraic tlieta function on
for all X eVl'^yeV^.
V{X^)
associated to L^. Comparison with 7.9D shows that when fc = C, we have the same theta function as the analytically defined "d^^^. We wish to prove the two fundamental identities for these algebraic theta functions:
(6.16).
(^'
IfQ=
^ \andxi
eV{X)^',X2eV{X)^',
then
^"'^^[a:i]i?«'^^[j:2] = cnst. ^^'^[ari,X2], (6.17).
IfQ' = ^A'Q'A, ^a,Q'
x-^A-^ y-A
andxe ^Cnst.
V^^.ye V^ then ^X^t/r(H)C/r(5)^"'^ R S
where R and S run through a system of coset representatives for T^ • ^A/{T^ n T^ . ^A) and T^ • A-^I{T^ T{R),T{S) §5
G G{L^^^)
Uxr(y)fix)
PROOF:
=
• A'^ D T^) respectively
act on functions on V{X)^
and
as usual (right action of
Xe{^,x)f{x-y)).
We shall consider only (6.17) as (6.16) is quite simple. Note that
by (6.14), we have an isomorphism
i^P'^r : f(L«') -^ equivariant for
J(V'A)*
f(LO)
* ^ ( L ^ ) —^ ^ ( L ^ ) . (6.17) states a relation between
2 distinguished elements in these spaces:
s e r(L^) c r(L^)
THETAS WITH QUADRATIC FORMS
111
and
t e r(L^') c r(L^'). The idea of the proof is this: the elements s and t are characterized as the unique elements of an irreducible representation fixed by certain maximal isotropic subgroups. To compare these elements we need to compare explicitly the representation spaces and the maximal isotropic subgroups of s and /. To do this, it is convenient to have the same Heisenberg group acting on both spaces. Define /?^ : a(L(^)) ^ G{L^) as follows: /?^(Arx . ry) = \TV{f'^){xQ) X £V^,y
^V^.
• rl^(/^)(y)
for all A G F ,
To show that (i^ is a homomorphism we must check that
=
e'-\v{f'^){xQ),V{f^)y))
e^"'\x,y),
but this is Proposition 6.13A; furthermore /?*' commutes with the involution and there is a commutative diagram: I ^ T
-^
aCLW) - ^ V{X^)
—^ 0
/?
1
1 ^ r ^ a(LW)) ^ v^(x«) — 0 where a^{x + j/) = Vf'^{xQ + y). In addition, aQ(TX'')
= TX'^
and /?<5<,'-<^>(x) =
xeTXK
Composing /?^ with the action of ^(L^) on r ( L ^ ) we get the desired action otg(L^^^) on r ( L ^ ) . Moreover, we define '??W = t'Ja'^a;],
for all s € f(L
TATA LECTURES ON THETA III
112
As in §5, we check that Ugd^ = d^
, for all g G ^'(L^'*)),^ G r ( L ^ )
where g{\S^^) acts on the space of functions on V{X^) as usual. Let HA be the composition
and define 4>A by the diagram
g(L(''))
"L
aCLC")
F(XW).
Putting this together we have the diagram: LC")
>
V{X'') (/Q)*(L
vO'
a«
•L«'
^ y(x'') L«
V{X'')
V(A)
V{X^) L«'
/'^
ViX'i')
^ y(X<3) The map <^>i is the dotted arrow LEMMA 6.18. x^A'^
The map HA preserves the involution and (f>Aix + 2/) =
-h yA where x G V^'' and y G V^.
P R O O F : HA
preserves the involution simply because /?^ , /?^ and j ( ^ ^ , L^ )
preserve the involutions in question.
=
{xiQA{QT\x2A)
=
{xi'A-\x2A). QED for the lemma.
THETAS WITH QUADRATIC FORMS LEMMA 6.19. IfU^
113
and U^' denote the actions of§{L^^^) on f (L^) and
on f (L^') induced by /?^,/?^' respectively and xj^'X : r ( L ^ ' ) —> f ( L ^ ) is the isomorphism induced by \j)'j^, then ^
"I^A^h^g^ = Uf^l^A^
B-
^%,
for all g € Q{\S''^), s G f ( L « ' ) , and
= ^?' o 4>Aforall s € r ( L « ' ) .
The proof is left to the reader. LetO^se
r ( L ^ ) and 0 7^ ^ E r ( L ^ ' ) . Put s' = ^ ^ t . In the classical
case, s corresponds to d^{Z^T)
and s' to d^ (ZA^T)^ and Proposition 6.6
describes the relation between them. Note that s is fixed by the subgroup
while, on the other hand, using Lemma 6.19 we see that 5' is fixed by = T{T^ ' ^A) • T{T^ • A - 1 ) = T ' . The following simple lemma tells
h-\T)
us how to get hold of s' given 5: LEMMA
T'nk
6.20. Let T and T he subgroups ofHeis{2g, A/) such that Tnk* =
= 1 and 7r(T) ancf 7r(T') are maximal isotropic lattices (i.e., compact
open subgroups) off\f.
Let 0 ^ s be a T-invariant vector in the Heisenberg
representation H o{JIeis{2g^l\j).
Then 5' = / J t ^ ^ 5 , where g runs through
a system ofcoset representatives ofT'/TfYT', ifandonlyif7r{T)n7r{r) PROOF:
=
9
is T'-invariant and is non-zero
7r{Tnr).
It is clear that 5' is T'-invariant. The important thing is that it is
non-zero when 7r(T) O 7r(T') = 7r(TnT')! Let s" = Y^Uhs' where h runs through a system ofcoset representatives of T / T f l T ' . Then
«" = E E ^"^^^ = E E ^('^W' <9))UgUHs h
g
9
9
h
h
because if ^^ $^ T fl T',^e(7r(ft),7r(^)) = 0. This shows that s' ^ 0. We don't need the converse.
h
QED for the lemma.
114
TATA LECTURES ON THETA III Applying Lemma 6.20 to the situation T -
T{T^)
•
T{T^),
V = T{T^ • U ) • T{Ti . A - i ) we find: / = const. X^X^^r(ii)^T(5)S. R s Taking d^ of both sides and using Lemma 6.19B, the fundamental identity 1 ? - ' ^ ' O j, = c o n s t . ^ R
Y. S
UriR)Ur(S)^''^^,
is proven.
QED
The algebraic development of this theory goes through if char, k = p and Q and A both belong to GL/i(Z[^]) and p f a. If p > 0 we may either ignore p-torsion to formulate 6.17 or assume that X is an ordinary abelian variety, in which case a Gopel structure at the prime p can still be put on {X, L). If 2 f a the assumption that L is even symmetric can be dropped for 6.1 to be true (assuming that L is still symmetric). COROLLARY
6.21. If L is a symmetric ample line bundle of degree one on
Xj then ^^^{x) = €'d^{—x) where e = 1 ifL is even symmetric and e = — 1 if L is odd symmetric (char, k ^ 2). When k = C and L is even symmetric, then X = XT for some T £ S^g and L is then the basic line bundle. Therefore
^2
= exp7ri^Xi{Txi
-f X2) - i}{Txi-^
X2,T)
which is obviously an even function of (xi, X2). However there seems to be no straightforward algebraic proof of this fact. PROOF:
Assume that L is even symmetric. Put h = 1 and Q — Q' = I
and A = —I \n 6.17. This gives t?^[—a:] — 6i^^[x] for some constant e, and therefore d^'lx] = cd^'l-x] = €'^'d''[x] showing that e = ± 1 . Now puth = 2,Q=
Id., Q' =r 2- Id., yl =: ( J
- 1 ) ^" ^•^'^•'
THETAS WITH QUADRATIC FORMS
115
r?6^T2/T2
r?G^T2/T2
(because Ur(2r,)^'' = ^''). Replacing y by —y ^^^s not alter the L.H.S. while the R.H.S. becomes
= ec
^ ( | ' ^ + 2/)^"[-^ + ^ ] ^ " [ - ^ + y]
Y »76iT2/T2
which is the old R.H.S. multiplied by e and this shows 6 = 1 . Take any ^ G | T ( X ) / T ( X ) , with C = ( 6 , 6 , - - - ) - Let M = T*^{L), By 4.22D, M is even or odd symmetric depending on whether e*(^) equals 1 or - 1 . Now 2;^L ^ 2;^M so that f(L) = f(M) and ^'(L) = g{\/l) and (T^{x) = e^{i,x)a^{x),
UO ^ s e r(L), then Ur(^^s = s' e r(M) and
therefore ^""'^(x) = const. C/^(^)i^'^''-(ar). Denoting i^"'^ by d^ as before, we have: , ? " • » = e ( i , x ) , ? " ( - ^ + ;c) ^«.M(_^) ^ , ( i ^ - x ) t ? « ( - ^ -x) = e{^,-x)e{C,
= e{i, -x)^-i^
x)e.{0^'^i-2^
+ x)
+ ^ + x)
(by the quasi-periodicity of 'd^) = e.iOe{^,x)^^i-^ =
+ x)
e.(0^''-'*{x).
This finishes the proof of the Corollary because any odd symmetric M = T^L for some x such that 2x = 0 and for some even symmetric L from Corollary 4.24.
QED
116
TATA LECTURES ON THETA III As in §5, 7?^'^(a?) can be described as a matrix-coefficient:
where i^ is the composite f ( L ^ ) —^ L^(0) - ^ Jb and 0 9»^ s G r ( L ^ ) , the subspace of f (L^) fixed by the subgroup ai^TX^) (Q ^ f(£Qy
C ^(L^'*)). Thus
determines the function t?^'^. Let S{V^^) denote the space of
locally constant functions on V^ that have compact support. Choosing an isomorphism / : S{Vi^) =. r ( L ^ ) , /i^ = i^ o f now appears as a finitely additive fc-valued measure on the Boolean algebra of compact open subsets of V^. Note that /i^ is determined only up to a scalar because / and i^ are only uniquely determined up to scalars. We want to compare (A)
A^^^, Ai^^ and /i^ where Q = ( ^ '
Q \
and (B) (A)
/i^' and //^ where Q' = ^AQA, Let Qi be (hi x ft,-)-matrices for z = 1,2 and let /i = /ii -f /i2- Choose
isomorphisms /,• : S{Vi *) = r(L^*). We then have the following chain of isomorphisms of ^(L('*^)-modules:
The last isomorphism comes from the fact that X^
= X^^ x X^^
and
L^ = PiL^^ (g) P2L^^ (which also shows that up to a non-zero constant multiple e^ = i^' ® ^^2). It follows that const. //^ = fi^' x fi^^ where /i^i X fi^^ is the unique Ar-valued measure on V^ such that {fi^^ x fi^^)(Fi x F2) = ^^^{Fi)'^^^{F2)
where Fi and F2 are compact open subsets of V/*^
and y^ ^ respectively. (B)
Let r = VA • r ( L ^ ' ) -^ f (L^). Then ^^ o r = const, e^'.
Choose
/ : 5(Vi'*) ^ f ( L ^ ) and / ' : S{V^^) ^ f (L^') to be g'(L('^))-module isomorphisms.
THETAS WITH QUADRATIC FORMS
117
Then
= const. (9 or o f — const./i^ o / ~ ^ o r o / ' . From Lemma 6.19A, r o U^'^g z=: U^ o r for all g G ^(L^'^)), so if F == / - I o r o / ' , F o Uh^g = UgoF for all g E ^(L^'*)). But such an F is unique up to scalars because <S(\^/*) is irreducible. Now f{x) I—)• /(A^^x) const. f{A^^x). subset U
is a candidate for this isomorphism so that Ff(x)
=
If xu is the characteristic function of a compact open
CV^,
= const.
fi^{Fxu)
= const. n^{xu o A~^) = const.
fi^ixAu)
= const.
fi^{AU),
We summarize: PROPOSITION 6.22. Given a Gopel structure {Vi, V2) on (X, L), every rational symmetric positive definite matrix Q determines, up to scalars, a finitely additive measure fi^ on the Boolean algebra of compact open subsets ofVi
such that (^^
^
Y then /i^ = const. //^^ x fi^\
A.
IfQ=
B.
IfQ' = ^AQA with A € GL/,(Q), then fi^\u)
= const.
n^{AU).
and
7. Riemann's theta relation Although the results of the previous section give us a huge class of identities satisfied by the theta functions, Riemann's theta relation is special in having some elegant reformulations. The purpose of this section is to explain these interpretations: in particular we give its formulations in terms of (i) the Heisenberg action on the tower of spaces r(L),r(L2),r(L^),..-, (ii) explicit formulas, and (iii) the measures on A^ that define d^ (see §5). THEOREM
7.1. Let X be an abelian variety over an algebraically closed
field and L. a symmetric line bundle of degree one on X. Take n > 1 and char k f n; then trivializing L on y{X) a subspace of the functions onV{X).
as in §6, we can consider r ( L " ) as Then
(a) r ( L " ) is the space of function generated by polynomials of degree n in
f(L). (b) r(L'^) is mapped into itself by G{L) acting by: forXek,xe
V{X), iui"^l^f){y)
= X"e{x/2, y)"/(y - x)
(c) r ( L " ) is irreducible under this action. PROOF:
Note that (c) implies (a) because the group action on r(L'^) is
just the n^^ symmetric product of the action on r(L). Choose a splitting V{X) = l^ 0 V2 into maximal isotropic subspaces for the skew pairing induced by L. Define g{L") = {im,xr.) I K
: mlL" S T ^ ^ m ^ L " } .
Then ^(L") acts on r ( L " ) and there is a map
a(L)
> a(L")
RIEMANN'S THETA RELATION
119
The kernel of this map is //„; hence this map is an isomorphism ^(L") = ^(L)//in. The commutator of ^(L") is e^{x, t/)", so ^(L") is still an Heisenberg group. Let Wi = T{X) D VJ; then Hn = l/nWi isotropic for e^(x,y)^.
® W2 is maximal
Lift Hn to Tin = T{l/nWi)T{W2)
and take the
quotient of L'* by Tin to get
Then A'(E) = 0 since 1/nWi^
L'^
—^
E
X
—.
X/Hn
W2 is maximal isotropic; therefore deg E =
1, and r(L'^)^** is one-dimensional. This implies r(L'*) is acted on irreduciblyby^(L"). LEMMA
7.2.
QED
Let 'd : A / —y k be a locally constant function such that
(a) d{x -h ik) = e^(k/2)e{k/2, (b) d{-x)
Vib E Z^^
x)d{x),
and
= d{x).
Then there exists a finitely additive measure /i on A^ such that for all X
=
i) t?(x) = I
^ e ( ^ ^ ^ ) e ( - * X 2 • u)d^{u), and
ii) For 6 G Z,
PROOF:
This can be verified by a straightforward calculation. We can also
proceed as follows: By Proposition 5.10 we know that the space of functions S generated by the action of the Heisenberg group on d is irreducible (the action is {UxT(k)f){x) = Ae(fc/2, x)f{x—k).
Under this action d is fixed by a
maximal isotropic subgroup; hence there is a unique (up to a multiplicative constant) map
rl>:Cl{I\])-^S sending 6, the characteristic function of Z^ to ?^. The action on Cj^(Aj) is ^r(yi,y2)/(^) = e(-^^^^)e(2/2, a:)/(x + t/i)• Define a linear functional ^ on 5
120
TATA LECTURES ON THETA III
hy i : f —)' /(O). Then £ corresponds, via rp, to a finitely additive measure on Ay. Since ip is equivariant
i9(x) = ^(C/.(_,)t?) = Jtp-\Ur^_,)^)dn = / Ur(^.x)6dn = I
e(-*X2 • w)e(-—-—)dn{u).
QED Let 1? satisfy properties (a) and (b) of the lemma above. LEMMA
7.3. If there exists a function xp : A / —>• k such that t?(xi,:C2)%i,2/2) =
(Bl)
E , e ( i i , / i , ) < - r i • ^ 2 ) ^ ( ^ ^ +V,X2 + y2m'-^
+ V,X2- 2/2)
then (B2)
il^{xi,x2)'(p{yi,y2) =
2-^E^,(ii./i.)eCC-^1)^(^1+ y i , ^ ^ + C W ^ i ~ 2/1, and vice versa: if there exists 'tp so that (B2) holds, then (Bl) does also. PROOF:
We often use the quasi-periodicity of d in the form
for T] G ^Z^, We only show that (B2) implies (Bl), since the proof that (Bl) impies (B2) is similar. We start with the R.H.S. of (Bl) and show after a short calculation that it is, indeed, ^(xi, X2)i9(t/i, 2/2)- First express ^^£i±ia ^ jj^x2-\- t/2)V'(^^V^ + ^) 2:2 - 1/2) in terms of i?'s by using (B2) after changing variables:
a;2 - • X2 + t/2
y2^
X2-
J/2
RIEMANN'S THETA RELATION
121
One obtains for the right side of Bl ^ ( 2 - ^ ) e ( - ^ r 7 • X2)eCC • ( ^ ^ y ^ + rj))dix^ + 2ri, x^ + Od{y,, t/2 + C) (by quasi-periodicity)
• i?(xi, 2^2 + 0 % ! , 2 / 2 + 0 = 2-^ ^
e(*C • ^^-y^)e(2*C ' ry)T^(xi, x^ -h C)^(2/i, 2/2 + C)-
Since C G |Z^/Z^, the sum over 77 is non-zero if and only if C € 2Z^. Hence the sum is, indeed, ^(ari, X2)??(yi, 2/2)-
QED
Choose a decomposition of A^^ =.Wi^
W2 so that e* vanishes on Wi
and on W2) and e(x^y) == e(*xi • y2 — *^2 * 2/i)- The main result of this section is: THEOREM
7.4.
Let d : f\.^ —^kbea
non-zero function
(i) d{x •\-k) = t^{k/2)e{k/2,x)d{x),
ykeT
satisfying
= VK
(ii) i){-x) = ^(x). Let fi be the measure on f\i associated to 'd as in Lemma 7.2. Then the following are equivalent: (A) Riemann 's theta relation holds:
«= 1
2-^ Y,
e,{r])e{-ri,xi)i){h{x)-\-rt)d{e2{x)
+ r])d{e3{x)-^
jffere
£l{x) = {Xi -\- X2-\- X3-}- X4)/2 £2{x) = {xi -\-X2- X3- x^)/2 4(x) = (Xi - X2 -h X3 - X4)/2 U{x) = {Xi - X2 - X3 -f X4)/2 and Xi G A/.
122
TATA LECTURES ON THETA III (B) There exists tp : A^^ ^
k so that (Bl) and (B2) hold as in
Lemma 7.2. (C) Under the action of the Heisenberg group Heis{2g,f\f) iU^X,y)f)ix) =
given by
Xe{y,x/2)fix-y)
the span of the set of functions {Ux,y^ • Ux'y^l is an irreducible
(A, y), (V, y') G Heis{2g, A/)}
Heis{2g,f\f)-module.
(C) For all n = 2^, Symm^'V, the span of
i=l
is an irreducible Heis(2g^^j)-module.
Moreover
Symm^V = 2 * {Symm^'^'^V). Here 2* is the map induced on functions by multiplication by 2 on f^f (D) There exists a measure v on Ay such that A*{fi
where A is the matrix A — ^ PROOF:
X fi) = u X 1/
1
1
..
We give all the details; thus the proof is long, but easy to under-
stand. We follow the diagram:
Step two
A C
>
Step one <=^
][
<^=>
B C
Step three <=>
D
RIEMANN'S THETA RELATION STEP ONE:
{A <=> B). Write
E(X,T/)
123
for the function on the R.H.S. of
{B2) so (52) says E(x,y) = rp{x)rp{y). This implies (*)
E(x,y)E(«,t;) = E(x,tz)E(y,t;).
On the other hand (*) implies E(x,t/) is the product of two functions
Since d can be expressed in terms of if), tp can't vanish identically, so this makes sense. Now (*) implies that if E = Vi * ^2? then V'l = ^^2 up to a constant. Hence we've shown (B) <^=> (*). The rest of step one is given to showing that (*) is equivalent to the Riemann theta identity. We write out (*). Set zi = {xi -h yi, (x2 + 2/2)/2)
Z3 = (ui -f ^i, (^2 + i^2)/2)
Z2 = (xi ~ yi,(x2
Z4 = {Ui - Vu(u2
- t/2)/2)
- V2)/2).
We get hiz) = {xi + uu{x2 + U2)/2)
isiz) = (yi + t^i, (2/2 + ^2)/2)
i2{z) = {Xi - t/1, {X2 ~ U2)/2)
£4{z) = (t/l - Vl, (2/2 - V2)/2),
(*) becomes [J2^C(^ ' ^1)^(2^1 + yui^2 + 2/2)/2 + rj)^xi
- yu{x2 - y2)/2 + T))]-
'[J2^C{C ' u,)^{ui + vu{u2 + t;2)/2 + C)^(^i - vu{u2 - V2)I2 + C)] = c E ^ C C ^ • ari)i?(xi 4- t/l, {X2 + «2)/2 + r7)i?(a:i - t/i, (^2 - t/2)/2 + r/)]• C ^ C ( C • yi)^(2/i + ^1, (^2 + V2)I2 + 0^(2/1 - ^1, (2/2 - ^2)/2 + 01
c
Here both r]X range over ^Z^/Z^. Now substitute z's into this, replace ^ by (0, C), ^ by (0, r}) (so we are summing over \{W2 H T)I{W2 H T)):
124
TATA LECTURES ON THETA III 4
+ ri)e{-r], (zi + 2r2)/2)e(-C, (za + z^)/2) =
YJlH'i r/,Ci=l
In this formula substitute zi-f/c for 2:1, 2:3-f A for Z3, multiply by e(—K/2, zi) and e(-A/2, Z3); sum over A, /c G (V^i nT)/2(H^i flT). The L.H.S. becomes, using quasi-periodicity,
E
.
e(-r7,
. -2:3 -f Z4 -f A
Zl-\- Z2-{- K
^
)e(-C,
^
.
.
.
.
) e ( - « / 2 , 2:i)e(-A/2, Za)-
e(K/2, zi + »7)e(A/2, zg + C)»?(^i + '7)'?(22 + »7)i?(z3 + C)'?(^4 + C)Simplifying, we get ^ e(K, »?)e(A, Qei-ri, (zi + Z2)/2)e(-C, (zg + Z4)/2)•^(^1 + r])i9{z2 + TJMZS + C)??(Z4 + 0 If we sum over /c, A, keeping rjX fixed, we see the sum is zero unless both TJX equal zero. Thus the L.H.S. is
We now evaluate the R.H.S. After summing over K,X, we have ei-n, {zi -\-Z2 + K)/2)e(-C, (zi - Z2 + «)/2)e(-K/2, zi)e(-A/2, zs)-
Y^ "•^'"•^
-Hhiz)
+ ^
•t^C^Cz) + ^
+ r,),?(^2(^) + ^
+ »?)•
+ Ot^C^Cz) + ^
+ 0-
Change —A/2 to A/2 using quasi-periodicity. This introduces the factors e{-X/2,{zi-Z2-Z3-\-Z4-\-K-X)/2-j-(:)e(-X/2,{zi-\-Z2-Z3-Z4-hK-X)/2-\-r)). Simplifying, the R.H.S. becomes ^
e(-A/2, zi - Z3)e(-K/2, zi)e(-A/2, Z3)e(-»?, ^ ^ ) -
77,^,AC,A
• e(-C, ^ ) e ( ^ , . ) e ( ^ , 0 ' ? ( ^ i W + ^
+ '?)•
RIEMANN'S THETA RELATION
125
The term e(—A/2, K) = 0 since A, K are both in Wi. The first three factors give e(—(« + A)/2, zi); the last two e-factors give a : = e ( / c - A ) / 2 , 7 7 + C). Fix K + X = u] sum over fi with «' = /c -f /i, A' = A — //. Note that the arguments in the ??-terms stay fixed. Since ^—^ = ^ — A we have
This does not stay fixed; we see that 'n'\-C must be in W2 H T, i.e., 77 = ^, to get non-zero summands. So, summing over fi gives for the R.H.S. 4 i/€(VVinT)/2(VVinT )
(:ei(W2nT)/W2nT
«=1
Notice that the sum can be viewed as running over all rj G ^T/T^
so
e{uX) = ^^iv) and e(—z//2,zi)e(—C,zi) = e(—77, Zi). This gives the Riemann theta identity and completes the proof (A) if and only if (B). STEP T W O :
Claim.
We begin by showing that (B) implies (C).
Equation (Bl) implies that the span of the functions of the form
U\^yd ' Ux'^yid is the same as the span of the functions of the form U\yij) where rp(xi,X2) =
ip{xi,2x2).
P R O O F OF CLAIM: We calculate
(**)
(f^Sy+y')/2)V^)(^) = <(y + y'y^' ^/2)'^(^
-iy+y')/^)
= e(y + y', x/2)V^((2ri - yi - y[)/2, (2x2 - ^2 - y'2)/^) = e{y + y', x/2)tl){{2xi - yi - y[)/2,2x2 - J/2 - ^2)On the other hand, using formula Bl:
126
TATA LECTURES ON THETA III {Uyd . Uy.d){x) = 6(t/, ^)^{x - y)e{y\^)^(x = e(2/ + t / ' , - )
^
- y')
e(-*r/• (X2 - 2/2))-
r?6l/2Z9/Z3
'^{{"^xi - 2/1 - 2/l)/2 + r;, 2x2 - 2/2 - 2/2) * V'CC-yi + yl)/2 + 77, - y s + 2/^). We can ignore the last factor since the variable x does not appear there. Since ^(^-!7,0)^(^) = ^(~*^ • ^2)V'(^1 + ^1, 2X2),
looking at (**) we see that e ( - S • X2)V^((2xi - yi - yi)/2 -f r/, 2x2 - 2/2 - 2/2) is in the span of Uy ^ip. Hence the span of the C/^^^^^'s contains the span of the UydUy.d's, From (B2) we have il){xi,2x2) = jp{yi, 2y2)""^2--^^e(*C • 2^1)^(2^1 -f yi, 2^2 + y2 + C)^(^i - 2/i, a?2 - 2/2 + C)C Now
e(*C 'Xi)'d{xi 4-yi,X2 + y2 + C)^(^i - 2/n^2-~2/2 +C), and this gives the non-trivial part of the formula for tp. Hence the span of the d^s contains the span of the U^'^^jp^s.
QED for Claim
The claim implies I span of n
C/A,,.^?
I = {span of ui'liiuiV^.^}
.
This last space, by the same argument as above with 'd and tp interchanged, is the same as the span of U[ ^d where d{x) — ^(2x), but the span of U\ l^d is just 2 * { span of
C^A,^^}-
This last space is an irreducible Heis(2^, A?)-
module. Therefore all these spaces are irreducible. We now show that (C) implies (B). Since r(Ti)r(T2/2) is maximal isotropic for the action of ^ ( ^ \ there is a unique (up to a non-zero constant)
RIEMANN'S THETA RELATION
127
invariant function under r(Ti)r(T2/2). Call this function rp. Let M be the subgroup fixing Ux,y^ ^xyd
inside r(Ti)r(T2/2). Then
^e(r(Ti)T(T2/2)/M)
where c(A, A', t/, y') is a constant depending on the indicated variables. Set = e{y + y\ x/2)d{x - y)d{x - y').
<j>{x) = {U,^y^){Uiy^){x)
The function <j){x) is an eigenfunction for PROOF:
Let z e.Ti,T2\
T{T\)T{T2).
then
U^f^^(j> = e{z, x/2)^e{y + y', {x - z)/2)d{x - y - z)d{x - y'= e(z, xl2fe{y
z)
+ i/', {x - z)l2)d{x - y)d{x - y')
' e*(|)'^(-'2^/2, X - t / ) e ( - z / 2 , X - y')
= e{z,y-\-y')(f){x). Thus the projection of (f) to the invariant subspace of r(Ti)r(T2/2) is zero unless y' = —y. Setting y' = — t/, we sum over ^T2/T2 to calculate this projection: EUlll^{Ur,y^Ur,-y^){x) Ee(77, x/2fd{x
=
- y - r})i9(x -^ y - rj) z=z
Ee(r7, x)'d{x + y-\- r])d{x - y-\- 'n)e{-T], 2x) = Ee(-r;, x)d{x + y + 'n)d{x - y-\-T]) = a{y)ip{x), where the sum is over r] G \T2lT2, and a{y) = c(l, 1,?/, —y). Notice that we have the R.H.S. of (B2).We now show that this is 'ip(x)ip{y). Indeed we assert that a(y)ip{x) is symmetric in x and y: a{y)ip{x) =
(quasi-periodicity) = ^
This gives (B2).
E
^ ( ~ ^ ' a!)i?(a: + y + r])d{y - x - TJ)
^ ( ~ ^ J X)}}{X
+ y 4- 'n)'d{y - x -\- rj)e{-r], y — x)
128
TATA LECTURES ON THETA III
STEP THREE:
We show that (D) => (B). Let h = 2 and write
f\y = AJ e Aj = v^i e V2 = v^ Set X = (xi,X2) G V^,y = (yi,y2) G V^. Let
<^=(;;). -=(; \ ) -
^-{ii)'
so Q' = *^Qyl. We start with a non-zero quasi-periodic function d and we define =
t^^
for
'd{xi,yi)d{x2,y2)
G K We deduce (B2) in two steps.
ONE:
We show d^ satisfies a relation of the form 6.17 for some function
Two:
We show there exists a function T/> on A / so that d^'
=
H^i,yi)'^{x2,y2)
S T E P O N E : Define
: Heis{2gh, (A,
Ay) —y Heis(2gh, )^(A,
yA
A/) ) •
One checks that this is an isomorphism of groups. Let Ck{V) be the space of fc-valued locally constant functions on V with Heis{2gh^f\j)
{U^,.f){x) =
acting by
\e{-,x)f{x-z).
Define a map Go from Ck{V) to itself by
GoU)
= f
x*A-^ yA
One verifies that Go is &(j>—Heis(2gh, A/)-map, so Ug{Gof) = Go(t/^(j)/). Since i? is quasi-periodic, the Heis{2gh,
Ay)-submodule F generated by t?
RIEMANN'S THETA RELATION
129
is irreducible by Proposition 5.10. Let F' = G^^(r) and let Go, when restricted to F', be denoted by G. Let d^
be a non-zero element in F'
fixed by (7(Z)^^^. Thus we have the diagram
A/)
Heis{2gh,
Heis{2gh,
CkiV)
Ck{V)
U
U r
F'
A/)
Lemma 6.20 now implies there is a relationship of the form 6.17 between G(»?«') and i?*? since »?0 is fixed by (***\ STEP T W O :
9*.<3'
yA
a(i)^^'':
= J2UriR)Ur^S)r'<^ R,S
AS in Lemma 7.2 we introduce measures.
Let a be the
unique H€is{2gh, f\f )-ma,p from F to S{VJ^), the space of locally constant, compact support functions on V/*; similarly let a' : F — i >• S{Vi^y be an iyei6(25'/i, A/)-isomorphism where S{V^y denotes a second copy of S{V^). Define linear functional on S{Vi^) and S{Vi^y by
£(/) = (a-7)(0) / ( / ) = [Go(a')-i(/)](0). Then £ and £^ are induced by measures on Vi", in particular, the measure fji from Lemma 7.2 satisfies
e{f) = J fdfi X dfi / € 5 « ) . Define the measure P by
i'{f) = Jfd9
fesiv.f^y.
TATA LECTURES ON THETA III
130
Define F by requiring that the diagram below commute G
a h\f 5(^1")
SiVr").
Since everything in sight is an irreducible Heis{2ghj f\f)-modu\e^
F is char-
acterized, up to a constant, by being a map oi Heis(2gh^ A/)-modules. One checks that the map from S{V^y
to S{V^) defined by
is a map of/fef5(2y/i, Ay)-modules, so we can conclude that this is, indeed, the map F, CLAIM: PROOF:
There exists a measure v on Vi so that 9 = v x v.
I-
' f{u)d9{u) =
eoGo{a')-^{f)
=
£oa-\Ff)
= l{Ff)dfi
X dfi
fA*{dfjL X dfi) /
•
— / f{du X du). This last equality is hypothesis (D). Since d^ is defined by z/ we have shown there exists tj) so that d^'
=
i^{xuyi)ip{x2,y2)'
We show the converse. Given Q,Q',A,d^,d^
and the function d
corresponding to the measure // and we get (***). Replace x^y by x^A and yA~^\ the R.H.S. is (B2). Use (B2) to conclude there exists xl) so that ^a,Q
_ ijj(^x)xl;{y). Let u be the measure corresponding to ip. By
Proposition 7.20, t?^'^ corresponds to A*(/i x //), but it also corresponds to 1/ X I/. So (D) holds.
QED for Theorem 7.4
RIEMANN'S THETA RELATION
131
Does a function ?? satisfying the equivalent conditions of Theorem 7.4 come from an abelian variety? This is almost true, but not precisely so. What can be proven is: ASSERTION
7.5. Let 'd satisfy the conditions ij, ii) and A) of Theorem 7.4
and also the non-degeneracy
condition:
Hi) For all x G Ay^, there exists rj G ^Z^^ such that i?(x + 77) ^ 0.
Then there is an abehan variety X defined over k, an even symmetric degree one ample line bundle L on X and an isomorphism
V(X) ^ f\f carrying T{X) to P^,e^
to e and e!r to e* such that d"^ for (X, L)
is equal to d. This assertion is in unpublished notes of the senior author, and is a straightforward generalization of the results of §10 in Equations defining abelian varieties III, Inv. Math., vol. 3, 1967. The results of that paper deal with i?'s on Q2^ satisfying the conditions of Theorem 7.4 and in this case, one can extend Assertion 7.5 to degenerate 'd's which don't satisfy iii). The result in §11 is: THEOREM
7.6. Let 7} :
QI^
—> k satisfy the conditions i), ii) and A) of
Theorem 7,4. Then there exists: a) a subspace W C Qll^ such that W^ C W, where W-^ = {y\e{x, y) = 1, all Let dim W/W^
xeW}
= 2h.
b) an element rjo G ^I-l^ such that e^{rjQ) = 1 and e*(x) = e{x^2r]o)j
alixew-^n^ll^,
132
TATA LECTURES ON THETA III c) an abelian variety X of dimension h and an even symmetric degree one ample line bundle L on Xj d) an isomorphism V2{X) -^ e^ to e\wxw
W/W^
carrying T2{X) to W n Z2^
aiicf ejr to the form
such that e) ^ = Oonql'
-{r]o-{-W + Zl'),
f) The function x H^ e(x, ^)i?(r;o -^ x) on W is invariant translation by W-^ and induces on WjW^
modulo
the algebraic theta
function "d^ ^ defined by X and L. It would be very nice if Theorem 7.6 were also true for the full adelic case of Assertion 7.5, but unfortunately, this isn't true: if ?^ is a nondegenerate theta function as in 7.5, then let U = Q2^ + Z^^ and let xu be the characteristic function of U. Then xc/^ silso satisfies the conditions of 7.4 but certainly isn't i9^ for any abelian variety. What we need are some further conditions on "d that imply: (A*) For every rational orthogonal hxh matrix A, the following identity holds:
for an appropriate constant c and where t] and ^ range over l(g, h) • 'A/Z{g, h)-*An 1(9, h) • *A-^/Iig,
I(g, h) and
h) • 'yl-i n l{g, h)
resp.
and: (C*) For all n > 1, Symm"F is irreducible as Heis(2^, A/)-module. This is an interesting topic for investigation.
8. The metaplectic group and the full functional equation of d In Chapter I, §7 we derived the functional equation in r for d{z^ r) by direct methods and the same procedure can be generahzed to get the functional equation for the many variab'le function d{z^T). We shall adopt (T) was characterized as a matrix-
another method however. In §3, i?"
coefficient of the representation 7i of Heis{2g,R): d"" xi
X2
and this combined with the fact that ej is fixed by
(T{I?^)
allows us to
deduce immediately the functional equation of i? in z: d{z, T) = exp(7ri*mTm + 2wi^mz) • i^(z -]-Tm-^n,
T).
In this section we shall construct a two-sheeted covering of Sp{2g^ R) which is called the metaplectic group Mp{2g^li) and show that there is a combined action of Mp{2g, R) and Heis{2g, R) on 7i. We then see that ?^(z, T), multiplied by an exponential factor, is a matrix coefficient of this representation, enabling us to write down the full functional equation of d in both X and T. Roughly speaking we obtain an action of Mp(2^,R) on W, the irreducible representation of Heis(2g^R)j
as follows: Let y G 5p(2^,R), so 7
defines an automorphism of Heis{2g^R) by (A,x) —>• (A,7(x)). Define a new action of Heis{2g, R) on 7i by iU'x,J) = {UxMr)f)By the Stone-Von Neumann-Mackey theorem there must be a unitary map A^ : H —)• H intertwining these two representations. The map A^ is determined only up to a constant; indeed, there is no well-defined map 7 —y Ay giving a group action of Sp{2g^R) on W; however there is an action of a two sheeted covering of Sp{2g^ R). This is the group Mp{2g^ R). We know that
r
X2
m
= (^r(..)/T,ez>.
134
TATA LECTURES ON THETA III
Let 7 C Sp{2g, R) act on H via Ay, then xi X2
{T) =
{AyUr^.^)fT,Ayez)
The functional equation falls out of this equation once we compute
Ayfx
and AyCj. Unfortunately, there are many details involved in carrying out this program, although the essential idea is so simple. In this section 7i will always be the unique irreducible representation of Heis{2g,R),
?^oo and H-oo as in §2, and friy)
= exp{7ri^yTy) G L'^{R^).
Because we are working with a fixed g we shall abbreviate Sp{2g,R),
and Mp{2g,R)
Mp{2g,R),
Heis{2gjR),
and Mp respec-
to Heis,Sp,Mp,
tively. PROPOSITION
8.1. Let U{7i) be the group of unitary isomorphisms
and Mp{2g,R)
= {A e U{n)\AU(^x,v)A-^
ofTi
= U(^x,yv) V(A,t;) G Heis, for
some 7 G Sp}. Given A G Mp, the corresponding j £ Sp is unique; denote this 7 by p{A). Then there is an exact sequence of groups: 1 —^ Ci —^ Mp-^Sp PROOF:
—y 1.
Let A G Mp, and let 71,72 G Sp be such that: C/(A,7,.) = AU^x,v)A-^ V(A,t;) G Heis.
Then C/(A,7it;) •
^(A^JV)
~ ^^' ^^^ therefore it commutes with Uh^'^h ^
Heis, showing that eijiv — 72^',^^) = 1 Vi/; G R^^, and therefore, that 7it; = 72?;, Vt; G R^^. This shows that p : Mp —> Sp is well-defined; we omit the trivial checking that Mp is a group and p is a homomorphism. The kernel of p = {^ G U{H)\AUhA-^
^Uh^^h
£ Heis} = C^ by
the irreducibility of 7^. It only remains to show that p is surjective. This follows from our remarks above explaining how Sp{2g, R) can almost act on 7i.
QED
THE FUNCTIONAL EQUATION
135
Next we shall write down explicitly members of Mp sitting above a generating set of Sp for the model 7i = L^(R^) where ^(1,^,0) is translation by X and t/(i,o,ar) is multiplication by the character exp(27ri^xy). LEMMA
8.2. I.
For all A G GLg{R), let y = (
^^_^ j ; for this 7 we
define
Pfiy)=fiA-'y)\det
A\-"\
II. For all symmetric C G GLg{R) let j = I Pf{y) = Ill
If7= [ ^
1; for this 7 we define
f{y)exp(-7ri^yCy).
~ ^ y then set
Pf{y) = f{y) = I
f{x)exp{-27ri'xy)dx.
In all these cases P G Mp, y £ Sp and p{P) = 7. PROOF:
It suffices to check that PC/(i^^)P"^ = ^(1,7^) for t; G R^ x 0 and
t; G 0 X R^ in all three cases. I. (Pf/(i,.,o)P-V)(2/) = (C^(i,x,o)P-V)(^-'j/)|det >l|-i/2 = {p-'^f){A-^y
+ x)\Aei
= f{A{A-^y-\-x))\A&i
A|-i/2
A\^l^\Aei
A\-^l^
= f{y + Ax) = {U(i,Ax,o)f){y)(PC/(i,o,.)P-V)(2/) = (f/(i,o,x)^"V)(A-ij/)|det = {P-^f){A-^y)exp = /(yl^-iy)exp
A\-''^
{2wi*xA-^y) • |det ^ | - i / 2 {2iri*CA-^x)y)
- (£^(l,0,M-ix)/)(2/)-
136
TATA LECTURES ON THETA III
II. (Pt/(i,,,o)P-V)(2/) = iUii,,,o)P-'f){y) := {P-^f){x
-expi-iri'yCy)
+ y) '
= f{x + y)exp[Ki\x
exp{-iri'yCy) + y)C{x -f y)] • exp{-7ri^yCy)
= f{x + y)exp{27ri^yCx + Tri^xCx) = {U(^i,x,cx)f){y)' Also (1,0, ar) = (l,7(0,x)) and clearly f/(i,o,a:) commutes with P so that PU(^l,0,x)P~^
= ^(l,0,a;) = ^(l,7(0,ar))-
III. is just the usual properties of the Fourier transform.
QED
Because the matrix representation of Sp{2g, R) comes from its action on the Heisenberg group, we must let it act on S^g by
j{T) = {DT-C){-BT
+ A)-\
which is the usual action after conjugation of 7 by (
j . We will see
below that this is necessary for everything to be compatible. In any case, 5p(2p,R) acts transitively on S)g and the stabilizer of il is: U{9)
{{B
~f)\'AA
+ *BB = I,UB
=
*BAy
Recall that the Lie algebra of Heis{2g^ R) acts on Tioo C "W; in particular, 6UA^ and bllB, denote the operators corresponding to the elements (l,e,,0) and (1,0,ef) if { e i , . . . , e ^ } is the standard basis of R^. WT denotes the span of the operators SUA, — YlTijSUsj for i = 1 , . . . ,^. Under the action of 7 G 5*^ on S)g given above,
WT
gets transformed into
^^(T)-
Because C • / T is the subspace of Tioo annihilated by WT (Theorem 3.2), if P G Mp and p{P) = 7, then P/T is annihilated by Wy(^T) and therefore: P/T
= (const, depending on P and T) • fy(T)'
We shall compute these constants explicitly:
THE FUNCTIONAL EQUATION THEOREM 8.3. Let P e Mp,
e"- - - . L^(W), fr - -'^'"^^^
= 7 =
p{P)
137
M
^
Recall that in
ThenWTeSig, PfT=C{P;T)fyiT)
where C{P]T)
is, up to a scalar of absolute value one, a branch of the -f A)]^^/'^
holomorphic function [det{-BT P R O O F : Let d
For PeGu Let
= {P e Mp\PfT
let Pfr = C{P; T)f,^p)T,
"iTef^g,
is continuous in T G f)^ and
C{P\ Tf\ det{-BT
PeG
V T G S^g}.
= const./^.^P^T,
C{P\T) G2={
onTeS^g.
+ A)\ = F{P] T)
is independent of T with values in C^ The Theorem is equivalent to the statement: d
= G2 = Mp (that
Gi = Mp follows from the above remarks, but we wish to deduce it from Lemma 8.2). Clearly Gi is a subgroup of Mp. To show that G2 is a subgroup of Gi it suffices to show that C{P\T) (i) C{P-T)C{Q-p{P)T)
and Aei{—BT -f A) are 1-cocycles:
= C{QP;T),
>/P,Q e Gi,
and (ii) d e t ( - 5 T + A) d e t ( - 5 ' 7 ( T ) -h A') = det{-B"T
(A
whereT=^^ here i ^
B\
.(A'
^ J and (^^,
^ j and f ^ ,
B'\(A
^,J(^^
B\
+ A'')
(A"
D J = (,G"
B"\
D"p
^ , j belong to Sp.
But (i) follows from the definition of C ( P ; T ) and (ii) is straightforward. Now f
^ 1
matrices C, and ( ^
j for ^ G GLgiR), i
j]
^^^ symmetric {g x ^f)-
^ 1 generate Sp and consequently the P G Mp
given in Lemma 8.2 I, II, III and A G C^ C Mp generate Mp and we shall show that all these elements belong to G2. This will conclude the proof of the theorem.
138
TATA LECTURES ON THETA III For P = A G CJ, A / T = A • fi,i^p)T and therefore we get C(P;T)
\,F{P;T)
=
= \\
For P as in I, {PfT)iy)
= fT{A-'y)\det
A\-'/^
= exp{wi*y*A-^TA-^y)\det = f'A-^TA-riy)\det = A(T)(j/)|det
A\-^'^
A\-"^ A\-'/\
For P as in II, Pfriy)
= exp(7ri*T/Ty) • exp (-Tri^yCy) =
fT-civ)-
Here C ( P ; T ) = F ( P ; T ) = 1. For P as in III, -,-1-1/2
T PfT = fT = {JfT)f-T-^ = d e t ( i
fy(T)
SO C(P]T) = [det(2^)]~^^^ with C{P]iI) = 1 and F(P;T) DEFINITION
8.4.
Let x ( P ) = F{P;T)
= {9,
= d e t ( - 5 T + A)C(P,T)2.
QED The
above shows that x • ^P —^ C^ is a character and X{X) = X\ VA G q C Mp. The metaplectic group Mp{2g, R) or Mp is defined as ker xNote that we now have an expression of 'd^[x](T) as a matrix coefficient of the combined representation of Mp{2g^ R) • Heis{2g^ R) mTi:
nx]{j{iI,))
=
C{P;iI,)-'{Uri^,yP^fir,,e:,)
where P G Mp{2g, R) and 7 = p(P). We shall topologize Mp and Mp in the following way: STEP I:
Let CT : Mp —y C* be defined by CriP)
= C{P\T).
Fix
TQ G ^g and give Mp the weakest topology so that p : Mp —^ Sp and
THE FUNCTIONAL EQUATION
139
CTQ '• Mp —)• C* are continuous. With this topology it will be shown that Mp is a HausdorfF manifold. Define g : Mp —^ C* x Sp by g{P) =
(CTO(P),P(P)).
Clearly the
topology on Mp is the weakest so that g is continuous. Also g is one-toone: if ^(P) = g{Q), then p(P) = p{Q) and thus Q = XP for some A G Cj:, but
CT{Q)
=
=
XCT{P)
implies now that A = 1 and P = Q.
CT{P)
Let f :C* X Sp —^ C* be defined by /(A, 7) = A^ det{-BTo 7 = ( ^
-f A) where
^ j . By Theorem 8.3, g{Mp) = / " H C I ) . Now | { 9^ 0 and CJ is
a submanifold of C* implying that g{Mp) is a submanifold of C* x Sp (and is thus automatically Hausdorff" because both C* and Sp are Hausdorff). STEP
II: We show that C{P\ T) considered as a function from MpxS)g —>
C* is continuous. Thus the topology on Mp is the weakest one so that p and CT are continous, VT G S^gC(P;T)^det{^BT-\-A)
= x{P) where p{P) = ("^
^V
Putting
T = To we see that x is continuous and therefore C ( P ; Tf
= xiP) deti-BT
+ A)-'
is continuous. To see that C{P\T)'^ has a continuous square-root
7i{P;T)
on MP X S)g, it suffices (from the lifting theorem and the fact that Sig is simply connected) to show that C{P;T)'^ when restricted to Mpx {To} hcis a continuous square-root, but by definition
CTO(P)^
= C(P;To)^ and
CTQ
is continuous. Thus we may choose a function 7i on Mp x S)g uniquely by demanding that W(P;To) = C(P;To) and H{P\Tf
= C{P,T).
Now fixing
P G Mp, the functions 7i{P;T)
and C{P;T) on S)g are both continuous,
= C{P',Tf
= C(P,To); from which H{P\T)
H{P\Tf C{P\T)
and
H{P,TQ)
=
for all T e ^g. This holds for all P £ Mp and thus W = C and
therefore C is continuous. STEP III:
The multiplication m : Mp x Mp —>• Mp is continuous. We
have to check that
140
TATA LECTURES ON THETA III (a) p o m is continuous, and (b) CTO O m is continuous.
But (a) follows from the fact that Sp is a topological group, and CTQ O m{P,Q)
=
CTO{PQ)
= C{Q]To)C{P;p{Q)To)
is continuous because C :
Mp X S^g —^ C*, p : Mp —y Sp and the map Sp x S)g —>• S)g given by (7, T) I—)• 7(T) are all continuous. STEP
IV:
We now show that h : Mp —>• C^ x Sp given by h{P) =
(x(P), p{P)) defines a connected two-sheeted covering of the Lie group C^ x Sp and thus gives Mp the structure of Lie group. h = {x, p) IS a, continuous homomorphism and ker h = ker xflker p = ker x n C J = ker(x|CJ) = {±1}. Now h = h'og where h' :C*xSp
—^ C* x
Sp is defined by /i'(A, 7) — (/(A, 7), 7) and / and g are as in Step I. Clearly /i' is a two-sheeted covering projection. Now (/i')~H^i ^ -^P) — 9{^P)
so
h' restricted to g{Mp) is a two sheeted covering: g{Mp) —>• CJ x Sp. It only remains to prove that Mp is connected; but this follows easily from the connectedness of Sp and CJ. The following is a more subtle fact: PROPOSITION 8.5.
Mp is a closed connected subgroup of Mp and p\Mp :
Mp —> Sp is a covering projection with kernel = { i l } . PROOF:
Except for the connectedness of Mp, everything else follows from
the preceding remarks. Put p|Mp = q. Recall that U{g) is the stabilizer in Sp of il G S)g. The coset space of Mp with respect to q''^{U{g)) is S^g by g —)• q{g){il).
Since S)g is connected it suffices to show that q''^{U{g)) is
connected. Here U{g) sits inside Sp{2g, R) as
It follows that CT for T = il^ i.e. C,/ : q~^{U{g)) —>• C*, is a continuous character and
THE FUNCTIONAL EQUATION We shall define det* : U{g) ^ CJ by det* ( J ^
141 ^ J = det(A -
iB)'K
Thus, q~^(U(g)) along with its topology is the fibre-product of det* and Sq:C*i-^
CI where Sq{\) = A^:
1-^
i U{g))
^
CI
and is therefore connected.
QED
In fact, 7ri(5p) ~ Z because Sp =. U{g) x S^g as topological spaces and S}g is a Euclidean space and det* : U{g) —y CI gives an isomorphism of fundamental group. Thus for each n > 1 there is a unique connected n-sheeted covering of 5p, and Mp is the unique connected two-sheeted covering of Sp. COROLLARY
8.6. The exact sequence 1 —> {±1} —^ Mp-^Sp
—^ 1 is
non-split and Mp = [Mp, Mp]. (z PROOF:
Embed U{1) in U{g) by z \-
0\ 1
\0 in Sp{2g^ R) as above. Then the sequence
, and embed U{g) 1/
l^{±l)^q-\U{l))-^U{l)-^l can be identified to i~^{=ti}—^q-^q—^1. To see this use the diagram in the proof of Proposition 8.5 and note that Cii restricted to q''^{U{l))
must be one-to-one (the kernel of g is {±1}
which is killed by Sq). This sequence is non-split (e.g., restricting to the torsion subgroups one gets 0^
Z/2Z - ^ Q / Z - ^ Q / Z —^ 0)
142
TATA LECTURES ON THETA III
and thus the sequence 1 —> {±1} —>Mp —>Sp —^ 1
(*) is non-split.
Because [5p, Sp] = Sp, [Mp, Mp] sits in the exact sequence 1 —y {±1} n [Mp, Mp] —y [Mp, Mp] —y Sp —^ 1. If {±1} n [Mp, Mp] is trivial, then Sp =. [Mp, Mp] C Mp and the sequence (*) splits. Thus {dzl} n [Mp, Mp] = {±1} and [Mp, Mp] = Mp. COROLLARY
8.7. For T e ^g, let U{T) = {y £ Sp\y{T) = T}. CT{Pf
= deeq{P),
QED Then
yPeq'\U{T)).
This is easy; the case T = ±il has already been checked while
PROOF:
proving Proposition 8.5.
QED
From now on, we shall denote elements of Mp by 7,6, • • •, assuming implicitly that they sit over corresponding elements 7, 6, • • of Sp. We shall now write down the functional equation for i^^[x](T) in T where x G R^^,T G S)g. Denote by a'{I'^^) the subgroup (1 x Z^ x 0) • (1 X 0 X Z^^) of Heis{2g,R),
and by Ty : Heis{2g,R)
automorphism {X,v) •—>- (A, 71;) where 7 G Sp{2g,R). ri,2 = {7 € Sp{29,R)\T,{
—^ Heis(2g,R)
the
Then
(T(Z2^)}
is easily checked to be ( _ fA {^"[C
B\ . Dj^^P^^^'^^\
.i^CA and *5D both h a v e \ even diagonal entries j '
Recall that Cej is the subspace of ?^_oo annihilated by C/j; ~ 1, Vx G (T{1?3)
C
lated by
Heis{2g,R). UT^{X)
— 1 VX
Thus, for 7 G Mp, with 7 G ri^2, T^z is annihiG
cr{l?^) and thus 762 = ^(7)^2 where 77(7) G C*.
It follows that r] : q^^{T\^2) —>• C* is a character. We can now write down the functional equation of d^[x]{T) in terms of 77:
THE FUNCTIONAL EQUATION PROPOSITION
8.8. Forx e R ^ ^ T G S)g,y e q^H^h^),
143 andy=
("^
^ V
we have: ^«[x](T) = ^det{-BT
4- Ay^'^d''[jx]{{DT
where the square root det{—BT + A)~^'^
- C){-BT
+
is determined to be
A)-^), C(j',T).
PROOF:
= ri{7){7U(i,x)7
T/T,ez>
'n{7){U{l,yx)C{j]T)fy(^T),ez) 7;(7)c(7;T)(C/(i,-^^)/^(r),e2)
=
rj{j)c{y;T)d-[jx](jT). QED
We immediately deduce the functional equation of d{z^T) in T: COROLLARY
8.9.
For all z e C^,T e ^g,7
^ q~^{^ia)^l
= (^
^ \
we have: r]{jydet{-BT-]-A)-^^'^'exp{7ri'z{-BT-^A)-^Bz)'dC{-BT-\-A)-^z,'y{T)). PROOF:
Starting from the functional equation for -d^, and using -d^ =
exp( iri^xi •^)?^(^, T) we see that we only need to check (A) if 2: = Txi -f X2, then
{DT - C){-BT
-h A)-^{Axi
+ BX2) + (Cxi + Dxi) = \-BT
+ A)-^z
and (B) \Axi
+ Bx2)\-BT
+ A)-^z -^xiz
= ^z{-BT
+ A)-i5z.
PROOF OF (A): {DT — C){—BT + A)~^ G iD^ and is therefore symmetric. Thus
144
TATA LECTURES ON THETA III (DT - C){-BT
= {-rB
+ A)-^{Axi
+ Bx^) + {Cxi + DX2)
+ *A)-\T*D-*C,-T*B
+ *A)(^
^)(l')
= V 5 T + yl)-i(T,l)T-^-7(^;) = * ( _ 5 T + >l)-^(Txi+X2). Let a = {^BT -\- A)'^,
P R O O F OF ( B ) :
Then a{-BT
+ A) = I which
implies a A = aBT -f 7. *(^xi + Bx2y(yz — ^xiz = ^{a{Axi + Bx2))z — ^xiz = ^{{c^A — I)xi + aBx2)z = \aB{Txi
+ X2))z
= ^Z'{-BT
+ A)-^
'Bz. QED
It is clear that 8.8 stands incomplete without a complete description of T]. However we discuss only ry^; this has the advantage of being a character on ri^2 and not just on ^~^(ri^2)PROPOSITION
8.10.
A. rj surjects onto the eighth roots of unity. B. If ker(r7)2 = A, then A contains r4 = { T G 5p(2^,Z)|7 = I{mod 4)}; in fact Anr2
= U'^
^^
eT2\Tr{D^Ig)
=
0{mod4)Y
C. The image of A in the composite A *^^ ri,2 -^ r i , 2 / r 2 = is precisely SO{2g, 1/2).
0{2g,Z/2)
THE FUNCTIONAL EQUATION
145
We only sketch the proof and leave the details to the reader. Let p be the image o f ( ^
^ | x / x . . . x 7 under the action of the natural
homomorphism from SL2{2) x . . . x 5^2 (Z) —>• Sp(2g, Z). It is easy to see that„2(^)^^2j^0 -1^.^2(j).^2(j) ^ 2 ( j ) ^ ^ 2 ^ 0 - 1 ^ The fact that the Fourier transform preserves ex quickly implies that V^ [ ^ i~^. Therefore 'r]^{p) = i~^ and the image of r; contains the eighth roots of unity. The proof of A will be complete if we show that [ri^2 • A] < 4. For ( "^ ^ J^_i j with A e SLg(l) the operator P from 8.2 is Pf{y) = f{A-^y)\detA\'i]
since ei = Yl^n in L^(R^), we have Pej = e i . The ^ j with N symmetric integral and even di-
operator associated to ( ^
agonal is multiplication by exp(7ri^yNy). The conditions on N insure that Pez = cz and hence both of these terms belong to A. Consequently matrices of the form I ^^
^ 1 with the same restrictions on A^ belong to A
(being conjugates of I ^
j ) . Let A' be the subgroup of ri^2 generated
rVi by the above three subgroups. Writing p as
0 0 0 1
0 0
0 0
h-i
0 1 -1 and do0
0 L 0 0J ing systematic column reduction, one sees that ri^2 = A'UA'pUAV^UAV^.
This finishes the proof of A. B. By column reduction again one checks that I SLg{I),
A = /(mod 2) and f
^ J and f
^
t /i-i ) ^^^^ ^ ^ j with A^ integral
generate a subgroup A" of r2 such that A" U A"p^ = r 2 . We have by our calculations in A that A" C A; thus we have
A"c(Anr2)cr2. But A" C A'" = { ( ^
^)
e T2\Tr{D - Ig) = O(mod 4 ) | which is a
subgroup of index 2 in r2, and therefore A" = A'". Also rj'^ip^) = - 1 , which implies that A H r2 = A'".
146
TATA LECTURES ON THETA III C. It follows from A and B that the image of A in 0{2g, Z/2) is a
subgroup of index 2, but SO{2g,I/2) and this proves the result.
= [0{2g,I/2),0{2g,I/2)]
always,
9. Theta Functions in Spherical Harmonics There is still another extremely natural and important generalization of the theta function! In §6, we introduced theta functions with quadratic forms Q, which arise inevitably either by considering the algebraic identities on -d or by seeking more general modular forms represented by theta series. In this section, we introduce theta functions defined using both a quadratic form Q and a spherical harmonic polynomial P. These can be motivated and then analyzed either by considering the derivatives of i? or by carrying further the representation-theoretic definition of i^, yielding still more modular forms given by theta series. We shall split our discussion in three; first defining these new functions by differentiating analytic theta functions, then by matrix coefficients, and finally by differentiating sections of line bundles on abelian varieties, leading to an algebraic theory. VIEWPOINT I: Differentiating analytic thetas As usual, T G ^y, Q is a positive definite symmetric rational (h x h)matrix, and Z is a complex {g x h) matrix. As in §6, let d'^[^]{Z;T)
=
Yl
x[^]{N)exp
iri Tr^ NTNQ ^2'N
Z)
where x[^]iN)
=
Oi{N-A^Iig,h)
— exp 27ri Tr^NB
otherwise.
Let R — C[zij; 1 < ^ < 5^, 1 < i < /i], the ring of polynomial functions on the Z-space of complex {g x /i)-matrices. For any homogeneous polynomial P e R, let P{d) = P ( a i ; ; ) . Then differentiating the
d^[^]
termwise, we get P{d)d^[^]{Z\T)
=
Y,
x[^]{N)P{27riN)exp7riTrCNTNQ-\-2'NZ).
N£Q(g,h)
Omitting the 27ri factors, we give these functions names:
148
TATA LECTURES ON THETA III
DEFINITION 9.1. For T,Q,Z
as above and for P £ R homogeneous and
A,Beqi9,h), (a)
,?^.«[^](^;T)=
(b)
dP'Q{Z;T) =
Yl
x[g]iN)P{N)exp7:iTrCNTNQ
+ 2*NZ).
^^'<^[yZ;T).
(c) i?^'«[^](T) = ,?^.'3[^](0;T). The above shows that when P is homogeneous of degree k, then d^-Q[^](Z;T)
=
i27ri)->'P(d)^<^[^]iZ;T).
We want to find out for which polynomials P £ R^ the t?^'^[ D ] ( ^ ) are modular forms (in a generalized sense which we shall explain later) for B\\ A, B E Q{g, h). The answer is: if and ony if P is pluri-harmonic in the following sense: DEFINITION
9.2.
Let S = (spg) be the inverse of Q.
Then P e R is
pluri-harmonic with respect to Q it and only if ^ a ^ - * p,q
V
= '
forai7(i,i).
J^
Let's work out some examples of what pluri-harmonic means: A.
When g = I, put zip = Zp. Then P
is pluri-harmonic <=> I T^^p^T—5— I P = 0.
But A 31 5]] Spq Qz %z ^^ J^^^ ^^^ Laplacian operator for Q (and hence invariant under the orthogonal group of the quadratic form Q). In this case P is pluri-harmonic iff P is harmonic in the usual sense. B.
When /i = 1, put z,i = Zi. Then P is pluri-harmonic iff ^^ ^
= 0 for
9
all i, j ; i.e., if P has the form: P = c-f/Jai>2^i for some c, ai, 02, • • •, % G C. 1=1
THETAS WITH SPHERICAL HARMONICS C. y/^
149
When h = 2 and Q = /2, put zn = Xi and Zi2 == Vi, and Zi = xi -f • yi, li = Xi - y/^
' Vi. Then i? = ^[zij] = C[xi, •••, x^, 2/1, • ' • , % ] =
C[zi,-.-,2:5;Ji,Z2,---,J^]. P € i^ is pluri-harmonic iff ^ f ^
+ ^17^ = ^
for all i and j . It is not hard to see that this is so if and only if
where hi and /i2 are polynomials and the Cij G C. Next we put g = h = I and (5 = 1, and use the results of Chapter I to get some idea of when 'd^'^ is a modular form. The functional equation of Chapter I, §7 says: d
(z,
T)
for all I
^2
= (cT + d) ^/^ • exp 7ri ( —
z
CT -\- d )
-
ar -\- b
CT -{• d^ CT + d
, 1 in some congruence subgroup of 5Z/2(Z) with the sign of
{cT + d) ^/^ carefully chosen. Differentiating with respect to z:
y/
i^,r) / ,x 1/9 /-27ricz\ -TTicz . {cT + c/)-i/2 . ( ^ ) • exp —r • i^ CT -{- d J
^ ^ CT -}- d
2
+(cr4-^)-^/2exp(^^)i^' CT -\- d
aT -{- b
z
CT -{• d^ CT -\- d
/ z ar + h\ \CT + d' CT + dJ
and
z
(e. + J ) - i / 2 f z ? ! [ i £ ) e x p : i ^ . , ?
aT -\-b
+(cr + rf)~^/2 ( _ _ _ ] exp —T-^ CT + d cr + rf +2(CT -h
rf)-^/2 /
-f(cr-hc?)-^/2gxp
gxp •
aT -\-b
CT -\- d^ CT -\- d
cr-j- d' cr + d
-TTICZ^
r • I?'
2
cr -h c?
cr + d cr -h rf ar + 6
z
CT + d^ CT -\- dJ
Putting z = 0, we get: ^'
(0,r) = (cr + c/)-^/V
0,
a r + 6'' cr -f rf
TATA LECTURES ON THETA III
150
+ (cr-fc/)-^/2.^// which shows that ??^'^
ar + b ' CT -^ dy
(r) is a modular form of weight 3/2, whereas
(r) is not a modular form. This is in agreement with the fact
Q^^/
that P{z) = 2: is pluri-harmonic and P{z) = z^ is not. In explicit series:
is a modular form, but X^xW-rz^-e-"^^ is not. Note that d^^^
(r) = 2_J^ ^^P Trm^r = 0 for all r so that the
modular forms obtained this way are not all non-zero. However, for any P, Q and T, d^^^(Z\ T) is not identically zero as a function of Z. And since A-\-B'T
are dense in C{g^ h), there are A and B such that ?^^'^
(T)^
0, i.e., for suitable A and B^ these series are non-zero. We shall now show that the functional equation of the 'd^'^
(T)
can be deduced exactly as above: by differentiating the functional equation r p"| of ??^ I ^ I (Z,T) . Having done this, we shall proceed to interpret the ^P,Q
(T) as modular forms in a suitable generalized sense. The two
functional equations are: THEOREM
9.4.
For rational (g x h)-matnces R and S,
{Z,T) = det{CT + D)-'''^ •i?«
exp{-7riTrCZ{CT
+
D)-^CZQ-^))
{\CT + D)-^Z,{AT + B){CT + D)-^)
THETAS WITH SPHERICAL HARMONICS
151
J in a suitable subgroup of Sp{2g,Z).
for all I THEOREM
9.5. Let P be a pluri-harmonic polynomial with respect to the
quadratic form Q. Then, for rational {g x h)-matrices R and S, (T) = det(CT+D)-'^/2^^',Q
^P,Q
with P'{Z) = P{{CT-j-D)-^Z),
for ^U ("^
({AT + B){CT + D)-^)
^ j in a suitable congruence
subgroup of Sp{2g,Z). Theorem 9.4 will be proved in Viewpoint II. First we need some generalities on pluri-harmonic polynomials. LEMMA 9.6.
For A € C[Xi, X2, • • •, Xn], we let A{d) denote the operator
^{dh>"'^dh)'
ForA^B
e C[Xi,X2,---,X„], {A,B)
=
{A{d)B){0)
is a symmetric non-degenerate bilinear form on C[Xi, X2, • • • ,Xn] which satisfies: {A, BC) = {B{d)A, C) for all A,B,Ce
C[Xi, X2, • • •, X„].
PROOF: (X^Xf^ • • • X ^ - , X f X f • • - X ^ - } = 0 if the exponents satisfy (/ii, /i2j • • -5 hn) ^ {gi,g2)'''i9n)
and equals hi\h2\ • • -/in- otherwise. This
proves that (A, B) is symmetric bilinear non-degenerate (in fact, positive definite when restricted to R[Xi, X2, • • • ,X„]), and similarly {A,BC)
=
{B(d)Ay C) is checked for monomials A, B and C, from which the lemma follows.
QED
LEMMA 9.7.
Let H C R = [zij; l
the space of
all pluri-harmonic polynomials with respect to Q and I C R be the ideal generated by the hij = ^^^P^^^P^^^
^^^ ^^^ ^ ^"^ ^ ^"^ ~ i^pg) — Q~^
^
p,g
before).
Then H = /-^ with respect to the pairing (,) introduced in the
previous lemma, and R = H^ PROOF:
{fhij,P)
I.
= {f{d)hij{d)P){0)
j , if and only if the hij(d)P
= 0 for all f e R, for all i and
vanishes along with all its repeated partial
derivatives at 0, i.e., if and only if hij{d)P = 0 for all i,j, i.e., if and only
152
TATA LECTURES ON THETA III
if P is pluri-harmonic. Thus H = Z-"-. The same argument shows that HR = 7|^ where HBJ = ^[zij] H H and /R = / n R[zij\. But (,} is positive definite on R[zij] and therefore 7R 0 HR = ^[zij]\ thus R = H 0 / . 9.8. If P is pluri-harmonic, then P{d)[g{Z)€xp
LEMMA
and P(d)g{Z)
QED
Tr{^ZCZQ-^)]
coincide at Z = 0, where C is any complex (g x g)-matrix
and g is any function analytic in a neighborhood of Z = 0. P R O O F : Let h{Z) = Tr{^ZCZQ-^)
and S = ( s p j = Q " ^
Clearly it
suffices to prove this for polynomials g{Z). Expanding, we see that: «^ 1 h{Z) ^ Yl
P{d)g{Z)exp
^P{d)9{Z)hiZr
n=0
when Z = 0 since only finitely many terms of the summation are non-zero. Therefore P{d)g{Z)exp OO
E
n=0
h{Z) at Z = 0 equals
^
^
(X)
7\'^P'3- n
=E
n=0
at Z = 0, provided we show Now
Z\i^i9)"P'9)
h{d)P =-. 0.
=E = E'^'kiY^
h{Z)
= (P'9) = P{d)9
ZijCikZklSij
ij,k,l
Zij ZkiSlj)
i,k
khik, i,k
and therefore since P is pluri-harmonic h{d)P = y^Cijfc(/iiJb(5)P) = 0, iyk
which finishes the proof.
QED
The following will be used in Viewpoint IL COROLLARY
9.9. If P is pluri-harmonic,
P{d)exp TrCZCZQ-^) PROOF:
= P{2CZQ^^)exp
Put h{Z) = exp TrCZCZQ-^).
h{Z + ^ ) = h{Z)h{A)g{Z)
Tr{^ZCZQ-^),
For any A G C{g,h), let f{Z) =
where g{Z) = exp Tr{2^ZCAQ-^),
Then
THETAS WITH SPHERICAL HARMONICS
=
P(d)h{Z)
evaluated at Z = A
P{d)f{Z)
evaluated at Z = 0
153
= h(A) • P{d)g(Z) evaluated at Z = 0, by the above lemma. But :^9{Z)
= ^ e x p 2{Y,^P,{CAQ-%,)
=
A repeated application of this shows that P{d)g(Z)
g{Z){2CAQ-')ij, =
P{2CAQ~^)g(Z),
and when Z = 0 this equals P{2CAQ-^). LEMMA 9.10. Let f{Z)
= f{AZB)
QED
andP'{Z)
= P^AZ^B),
where A and
B are complex (g x ^f) and {h x h)-inatnces respectively. Then P{d)f{Z) {P\d)f){AZB).
In particular, {P, f) =
=
(P'J).
We omit the proof. COROLLARY
{A,B)P(Z)
9.11.
For the action of GLg{C) x 0{Q) on R given by
= P{A^^ZB)
for all A G GLg{C), B G 0(Q),
P e R, the
space of pluri-harmonic polynomials, H, is an invariant subspace. PROOF:
By 9.7 the orthogonal complement of H is the ideal I generated by
hij = Yl^pq^ip^jq- ^^ ^y ^-^^ ^^ suffices to check that hij{AZ) and belong to / for all A G GLg{C), B G 0{Q), where hij{Z) =
/.SpgZjpZjg. p,q
hij{AZ) = 2_^ ClilZipajmZmqSpq
ZtpZmqSpq i,m
hij{ZB)
\pjg
= 2_^ ZithtpZjm^rnqSpq l,m,p,q
>
hij(ZB)
TATA LECTURES ON THETA III
154
But ^BQB = Q] therefore B-^Q-^ Therefore hij{ZB)
^B'^ = Q - ^ thus Q-^ = BQ'^
^B. QED
= Kj{Z).
We shall now prove Theorem 9.5. Let P be a homogeneous pluriharmonic polynomial of degree k. Apply the operator (21^1)'^P{d) to Theorem 9.4 and put Z = 0. By Lemma 9.8, we get: ^P,Q
(T) rr {27ri)-^ det(CT -f D)-'^/^.
R {'{CT + D)-^Z, {AT -f B){CT + S evaluated at Z = 0. By Lemma 9.10 this simplifies to .P((9)i^^
det(CT+D)-'^/2^^',g where
P\Z)
{{AT-\-B){CT
+
D)-') QED
= P{{CT-^ D)-^Z).
In the functional equation for d^'^,d^'^
D)-^)
is related to d^ '^ for a P'
obtained from P by the action of GLg{C) on the space of pluri-harmonic polynomials. A clearer way to express what's happening is to group the i^^'^'s together into vector-valued functions which transform into themselves if we view these vectors as points in a homogeneous vector bundle over S^g. Recall that a homogeneous vector bundle E on S^g is a holomorphic vector bundle over S)g together with a lifting of the action of Sp{2g, R) from S)g to E. Actually, we want to allow for the usual ambiguity of sign in the functional equation, so we generalize this slightly and ask for an action of the double cover Mp{2g,R) of Sp{2g,R) on E which makes the projection Fi -^ S)g equivariant. As usual, these bundles are obtained from finite dimensional representations of the subgroup of Sp{2g,R) (or
Mp{2g,R))
which fixes a base point, e.g., ilg, in S)g. This subgroup is the unitary group U{g) and its finite dimensional representations extend to holomorphic representations ofGLg{C). are given by:
Then homogeneous vector bundles for Sp{2g, R)
THETAS WITH SPHERICAL HARMONICS where p : GLg{C) -^ Aut(E)
155
is a representation of GLg{C) and Sp{2g,R)
acts by y{X,T)=:{p(CT-\-D)X, for all X e E,T G ^gH
(AT-\-B)(CT
= (^
+ D)-^)
^ ) G Sp{2g,R).
As in §8, to be con-
sistent we are sometimes forced to let Sp{2gyR) act after conjugating by -/
0^
, in which case it come out as:
7(X, T) = {p{-BT
(DT + C) • {-BT
^A)'X,
+
A)-'),
To include metaplectic actions, we need to define a double cover GLJ(C) ofGL^(C) by: G i J ( C ) = {{a,A)\a'^ = d e t ^ ,
a € C*,
Ae
In what follows, a will simply be denoted as y/det(A).
GLg{C)}. Then if
p:GL^{C)-^Aut{E) is a representation, E^^-^ = E x S)g as before and Mp(2g, R) acts by: 7(X, T) = (p(C(7; T)-\ where j maps to I
i-BT
+ A)) • X, (DT + C) • ( - 5 T + A ) - i )
n ) ^ •5'p(2y,R) and C(7;T) was defined in (8.3).
The simplest example is given by the one-dimensional representation in which p{a,A) = a = y/detA.
This defines a homogeneous line bundle on
S)g which we write as L^/^, i.e., L^/^ = C X ^^ with action : 7(x,T) = ( C ( 7 , T ) - ^ . i . , 7 ( T ) ) = ( d e t ( ~ 5 r + Ay^
' X, {DT - C){-BT
-f A)-^)
Its square L is the homogeneous line bundle for Sp{2g^ R) associated to the determinant. As usual, a vector-valued modular form on ^g is a holomorphic section of such a bundle E^^^ which is invariant under some F C Mp(2y,R)
TATA LECTURES ON THETA III
156
whose image F in 5p(2^,IR) is a congruence subgroup. Such a form is a holomorphic map f : f)g —^ E such that f{T) = p{C{r,n
{-BT
+ A)-')
• f{{DT - C){-BT
+
A)-')
for all 7 G f. Now let W be some GLg(C)-sta.hle subspace of H, the space of pluriharmonic polynomials. Define dw
•
•
^
^
W* as follows:
dw I " I {T)] (P) = d^''^
(T)
for all P G W^ C H. Theorem 9.5 now reads as: dw
{T){P) =
where P'(Z)
dei{CT+D)-''''^-^w
= P{{CT + D)-^Z).
on H^ C H given by ip{A)P){z)
{{AT + B){CT +
D)-^){P')
Denoting by p the action of GLg{C) = P{A-^z)),
P' is simply p{CT + D)P.
The action p* of GLg(C) on W* is given, as usual, by: (p* {A)£){P) = £(p{A)-'^P) for all A G GLg{C), £eW*,P£W.
The above formula now
reads: •0
{T) =
w
det(CT-hD)-'^/V*(CT-f D)-ii?w
((AT4-5)(CT + D)-i).
Combining this with the remarks at the beginning of the section, we have: T H E O R E M 9.12.
The
^w
(T) defined above is a modular form with
values in W* 0 L'*/^. For any W and T, it is non-zero for suitable R and S. This is the "modular form interpretation" of Theorem 9.5. We want to give an interesting application that shows the usefulness of these series. The following theorem is a version of results of Freitag and Stillman:
THETAS WITH SPHERICAL HARMONICS THEOREM
157
9.13. For all g > 2 and I < r < g — 1, there are congruence
subgroups r C Sp{2g, Z) and T-invariant non-zero holomorphic k-forms on
This is a basic insight into the complex geometry of the Siegel modular varieties S)g/T: for r = g — l^g — 2," -, this means that we get ^f-forms, (2g — l)-forms, (3^ — 3)-forms, {4g — 6)-forms,- • • on these varieties. The first step in the proof is to identify holomorphic forms as sections of a homogeneous vector bundle: LEMMA 9.14. If V denotes the g-dimensional identity representation of GLg{C)y then there is an Sp{2g^R)-invariant
isomorphism:
where Q^ is the cotangent bundle or bundle of 1 forms. PROOF:
Use the elementary identity:
{A(T + 6T) -h B)(C • (T -f- 6T) + D)'^ = (AT + B){CT -f D)-^ -h ^{C{T + ST) -f D)-^ • ST • (CT +
D)-^
This implies that a tangent vector to S)g, given by a symmetric gx g matrix X, transforms by the rule: y{X, T) = C{CT + D)-^ • X • {CT -h Dy\
{AT -f B){CT +
D)'^).
Therefore the dual action on cotangent vectors is given by: j{w,T)
= {{CT -\- D) ' u; '\CT
-]- D),{AT + B){CT -^ D)-^)
where LJ is also given by a symmetric g x g matrix, (paired with a vector X via Tr{u) • X).) But V is defined by the action
j{x,T) =
{{CT-^D)'X,{AT-^B){CTi-Dy^)
where x is a column vector. So if we write the elements of Symm? (V) as T>Xi 0 *x«, then we get the same rule: f{^Xi0*Xi,T)
= ((Cr+£»)(^x,®'ar.)-'(CT+£>),(^T+B)(CT+D)-i). QED
158
TATA LECTURES ON THETA III
COROLLARY 9.15. There is an isomorphism of homogeneous bundles:
Now recall the classification theorem for irreducible representations p of GLg(C):
for every sequence di > c/2 > ••• > dg, there is a unique
irreducible representation E of GLg{C) which contains a vector x such that /aiiawciig\ I 022- • • Cl2g
(*)
\
O
X - (ai/'a22''^ • • '(^gg^^) ^
annl •
^99
and every irreducible representations arises in this way. (di, • • •, c?^) is called the hightest weight vector of E. The representation A^^Symm^iy))
of GLg{C) has a rather compli-
cated decomposition into irreducibles. However, if k =
g{g-^l)/2-r{r+l)/2,
there is one quite simple piece: namely, if ei, • • •, e„ are the unit vectors in V, consider x=
f\
(eioej).
l
Here CiOej G Symnn?V and exactly k terms are wedged. It's easy to see that X satisfies (*) with di = • • • = dg_r = ^ -f 1 and dg^r-\-\ — - - - — dg ^ g — r. This proves: LEMMA
9.16. If E^"^^"") is the irreducible representation ofGLg{C)
with
highest weight di = - - - = dm = n, dm-\-i = - - • = dg = 0, then there is an embedding of homogeneous bundles: (
, 1N
g(g-H)
r(r+l)
THETAS WITH SPHERICAL HARMONICS
159
Finally, we need some pluri-harmonic polynomials in order to get global sections of this bundle via t h e t a series. Take Q to be the identity m a t r i x of size 2(g — r) x 2{g — r ) , so our harmonic polynomials P are functions of a. g X 2{g — r ) - m a t r i x Xij.
By the examples given in the beginning of the
section, all polynomials:
are pluri-harmonic. In particular, look at ir+l
'
p{x)
det
{Xi^2j-i
+ A /— - T • ^i,2j)
r+l
Under the action of GLg{C),
it is easy to see t h a t P satisfies the highest
weight condition (*) with di = • - - = dr = 0, dr+i = - - - = dg = —(r -f- 1). Therefore H contains the corresponding irreducible representation W of GLg{C).
But W* has highest weight di = • • • = dg^r = {r -\- 1), rf^-r+i =
"• = dg = 0, i.e., W* ^ £'(i/-^r+i)
Proposition 9.12 plus Lemma 9.16
therefore imply T h e o r e m 9.13.
V I E W P O I N T II: R e p r e s e n t a t i o n t h e o r y Let K be a real vector space with a non-degenerate alternating form A and let l y be a real vector space with a positive definite quadratic form B.
T h e n V <S> W has a natural non-degenerate alternating form A <S) B:
(A (g) B){v 0
K;,
i;' 0 w') = A{v, v') • B{w, w').
0{W, B), then ^ ® cr G Sp{V ^W,A® Sp{V, A) X 0{W,B) Let Tiv and Tiyi^w tively of Heis{V) action of Mp{V
If ^ G Sp{V, A) and a G
B). This gives a homomorphism —^ Sp{V ®W,A^
B).
be the irreducible unitary representations respec-
and Heis{V
0 W) such t h a t Ux = X - Id, VA G CJ. The
<S>W,A<S>B) on 7iv<^w restricts to an action of a double-
covering of Sp{V, A) X 0{W, B) on Hy^w-
In fact, when h — dim W is
160
TATA LECTURES ON THETA III
odd, the double covering is almost Mp{V^ A) x 0(Wj B) but not quite, and it is {±1} xSp{V,A)
X 0{W, B) when h is even. To see this, note:
(a) Let V — Vi 0 ^2, with Vi and V2 isotropic spaces, then V ^W (Vi 0 W) 0 (V2 0 W) and the Vi^W
=
are clearly isotropic spaces. In
the model Hv®w = L'^{Vi 0 W), for a G 0{W,B)J
G L^{Vi 0 W),
p(a)f = / o (Iv, (8) cr-^) defines an action of 0{W, B) on L'^{Vi (g) W). From Lemma 8.21, it follows that P{(T) G Mp{V (g) 1^, A (g) 5 ) if (T G 50(Vr, 5 ) or if a G 0{W, B) and dim Vi is even. (b) Now put W in standard form: W — W^ with B{x,y) = ^xy. Then ^2(1^1 0 H^) ^ L2(I/I 0 VI . . . 0 F I )
For 7 G Mp(V^i, A)^ define ^(7) = 7 0 7 0 • • • (g) 7. This defines p : Mp{V, A) —y Mp{V 0 ly, A 0 5 ) . In any case, (a) and (b) combine to give — . Mp{V®W,A^B)
p:Mp{V,A)xO{W,B)
The complete decomposition of Tiy^w Mp(V,A)
X 0{W,B)
^
Aut{nv^w)^
under the action of the group
is studied in Kashiwara and Vergne, On the Segal-
Shale-Weil representation and harmonic polynomials, Inv. Math., Vol. 44, 1978. Let 2g = dimV and h = dimW. Their main result is that we have two decompositions. Firstly, the space of pluri-harmonic polynomials decomposes:
(9.17)
H=:0(^,0F,) a
where Ea are distinct irreducible GL^(C)-modules and Fa are distinct irreducible 0{W, 5)-modules, while (9.18)
Hv^w
=0
[r{S)g, E ( « ) 0 L^^^) 0 Fa
THETAS WITH SPHERICAL HARMONICS
161
(here F refers to the space of X^-sections of the homogeneous bundle as defined in Viewpoint I, considered as a representation of Mp(VjA).)
They
also give a detailed algorithm for describing which representations of GLg and of 0{W, B) are paired in this sum. We do not discuss all this but go just far enough to write down the r /?i functional equations for 'd^'^ c • ^^^^ ^^'^ involve the transformation laws for P(y)exp iri^YTYQ
G WV<^W' ^
and the invariance of e
As we have seen in §8, the functional equation for t^ comes from (a) defining ?? as a matrix coefficient of fx = e^^^ ^^ with respect to e[^], and (b) studying the action of Sp (or rather Mp) on / T and e[ ]. We proceed in exactly that fashion here. Of course we do not want to determine how all of Mp{V 0 W) acts, rather just how Mp{V) acts on where T* corresponds to T 0 Q as in §6 (note that there is
P{Y)fT*{y),
an embedding i : ^y —^ ^v^w
sending T to T*). We shall now proceed
to work everything out in detail in matrix notation. I.
Let V = W^W
and W = W', the members of V and W being thought
of as columns and rows respectively. Then V ^W
is naturally identified
with IR(y, h) 0 R(^, h) where R{g^ h) = the space of all {g x /i)-matrices with real entries. Let A{x^y) = ^Xiy2 —^X2yi where x = (^1,2:2) and y = (yi,y2) and xi,X2,yi,2/2 € R^. Let B{x,y)
= x - Q'^ -^y where Q
is a positive definite rational symmetric matrix. Then (A 0 jB)(X',y') = Tr{CX[Yi
- 'X'2Y() • g - i ) , where X' = (X{,X^) and Y' = {Yl,Y^)
and
X ( , y / , X 2 , y 2 ^ ^(Oi ^)' However we want A (Sf B in standard form, so it is prudent to put ( X J , ^ ^ = {XiQ,X2)
{A ® B){X'X) II.
so that the pairing is
= A{X,Y) = TrCXiY2 -
'X^Y,),
Let Sp{2{g^ /i);R) be the group of all linear transformations from the
space U{g,h)
0
R{g^h) to itself that preserve the pairing A{X^Y)
=
162
TATA LECTURES ON THETA III
Tr{^XiY2-^X2Yi). (c
D)
Wewntedownj
:Sp{2g,R)—^
^ ^ ^ ( 2 ^ ' ^ ) ' ^^^" m(X[,X!,)
= {AX[ + BX!,,CX[
Changing coordinates by (Xi,X2) = {X\Q,X2), i ( ^ ) ( X i , X2) = {AXi + BX2Q-\ III.
We have
+ DX!,).
we get: CXiQ + DX2).
Mp{2g,R)
-^
Mp{2{g,h)-R)
Sp{2g,U)
-^
Spi2ig,h);R).
1
If^ =
Sp(2{g,h);R),
i
We compute p explicitly on a set of generators for the model of the representation i^(R(fif, h)) S Tiv®w- For 7 G 5'p(2jf, R) and 7 G Mp{2g, R) over it: (a) If 7 -
A
I 0 n
0
(M^-1
/'
(p(7)/)(>') = / ( ^ " ' > ' ) ( d e t y l ) - ' ' / ^ where C(7;T) = (det ^ ) - i / 2 . (b) If 7 = I
I with C symmetric and C{'};T) = 1, then pi7)fiY)
(c) 7 = ( 5
= / ( y ) e x p 7rjTr(*yCyQ).
~ / Y then i(7)(Xi,X2) = ( - X 2 Q - i , X i Q ) , hence p{j)f{Y)
= /(yQ) • r''/2
where
r^^^ = C{f,
il).
(d) If 7 = - 1 , 7 = Id. G 5p(2if, R), then P(T)
= (-l)^
These follow immediately from Lemma 8.2 and the formula in II above. IV.
We discuss i : S)g —> ^{g,h)- Let T E S)g. The complex structure on
W e R^ is given by /i : R^ 0 R^ —^ O , h{xi,X2) = Txi
+ X2. It follows
that the complex structure on R(g,h) 0 R(g,h) is given by (X'^^Xl^) •—^ TX[
H- X!^. Putting (X{,X^) = ( X i g , X 2 ) , the isomorphism R(i^,/i)0
THETAS WITH SPHERICAL HARMONICS
163
R(^, h) —y C(g, h) given by (Xi,X2) i—^ TXiQ -f X2 gives the complex structure. This corresponds to T 1—> T* of §6. Thus €
fi^T)iY)^exp7tiTr'YTYQ V.
L\R{g,h)).
We now come to the main calculation: the transformation laws for
fi(T) e L^{Rig,h)),
and more generally, of P{Y)fi^T){Y)
E
L\H{g,h))
where P is pluri-harmonic with respect to Q, under the action of the metaplectic group Mpi2g,R). (a)
If 7 = I
Put f(Y) = P(y)/,(T)(y).
t . _i I, then apply Ill(a) from above: = (det
pi7)fiY)
A)-''''fiA-'Y)
= (det A)-'''^P{A-'^Y)exp
TriTrCY*A-^TA-^YQ)
= (det ^ ) - ' ' / 2 p ( ^ - i y ) / , , ( T ) ( y ) . (b)
° j and C(7;T) = 1, then
Uy=(^
pi7)f{Y)
= P(y)/,(T)(y) • = P{Y)U^T-C){y)
(c)
If 7 = ( ^
expi-mTr'YCYQ) =
P{Y)fi,iT){Y).
~ ^ \ then we have defined p{^)f{Y)
We compute this. For all T € Sjgjriv)
=
i-^l'^J{YQ).
= d e t ( ? ) - i / 2 . f_T-i{y).
This
holds for any jr, and in particular, l^r){Y)
= det(^)-i/2exp;riTrCy(-T-^)yQ-i) = d e t ( ? ) - ' ' / 2 . det (Q)-^/2 • e x p « T r ( ' y ( - r - i ) y Q - i ) .
Assume now that P is a homogeneous pluri-harmonic polynomial of degree k. Since f{Y)
= P{Y)fi^T){Y)
= {27ri)-^P{27riY)fi^T){Y),
and since the
Fourier transform interchanges differentiation and multiplication by the variable, it follows that / ( y ) = (27rO-*P(a)/;(T)(y)
TATA LECTURES ON THETA III
164
= (27ri)-*^P(a)exp7r2Tr(*y(-r-i)yQ-i)det(?)-^/2 .det
(Q)'^^^
• det (Q)-^/^
= P{-T-^YQ-^)expmTrCY{-T-^)YQ-^)det{^)-^^^ from Corollary 9.9. Therefore
p{P)f{Y) = i-^l-'JiYQ) = i - ' ' / 2 p ( _ j ' - i y ) e x p ( ^ i r r ( ' Q ' y ( - T - i ) y ) d e t ( ? ^ ) - ' ' / 2 .det {QY^I^ = /i^(T)P(-T-iy)det(-T)^. Finally, (d) If 7 = - 1 , 7 = Id. € 5p(2<7,R), p(7)/.(T) = {-^f
fi{T).
Using exactly the same method as in Theorem 8.3, we deduce: PROPOSITION
(9.19). For a i i 7 = ( ^
G S)g and P
^ Sp{2g,R),T
D)
pluri-harmonic, p(7)fi(T){Y)P{Y) VI.
= U^^T){y)P{{-BT
+ A)-'Y)
We now investigate how Mp acts on e
R,S e
Now e
-f
det{-BT G ^v'^w
where
is defined as a distribution by
{N)f{N) N£Q{g,h)
where S
{N) = 0 if
N-R^I{g,h)
= exY>27r iT r^ N S
otherwise.
We have: (/,e
) rz
Y,
+ N)S • f{R + A^)
exp27riTr\R
Ne2(g,h)
0 -
(/,f^(i,0,-5)^(1,-ii,o)e
= (^(l,il,0)^(l,0,5)/, e
).
Ay^'^
THETAS WITH SPHERICAL HARMONICS win be denoted by ez.
As usual, e
We shall show that the 7 G Mp{2{g,h)\R) R
165
that satisfy 76
form a group G so that q{G) C Sp{2{g, h); R) contains a congruence
subgroup. Let A = {je
Mp(2{g, h); R)|7ez = ej}.
By 8.10, q{A) D the principal congruence subgroup of level 4. Let A' =: {7 G A 1^(7) = 1 (mod 2 P ) } where k is chosen so that kR and Ar5 are integral. Then, for ieA',q{j)
=
jeSp{2{g,h);I), = 7^^(1,0,-5)^(1,-ii,o)ez
je
= ^(1,7(0,-5))^(l,7(-il,0))7ez = U(^l,2kZ,2kW-S)U(i^2kZ'-R,2kW')^I. = t^(l,0,-5)^(1,-71,0)^2 = e From the explicit formula for Sp{2g,R) —^ Sp{2(g^h);R),
we see finally
that
= 1 (mod 2k^n)} where n is chosen so that nQ
and nQ~^ are integral. VII.
The functional equations now follow easily: the relation ^P,Q
(T) = (p(y)/,(T)(y),e
>
combined with V and VI above allows us immediately to deduce Theorem 9.5.
166
TATA LECTURES ON THETA III Since )= exp iriTr*XiZ • t?<^
{Ui,x,,x,)L(T){Y),e
(z,r)
where Z — TXiQ-\-X2j using the reults of V and VI we can deduce Theorem 9.4 exactly as in 8.8. The idea of treating theta functions as matrix coefficients is due to Weil in his classic paper, Sur certains groupes d'operateurs unitaires, Acta Math., vol. I l l , 1964. From this point of view, it is natural to view the whole construction of theta functions as embodied in a basic map, which is often called the Weil map. Abstractly, the situation is as follows: suppose G = any algebraic group defined over Q, and suppose that one comes up somehow with a representation r:Gu
—^
Aut{n)
and a vector e G '^-oo
such that
^(e) — e,
all 7 G some arithmetic subgroup T C GQ.
Then define the Weil map: w'.Tioo —
C(r\GR)
by H/)(T)-(KT)/,e), In other words, one gets automorphic functions of some kind from any such representation r and vector e. Moreover, if Tij = {e'|r(7)e' = e'
for all j in some congruence subgroup},
then we can extend w to: w.Hoo^'Hj—^
IJ [congruence subgroups]
r'cr
C(r'\G85)
THETAS WITH SPHERICAL HARMONICS wif®e')(j)
=
167
iriy)f,e').
This is almost exactly what theta functions do for us: let G — Sp(2g), H =
L'^(R^^'^^)
and let p be the 2-valued representation: p:Sp{2g,R)
—^ Aut{H)
(single-valued if h is even). Let ti — e
. Then for some congruence
subgroup r C Sp{2g^ R), lifted to Mp, we get w : 5(R(^'^)) —y C{T \ Mp{2g, R)) ^ / ) ( 7 ) = (/>(T)/,ez) and using the e (9.19)
's, we can extend this map to:
w : 5(R(^'^)) 0 S{f\^/^^^) -^\JC(T\ r
Mp{2g, R)).
This can be reformulated adelically if one wants. In order to get classical modular forms from this map, we need only to specialize to particular elements / G S{R^^'^^) and then combine the resulting functions of 7 with elementary factors (to make them right invariant by the maximal compact subgroup U{g) instead of left-invariant by F): start with
where the Pa are pluriharmonic and transform under GLg by the representation T, i.e., Pa{AZ) = Er„^(A) • Pp{Z). Let
//>„,Q(^)
=
Fp^,Q{Z,iIg).
Then w{fp^,Q®e
)(7) = (K7)/p„,<3.e
= det(A - iB)-^l^
E/j rafi{{A -
which we may invert to show:
) iB)-'){Fp,,Qi;yiiI,)),e
),
TATA LECTURES ON THETA III
168
det{A - iBf^
^
r«/?(A - iB)w{fp^^Q 0 e
= (^P.,Q(-,T(^/<,)), e = ^P-^Q
{j{ilg))j
)(T)
) a modular form on ^^//iT.
I don't want to pursue the applications of these ideas any further, but I would like to make a few conjectures which highlight exactly how little beyond the definitions we know. There is a vast unknown area concerning the image and kernel of the Weil map. Let's specialize to the very simplest caise g = 1 , Q = : / ^ , P = 1, so / = e-'^^^i^ '+^^). Then (a - ib)^''^
'w{f^e
) 7 = d^>^^
(7(0).
Fixing / , but letting r and s vary over /i-tuples of rationals, this extends to a map
{a - ib)^^^ ' w : S{f\^j) —^ [j
( modular forms on S) of wt. /i/2, w.r.t. congruence I >. subgroup of level k of 15L(2,Z)
Explicitly, this is the map
where x is supported on ^1.^ and is constant modulo mZ'*, for some integer m. Here are two questions: QUESTION
1: Does the range of this map include all cuspidal modular
forms if h even, h > 4? QUESTION
2: Is the kernel of this map spanned by the obvious relations
xoA-x, Aeo{h,q)?
THETAS WITH SPHERICAL HARMONICS
169
To go a little further, the question arises whether pluriharmonic P's actually add new scalar modular forms, or just vector-valued ones. To get scalar modular forms, we need pluriharmonic P's such that: P{AZ) = det{AY • P{Z) for some Z and all A G GL{g). The first case is h = g, P{Z) = det Z. Then we get the family of power series (for Xi ^ <^(^/))*
which are modular forms for the homogeneous line bundle defined by the representation (det)^"*"(^/^\ i.e., L^^"^^)/^. On the other hand, multiplying 5f -f 2 ordinary theta series, we get the modular forms: /2(T)=
Yl
X2(iV)-e"^^('^^-^)
which are also sections of L^^"*"^)/^. QUESTION
3: For which xi and X2 are / i and /2 equal?
This has a long history: for p = 1, a particular case is the famous identity of Jacobi
proven in Ch. 1. For g = 2^3 and 4, Riemann and Frobenius found generalizations of this identity (some proofs were discovered in his unpublished papers by Edwards!). Fay {On the Riemann-Jacobi Formula, Nachr. Akad. Wiss. Gottingen, 1979) found a generalization to ^f = 5 analyzed the situation for higher g and found that not all forms of type / i were equal to some /2.
170
TATA LECTURES ON THETA III VIEWPOINT III: T h e algebraic version A purely algebraic method of defining the analytic modular forms
^P,Q
(T), for pluri-harmonic P and positive definite rational Q can
be found by following the beautiful ideas of I. Barsotti, contained in his paper, Considerazioni suUe funzioni iheia^ Symp. Math. 3 (1970), p. 247. In characteristic p, the story is more complex and has its own twists - as does any construction involving differentiating - so here we develop the method only for characteristic 0. Barsotti's theory has been developed in many interesting ways by Cristante, to whose papers we refer the reader. First, recall the basic result: THEOREM
9.20. Let X he an abelian variety and let pi^ ^^ : X^ —^ X
denote the map (ari, • • •, Xn) -^ Xij -f on X. onX
h Xi^^. Let L be any line bundle
Then there is a canonical isomorphism xp between the line bundle xX
xX:
L =
P123^<SIP12^~^<^P13^~^^P*23^~^<^PIL(^PIL(^PIL^L{O)-^
and the trivial line bundle. This is an immediate consequence of the theorem of the cube: if we restrict L to X x X x (0), we get PI2L (8) Pn^"'^ 0 P I L " ^ 0 p^L'^ 0 piL 0 p^L 0 L(0) 0 L(0)"^ which is trivial. The same holds on X x (0) x X and (0) x X x X, so L is trivial. Moreover, at (0,0,0),
L is defined as the tensor product
L(0)^ 0 L(0)"^, i.e., it is canonically trivial, so the "trivialization" of L is canonical too. Barsotti's result concerns trivializing an arbitrary line bundle L over an abelian variety X in a neighborhood of O G X in the sense of formal power series. Over an arbitrary field Ar, since X is smooth at O, we can start with arbitrary functions xi,-- - ,Xg in the local ring Oo,x of rational
THETAS WITH SPHERICAL HARMONICS functions regular at O. Then \{xi{0)
171
= 0 and the differentials dxi, • • •, dxg
are independent at O (algebraically, this means x,- G A^o,x and Xi G - ^ o , x / - ^ o x ^^^ independent over k), an arbitrary function / G C9o,x can be expanded as a formal power series in xi^- - - ,Xg. The algebraic way of saying this is that the completion Oo,x of ^o,x
is isomorphic to the
ring fc[[xi, • • •, Xg]] of formal power series in ari, • • •, ar^. When X is abelian and char ^ = 0, there are power series coordinates ^ir''
j^g ^ ^o,x
in terms of which the group law on X is just addition,
i.e.,
do,x=k[[tir".tg]] and ti{P-^Q) = U(P) + ti(Q) (for any i^-valued points P, Q of X factoring through Spec(Oo,x))'
In
other words, these ti play the role algebraically of the linear coordinates iir '' i^g on the universal cover C^ of X, when k = C We won't develop this theory here at any length, except to show how the ti are constructed: starting with any local coordinates xi^ — '^Xg in Oo,x, the dxi span the cotangent space to X at O, and their duals d/dxi span the tangent space to X at O. Translating by the group law, d/dxi extends to an invariant vector field Di on X. Algebraically, Di is a derivation from Ox to Ox- We form the expression oo
which defines a ring homomorphism from Oo,x to
C>O,A'[[^I,
* • * j^^?]]? hence
to fc[[/i, • •' ^tg'W (by evaluating functions on X at O). This is an isomorphism of Oo,x with Ar[[/i, • • -jt^]] and the inverse images of the ti are the desired additive coordinates. To see what we're doing, recall the dictionary:
172
TATA LECTURES ON THETA III
vector field X
<—^
derivation D of a ring
1-parameter group of automorphisms generated by X
<—^
automorphisms e^^ of a ring.
Thus TitiDi are the invariant vector fields on X and ^ is dual to the map C^ X X —y X obtained by integrating simultaneously the commuting vector fields Di. Thus C^ X (0) ^
C^ X X —^ X
gives additive coordinates, and this is dual to (eval. at O) o xj;. Now fix such additive coordinates
9.21. Suppose X is an abelian variety in char. 0 and L is any
line bundle on it. Let Oo,x be the completion of the local ring of X at 0. Then there is an isomorphism <j) : L (8)Ox Oo,x - ^ such that the induced
do,x
isomorphism
0 : L 0 Oo,xxXxx
-^
Oo,xxXxX
is the completion of the canonical trivialization ip of Theorem 9.20. Moreover, any two such
coordinates
at O E X.
We can re-phrase Barsotti's theorem in terms of formal power series as follows. Let
(i>:L^do,x
—^ do,x
THETAS WITH SPHERICAL HARMONICS
173
be any formal trivialization of L at 0. Then using (j) we obtain trivializations of all bundles p*^ •• t„L at (0,0, 0), and hence a trivialization <^ : L (g) Oo,xxXxx-^
Oo,xxXxX'
Our aim is to choose <^ so that (f) equals the canonical trivialization ip. Denote the power series tj; o <j)~^ by
f{xyy,z)
edo,xxXxX'
What we must show is that if (f) is modified by multiplication by some 9{x) G Oo,x, ^(0) ^ Oj then / becomes 1. But changing cj) by g{x) changes Pi23^ by g{x-{'y-\-z),
Pi2^ t)y g{x-\-y)^ etc., so it changes the trivialization
ofLby g{x + y -{- z)g{x)g{y)g(z) 9i^ -f y)g{x + z)g{y + z)g{0)' We need to show that for suitable g:
fi^^y.z) =
g{x-hy + z)g{x)g{y)g{z) 9{x + y)9ix + z)9{y + 2:)9{0)'
Obviously we need some identities on / to do so. What can we say about / ? We claim: (a)
f{x,y,0)==l
(b) / ( x , y, z) is symmetric in x^y^z (c) f{x, y,u-^v)'
f{x, u, v) = f{x, y-\-u,v)'
f{x, y, u)
(a) follows because the factors in L | ^ ^ ^ .Q. all cancel out, so both the restriction of ip and trivialization induced by (j) are canonical, hence equal. (b) follows similarly because both the canonical trivialization ip and that induced by (j) are invariant under permuting the factors of X x X x X. To prove (c), we consider P1234L onXxXxXxX.
Now Pi^^-.^if,
denotes the appropriate projection from X x X x X x X. Let's abbreviate p*
J- L to L,j^...^,-^. Then we will construct a diagram of bundles on X x
X X X X X:
174
TATA LECTURES ON THETA III
L1234 0 L234 0 L;"2^ 0 L]"3^ 0 L^^ 0 L? 0 L2 0 L3 0 L4 0 L ( 0 ) - ^
ay
\b
Li340L34^0L^3^(g)Lf4(8)Li0L30L4
Li230Lj3^0L^2^0L]"3^(g)Li0L20L3
c\
yd OxxXxXxX^l-{0)
Here (/is j u s t the trivialization ip of 9.20, and c is the analogous trivialization on the 1st, 3rd and 4th factors, i.e., m a p c = (pi,P3,P4)*V' m a p d=
(pi,P2,P3)>.
But consider the morphism (Pi,P2,P34) :X
xX
X X xX
—. X X X
xX
(i.e., (x, y, w, v) — ^ (x, y,u -{• v)). T h e n (pi,P2 5P34)*V' is a trivialization of the bundle: L1234 0 L[2^ 0 L'^^ 0 LJ3\ (g) L i 0 L2 0 L34 0 L ( 0 ) - ^ T h u s , checking t h a t the factors cancel appropriately, we see t h a t m a p a = (pi,P2,P34)*V'Likewise: map 6 = (pi,P23,P4)>. Now the diagram must commute, because it certainly commutes up to an a u t o m o r p h i s m of O x x X x X x X 5 i e . , up to a scalar, and, at 0, all 4 bundles have fibre L(0) and the m a p s are the identity. Finally, the trivialization (j) of L 0 Oo,x
also gives us a commutative diagram, and comparing the two
diagrams, we get the identity /o(Pl,P2,P34) •/o(Pl,P3,P4) = /o(Pl,P23,P4)-/o(Pl,P2,P3) which is (c).
THETAS WITH SPHERICAL HARMONICS
175
This is the first step of the proof. The second is the lemma: LEMMA 9.22. Let x, y, z stand for g-tuples of variables (xi, • • •, Xg), etc. Then for all power series / ( x , y) over a field k of char. 0, if f satisfies: a) /(0,0) = 0 f{y, z) = fix -f y, z) + f{x, y),
b) f{x, y-\-z)^ then f has the form:
^x 'A'y-\-
g{x-\-y)-
g{x) - g{y)
for some skew-symmetric gxg matrix A and some power series g{x) without constant or linear terms. P R O O F OF LEMMA:
Note that because of (b), / can have no linear terms.
/ can have an arbitrary bilinear term however, which we can express by ^xAy,A skew and by g(x + y) — g{x) — g{y)^ g quadratic. So let's assume these parts are dealt with. We now assume / has no linear or bilinear terms. Take identity (b), differentiate with respect to zi and set 2: = 0. If /2,i denotes the partial of / with respect to the i^^ component of its 2^^ argument, we get: /2,f (^, y) + /2,e(y, 0) = /2,i(x + y, 0). Write hi{x) = f2,i{^) 0), so that (*)
f2A^^y)^hi{x-\-y)-hi{y).
Now if hi J denotes the partial of hi with respect to its j ^ ^ component ^ ^ (/(^) y)) = h^j{x - f t / ) and
hij{y)
= hj^x -{-y) - hj^y).
Therefore hiji^) = hjj{x) + hij{0) - hj^O).
176
TATA LECTURES ON THETA III
But by the absence of bilinear terms in / , differentiating (*) with respect to Xjj and evaluating at 0, we get:
Thus hi J = hji. Therefore there exists a power series g{x) such that
But then (*) says ^/(x,y) = ^^(x4-y)-^^(y), or
alii,
/ ( x , y) = g(x -\-y) - g{y) - h{x),
for some h{x) with /i(0) = 0. Putting this back into (b) and setting y = 0, we get immediately h{x) = g{x).
QED
Finally we put these together. Let /(x^y^z)
be the power series sat-
isfying (a), (b), (c) in the beginning of the proof of 9.21. Let P{x^y^z)
=
log(/(x,2/, 2:)) to make everything additive. Considering x as constant, /* satisfies the conditions of the lemma. As the lemma is just a formal manipulation of power series, it tells us that (**)
r (^, y, ^) = *y • M^) • ^ + 9ix^ 2/ -f ^) - g{x, y) - g{x, z)
for some skew matrix A with power series entries and some power series g{x^ y) without linear terms in y. Since /* is symmetric in y and z, A{x) = 0. Now let /3 J- be the partial of /* with respect to the i^^ component of its y^ argument, ^2,1 similarly. Differentiating (**) with respect to Zi and setting z = 0, we get: fsA^^ y^ 0) = g2^i{x, y) - g2^i{x, 0) = g2^i{x, y). But by the symmetry of/*, fsd^,
V, 0) = fsiiV) ^j 0)- Likewise, /* satisfies
the cocycle condition in its 1st two variables, so /g ,• does too: fsA^
-f y, z, 0) -f fliix,
2/, 0) = /3%-(x, t/ + z, 0) -f /3* ,(y, z, 0).
THETAS WITH SPHERICAL HARMONICS
177
Therefore g2^i is symmetric and satisfies the cocycle condition so by the lemma again: (***)
92,i{^^ y) = hi{x -f y) - hi{x) - hi{y)
for some hi without constant or linear terms. But Q Q (g(^. y)) = •Q-i92,ii^' y)) = hi,j{x + y)and
= '^i92j{x,
hijiy)
y)) = hj^i{x -\-y) - hj^i(y).
Therefore setting ?/ = 0, we get hij{x) = hj^i{x) + hij{0) - hj^i{0) =
hj4x).
Therefore there is a power series h(x) such that hi{x) = ^h{x),
h{0) = 0.
Therefore by (***) c -^,(9(x,y) dyi
~ Kx + y) -f h{y)) =
-hi{x),
or g{x, y) = h{x + y) - h{y) - ^
hi{x) • y,- - k(x).
Putting this into (**) and recalling that g{x,y) is a power series without linear terms in y, we see that f{x,
y, z) = [h{x 4- y + z) - /i(y -f 2:) - k{x)] - [h{x -f y) - h{y) - k{x)] - [h{x + z) - h{z) - k{x)] = h{x -\-y-\- z)-
h{x-\-y) - h{x + z) - h{y + z)
-f fc(x) + /i(y) + /i(z). Setting y = z = 0, we see that h{x) = k{x) and that if g*(x) = e^^^\ then ,.
. _
^^""'^''^
as required.
(y*(x-fy+z)-/(x)-(y*(y).j;*(z) g*{x + y)^g*{x-^z)^g^{y
+ z)
QED for Th. 9.21
178
TATA LECTURES ON THETA III Barsotti's basic result leads directly to the algebraic construction of the
theta functions with pluri-harmonic polynomials. Now analytically these functions are vector-valued modular forms in T, when z = 0, so what do we expect to be able to define algebraically? Firstly, we need an analog of homogeneous vector bundles E on S)g. These are functorial ways of assigning a Ar-vector space: E(X,L) to a pair: X = abelian variety over k L = ample, degree 1, symmetric line bundle on X, which, moreover, "glue" together to vector bundles: E(A',£)over 5 whenever X = abelian scheme over 5 £ = relatively ample, degree 1, symmetric line bundle on A!. Then an E-valued algebraic modular form <> / is a functorial rule for defining elements {X,L)€E{X,L) which glue together to sections: {X,C)eTiE{X,C)). We don't want to develop the abstract theory of these at all, but only use these definitions as a setting in which to draw out the consequences of Barsotti's theorem. Let's first of all consider the case g = I. Start with an elliptic curve X/k.
Then r ( X , L) is one-dimensional and let ^ G r ( X , L) be a non-zero
THETAS WITH SPHERICAL HARMONICS
179
section. Let i G Oo,x be an additive coordinate. Then the theorem tells us that there is a canonical trivialization L(8)Oo,x
-^Oo,x
unique up to (^* = c-e^* .(^ for a,c E k^c ^ 0. Therefore d{t) = {'d) is a power series in /, unique up to transformations.
i.e., modulo representations of the multiplicative group Gm' ^ •—^ c ' d,
c E Gm
and the additive group G^: d I—y e^*' 'd,
a GGa.
Expand
Z
D
z4
Let i^(i^05 • •'j^e) be any polynomial which is homogeneous of degree h. Then polarizing R, write R{^o, • • •, t?e) = R{4'\
•••, ^i'^l • • •; ^i''\ • • •, ^'l'^)
where R is linear in each set of variables ??Q , • • •, t?e . Then >2l=-=^fc=0
-21= ••=-2^=0
TATA LECTURES ON THETA III
180
where P{xi, • • - ,Xh) — R{l,xi,
• • •, xf; • • •; 1, a:/i, • • -jX^). Now we can also
expand
E-'JIIik! giving a representation of the additive group in a on the vector space of sequences (i?0) "^ir '')- Then i2(^S,...,^:) = 5 ( e - ? ? ( 2 i ) , - . . , ^ ( e - ? i ? ( z i ) ) ; . - . ) 2 l = . . = Zfc=0
-
<
•
' dzh Ke-'i^CzO-.-e-'^^^fc))
Therefore R{^o, ••-,»?«) = -R('?5, • • •, t?e) for all a if and only if
is independent of a. By Lemma 9.8, this means that P is a harmonic polynomial. Taking low degree harmonic polynomials P we get the following R's: P
R
1
^0
z? - 3 z i z | zt - Qzlzl + zt z\ - 10zfz| + 5ziz|
2(l?4'?0 - 37?|)
zlz2-^z\zl-\-Zizl
Thus any harmonic polynomial P{zi^ - - '•,Zh) in h variables defines by (f) an algebraic modular form. How does this work: first of all i?o is just the image of ?? G r ( X , L) by evaluation at 0:
r(x,L)
L(0).
THETAS WITH SPHERICAL HARMONICS
181
To put this in terms of modular forms, the 1-dimensional vector spaces Hom(r(X,L),L(0)) glue together into the basic "theta" functorial line bundle on i^i, which we will write £^(X, L). Then 1^0 is a section of this line bundle CQ\
^0 e nce). How about ??i? z?i is the differential of i? at 0, i.e., the image of T9 G T{X, L) by: r(X,L) —+ L(0)(8)fi^(0). Thus if uj is the functorial line bundle defined by a-(X,L) = fi^(0) or
uiX,C) = n\,siO)
then
Similarly, 'dk looks like a section of Ce 0 LJ^ - except that it depends on the trivialization . U P is homogeneous of degree e, then one checks that:
''(^•••••5l:'*''-^>'» This still looks a bit more complicated than the analytic theory. The final step is to construct a canonical isomorphism of the line bundles
over some finite covering of the moduli space i^i, i.e., for ( ^ , £)'s with some finite extra structure. In terms of this fundamental isomorphism, a section of Cg 0 u^ becomes a section of >C^"^^^, which we call a modular form of degree h/2-\-e. This is an algebraic construction of the modular forms ^^'^ for 5f = 1, Q = identity h x h matrix.
182
TATA LECTURES ON THETA III For p > 1, the same construction works. Barsotti's theory gives us a
nearly canonical trivialization: (^ : L (g) do,x
- ^ Oo^x
hence a power series -0 = <^(i^) unique up to multiplication by a quadratic exponential. As above, for any pluriharmonic polynomial P{zij) in a, g x hmatrix Z PI z,,=0
is a polynomial in the coefficients of i? invariant under such transformations. Thus we can construct algebraic modular forms for ^f > 1. Where do they lie? As in the case g = 1, £9(X,L) = Hom(r(X,L),L(0)) defines a functorial line bundle for ^-dimensional abelian schemes and
defines a functorial vector bundle of rank g. They are connected by a canonical isomorphism: Cj ^ A^a; defined for (A',£)'s with some finite extra structure (cf. Morel-Baily, Pinceaux de Varietes Abeliennes^ Asterisque 129, 1985). If P is homogeneous of degreee e, then it is easy to see that
eT{C^^Symm%uj)). z.,=0
These sections give an algebraic construction for the modular forms i^^'^'s with Q = h X /i-identity matrix (which can readily be generalized to arbitrary Q).
10. The homogeneous coordinate ring of an abelian variety One of the main applications of theta functions is to provide explicit bases for linear systems r ( X , C) on abelian varieties X. The goal of this section is to study the consequences of having such explicit bases. Now one of the main elements of structure of these vector spaces is the set of multiplication maps: T{X,C)(S>T{X,M)
—^ r(X,£(g)7W)
and, in particular, the ring structure on oo
R{x,c) = ^r{x,c"). n=0
The theta identities described in §6 and §7 allow us in many cases to express these multiplications in terms of the theta bases with coefficients given by the values at 0 of other theta functions. Moreover, we can hope that the polynomial identities defining the ring, i.e., the kernel /„ of:
^:5"(r(X,£))—^r(X,£'^) will be spanned by linear combinations of suitable theta identities as found in §6 and §7. In this case, the whole algebra of the equations defining abelian varieties can be determined from the theory of theta functions. Let us now be more precise. The basic case is where C is an ample symmetric degree 1 line bundle on the complex abelian variety XT- Then a basis of r ( X , £") is given by: ^
{nz.nT),
ae
n
-V/I^
An analog of Cor. 6.10 will give the multiplication map explicitly in terms of this basis: to determine
(*)
r(£") 0 (r(£^) —^ r(£"+^)
184
TATA LECTURES ON THETA III
(\
-mV
0\
(n
(I
0 m
n 0
we use the equivalence of the quadratic forms Q' =
-m\_fn -f m 0
and Q
nm{n -f m)
and deduce formulae of the type (mz, mT) =
{nz,nT)'d
E
{{n + m)z, (n + m)T) • ^
0
n-|-m
0
'
(0,(n +
m)T),
where the sum is over all 77 G ^}_^Z^/Z^. (This is Remark 3 preced1 —m' , AT r= 0, Z = (nz.mz), Q ing Cor. 6.6. Set A = ^1 n n 0 n 4- m . . 1 and Q' = .) Note, however, that dif0 m 0 nm[n •\- m) J ferent theta functions are used for each n making the story a bit complex. We do know that the two theta functions i){nz,nT)
and
e{n(^ z.nt'^T)
are essentially the same (because the two quadratic forms nX^ and nPX'^ are equivalent over Q and we use the second fundamental identity). Therefore, the full description is in terms of the entire family of theta functions ^^{nz^nT)^ for all square-free n. One way to get a handle on this algebra is to stick to the linear systems
r(x,£2") and the products: (**)
r(x, C ) 0 r(x, C ) —^ r(x, c' )
In this case, all bases are written using one of two theta functions 1^(2:, T) or 'd(2z^ 2T), depending on whether n is even or odd and the multiplication comes from use of the simplest identity: 1 1
-1 1
1 0 0 1
i-'Xii)
THE HOMOGENEOUS COORDINATE RING
185
applied in Cor. 6.8 and §7. In the paper Equations Defining Abelian Varieties, this was the approach taken. In particular, this approach allows one to prove that (**) is surjective and to give its kernel explicitly if n > 2. Subsequently, the questions of when the more general map (*) is surjective and the description of its kernel have been investigated at length by Koizumi, Sekiguchi and Kempf. Two methods have emerged: the explicit description of the map via theta functions and the use of more abstract cohomological arguments together with the use of the finite Heisenberg group. The latter gives more powerful results and we want to show here how these methods work. We follow Kempf, Linear Systems on Abelian Varieties, Am. J. Math. (1989), closely. We will first prove: THEOREM
10.1. If C is an ample line bundle on an abelian variety over
any field k, then
r(£")0r(£^)-^r(£"+^) is surjective if n > 2, m > 3. The main new tool we need is the concept of the dual abelian variety X.
A general reference is D. Mumford, Abehan Varieties, Oxford Univ.
Press. X classifies the line bundles on X which are deformations of the trivial bundle. In fact, there is a "universal" bundle V over X x X called the Poincare bundle such that the restrictions Va ofV to X = X x {a}, for various a E X, run through all such line bundles exactly once: i.e. i)Va = Vb
iff a = 6
ii) all deformatins of the trivial bundle occur. If C is any line bundle on X, then {T*{C)}xeX
gives us a family of de-
formations of C itself, parametrized by X. Thus {T^{C) (g) C~^}:cex is a family of deformations of the trivial bundle and we can define a map: <j>c : X — y X
186
TATA LECTURES ON THETA III
by (pcix) = [point a of X such that T*C (g) C~^ ^ Va] This map <j)c is a morphism of varieties, and it can be used to construct X. In fact, if C is ample
= Ki'^) SO we may define X as the quotient X/K{C).
(In char, p, w must use the
full group scheme K_{C) defined at the end of §3). Finally, the addition on X is a result of the tensor product operation on line bundles, i.e..
In the above construction, the "theorem of the square":
T:^yC^T:C®T;C®C-^ implies then that (j)c is a homomorphism:
^ {T^C (8) C-^) 0
(T;£
0 C-^)
= T*^yC (g) C~^
(by the theorem of the square)
hence c{x + y). Our first step in proving 10.1 is the LEMMA
10.2. Let R^ S be invertible sheaves on X such that T(R) ^ 0 and
r ( 5 ) ^ 0, and R(^ S is ample. Then Y, r(i? 0 Va) 0 r ( 5 O V^a) —^ T{R (g) 5) aex is surjective.
Indeed, there exists an open dense subset U so that if you
sum over a £ U the assertion remains true. This result is shown in D. Mumford's lectures in the C.I.M.E. Summer School, 1969, Questions Concerning Algebraic Varieties (these notes, unfortunately, are almost unobtainable).
THE HOMOGENEOUS COORDINATE RING PROOF:
187
Let W denote the image; we show that W is invariant under the
action of the Heisenberg group Q{R^ S). This suffices: Since T{R^
S) is
^(i?05)-irreducible either PF = (0) or V^ = ^(i^(g)5). The first possibility does not occur since T(^R) and r ( 5 ) have non-zero elements. Let r G T{R^Va),se
T{S 0 V - a ) with divisors D, E. Thus r 0 5 has
divisor D + £". If R{x)-^
= R0(T^R<^R^^)<^Va
Similarly, T ; ( 5 0 V ^ a ) = S®V-^.
= R®V^^(^^)^Va Thus T ; r = r' G
=
R^V^.
T{R0P;?),
T ; 5 = 5' G r ( 5 0 V.p) and r' • s' has divisor T!^D + Tl^E.
QED
Using this, we follow Kempf and prove: THEOREM
10.3. Fix y E X. For x in any dense open set of X, the multi-
plication m(x) : r ( £ 2 0 7?_^)0r(/:^0T>^+y) —y
T{C^®Vy)
is surjective. If dim T{C) = 1, then m{x) is an isomorphism. PROOF:
If dim T(C) = 1, the source and target of m(x) have the same
dimension; hence surjective implies isomorphism. The proof goes as follows. First we find a group /C acting on the domain and range of m{x) in such a way that m{x) is equivariant. The key point here is to show that the target has a finite number of maximal /C-invariant subpaces, so as x varies the images of m(x) cannot span the target unless almost all m{x) are themselves surjective. Let g = QiC^), Then G acts on C^®V-x' note that C'^^V.^ ^ T*{C^) for some z E X since c(S>2 is surjective, hence if <^ G G, then (p acts on
188
TATA LECTURES ON THETA III
£^ ®V-x by T*{(f). Similarly G acts on C^ ^Vx+y
Tensoring these, we
get an action of G on C^ <S> Vy so that 7n(x) is equivariant. We restrict the action of Q to an abelian subgroup.
Let A'o be a
maximal isotropic subgroup of K{C) and set
IC=:{xeX\2xeKo}. CLAIM:
e^^(A:,/C)
C A^2-
P R O O F OF CLAIM: e^^fC, fC^ := e^{2IC,2IC) = e^{KoJ
This has two consequences. First it implies that /C is maximal isotropic in /\(£^) since e^ (/C,/C) = e^ (fC^JC)'^ = 1 and the rank of /C, as a finite group scheme, is correct. Second, note that the action of G{C^) on the space r ( £ 2 0 7>_^) ^ r ( £ 2 0 Vy-^) and on r ( £ ^ (g) Vy) factors through G{C^)/fi2 = G^' The claim implies that we can split G' over /C: 1 — G^ ^
e?' ^ / C ( £ ® 2 ) _ ^ 0 K \ \ \
Thus /C acts on the domain and range of m(x). We now study m(x) as a map of /C-modules. LEMMA
10.4. If/C C K{^) is a maximal isotropic subgroup then there are
only a finite number of maximal IC-invariant subspaces W in T{X, C). PROOF:
The irreducible Heisenberg representation of 6{C) can be con-
structed as the space of functions on K{C)/JC} By the Heisenberg property, ^If K{C) is an ordinary finite group, this is clear. In the case where K{C) is a group scheme^ we prove this by considering a splitting a of G{L) over /C and showing that V<j = < functions / on G{L)
f(Xx) = Xf{x),XeGm^ \ f{x'a{k)) = f{x), all ibG/C J
is the Heisenberg representation of G{L), cf. Moret-Baily, Pinceaux de Varietes Abeliennes, Asterisque 129, 1985.
THE HOMOGENEOUS COORDINATE RING
189
this is the dual abelian group ICio tC and K operates here by multiplication by characters. Therefore r ( X , C) is isomorphic to the afRne ring of fC and we need only show that this ring contains only a finite number of maximal subspaces invariant under multiplication by all characters. But a subspace is invariant like this if and only if it is an ideal in the afSne ring of /C and any finite dimensional commutative ring contains only a finite number of maximal ideals. Note that this proof is valid in char, p where Q{C) and K{C) may be group schemes. One should notice that even if, e.g., /C consisted of only one point, so K had only the trivial character, for W C r(AC) to be invariant under the character action of the scheme /C still implies that W is an ideal. This is because the ring multiplication in r(/C):
(*)
v{K) 0 r(^) - ^ r(£)
is dual to the character action of the group scheme /C on T{JC)'. V{K) —^ V{K) (g) T{K) via the duality of r(/C) and T{JC), More explicitly, let g be an i?-valued point of /C for some A:-algebra R. Then g is given by a A:-homomorphism
r(/c) —^ R, which may be viewed as an element of Hom(r(/C), k)<^k R which is r(/C)(g)jk R. In this way, the set fC{R) of i2-valued points is a subset of T{IC) <^k R, and using the ring multiplication (*), we get an action: )C{R) X ( r ( £ ) ®k R) -^
( r ( £ ) 0ib R).
Since, as R varies, the points fC(R) span T{JC), we see that a /C-invariant subspace of r(/C) is indeed an ideal.
QED
190
TATA LECTURES ON THETA III Apply this to our K in K{C^).
It follows that for each x the image
W{x) of m{x) is either all of r ( £ ^ 0'Py) or is contained in one of a finite set f^i J • • •) UN of subspaces of T{C^ ^^y)r ( £ 2 (g) V_^)^
Now as x varies, the vector spaces
(resp. r{jC^ (g) V^+y))
fit together into vector bundles, and the maps m(x) vary continuously. Therefore, for all x in a dense open set, m(a?) will have a fixed maximal rank. Therefore, either W{x) = r ( £ ^ 0 7>y) for all X in a dense open set, or W{x) C some fixed t/,-, all x. But Lemma 10.2 says that
X
Therefore W{x) must equal T{C^ ®'^y) for almost all x. QED for Theorem 10.3. To prove Theorem 10.1, note that by Theorem 10.3,
r(£2) 0 r(£' (g) v^) -^ r(£^ 0 v^) is surjective for all x in a dense open set U oi X. (Take t/ = 0 in 10.3 and note that the maps
r(£2 0 v^^) 0 r(£2 0 v^) —. r(£^) r(£2) (g) r(£2 0 7>2x) —^ r(/:^ 0 7^2^) are just translates of each other, so have the same rank.) By Lemma 10.2,
Y, r(^^ ® ^x) 0 r(£ 0 p^) —^ r(£^) is surjective. But it factors through r(£^ ) 0 r(£^)! This proves 10.1 for n = 2, n = 3. The higher cases follow similarly. In particular, 10.1 implies:
QED for Theorem 10.1.
THE HOMOGENEOUS COORDINATE RING COROLLARY 10.5.
IfM
- C^
191
and
k=0
then R is generated by r ( X , M)ifn>3,
hence M is very ample.
We now turn to the relations in the homogenous coordinate ring of an abelian variety. The main result is this: THEOREM
10.6. If C is an ample line bundle on an abelian variety X over
any field k and M = C^, let I be the kernel of:
0snr(x,A^)]—.0r(x,A^*). k=0
k=0
Then the ideal I is generated by its quadratic and cubic polynomials I2,13 if m > 3 and by its quadratic polynomials alone if n > 4. In fact, if n > 4 and n is even, then we can give a basis of I2 by theta relations resulting from the Riemann theta relation. The simplest case is the pair of quadratic equations defining elliptic curves embedded in P^ by T(Xj £^) with deg £ = 1, given in Chapter I. Let's describe these quadratic relations for general g when X = XT is a principally polarized complex abelian variety and C is the basic degree 1 line bundle. Then as above (nz,nT),
t9
ae-I^ n
is a basis of r ( X , Ai), and instead of writing multiplication by d
(nz, nT) • d
(nz, nT)
^^ + 1 (0, 2nT) • I? ^ + ^ (2nz,2nT) 0
as above, we replace a and 6 by a + ^1, ^ -f ^1 and multiply by a character
192
TATA LECTURES ON THETA III
of r/i. This gives us (compare Ch. II, (6.5), {^.Q) and (6.7)): y ^
a-\-r]
^4iri*cia+r))^
L
£^13/29
0 -
(nz^n
r) .1?
J
L
= e^'^'*"-^ • J2 e^'''""' • ^ .
^
e^'^*'"'^-??
0
r^e^i^z^ = 1?
r
a—h c
'(r)^d
L ^
0 -
{nz.uT) J
(0, 2nT)
0
L
r)£\Z9lZ9
(*)
'6 + ^
'
••
(2nz, 2nT) ••
(n.,fT)
for any c G ^Z^/Z^. The last equality is verified by using the standard Fourier expansion for d. Here we use the alternate basis: d
Z
n
z
of r ( X , M'^), Since T{X, M) 0 r ( X , A^) maps onto T{X, M'^), the RHS's of (*) must span r ( X , A^^), so taking d = a + 6, a Corollary is the nonvanishing result: For all d 6 - Z ^ cG - Z ^ ^ n 2
n 1 (0, - T ) ^ 0 for some 6 G - Z ^ . 2 n
d-26 c
Now since the equations (*) are a basis for the full multiplication table T{X,M)
® T{X,M)
—y T{X,M'^),
they tell us that a basis of the space
I2 of quadratic relations is given by t?
a'-6' c 'd
(0,?T)^e .^
a—h c
0
(n2:,nT)i?
b-\-T]
0
(0,5,T).^e- .^ ^'^^](nz,nT)T?
(nz,nT) 6' + /? 0
if a + b = a* -\-b^ mod Z^. We multiply this by
a + b (nzo,-T) c
= e-^
a -h 6 + 2d c
(O.fT),
{nz.nT),
THE HOMOGENEOUS COORDINATE RING where ZQ = T - d^ d £ ^V\
193
this factor can be made non-zero by suitable
choice of d. Then using (*) and putting UZQ into the characteristic, the identity becomes:
(10.7)
where Sr^ = (~1) ^'^^^'^'^^^, In this form, we have a basis of I2 whose coefficients are the value at 0 of the basis of T{X^M).
Using algebraic theta
functions, this result extends to any field k if char(Ar) f n. Also note that if we set z = 0 in (10.7), we get quartic identities on the theta-nulls d a (0,nT): these are just another form of Riemann's quartic theta re0 lation Our next goal is to describe Kempf's proof of Theorem 10.6 (implying that in the case n even, n > 4 the relations above generate the whole of / ) . It is based on ideas similar to the proof of Theorem 10.1, except that Heisenberg groups are not used, but instead we use more cohomology. The point is to look at families of maps where the line bundles involved are tensored with 'Pa's, and put these together into maps of bundles over the space X of all a's. At a key place, we will make use of a basic calculation of the higher cohomology, groups oiV itself, which can be found in Mumford, Abelian Varieties. First, some notation: If 1^ is a vector space, V is its dual and if S is an Oy-sheaf for a scheme y , then S^ = Hom(5,C>y). Let w and 9 denote the projections of X x X to X and X respectively. As we will be dealing
194
TATA LECTURES ON THETA III
with many powers £^* of the basic ample £ (with ^,- > 0), we abbreviate this to Ci. Finally, the family of vector spaces
fits together into a vector bundle >V^(£,) on X. Formally:
Since tensor product maps £ i 0 Va times £2 0 ^-a to £1 0 £2, we get a map of bundles: M : W + ( £ i ) ( g ) > V - ( £ 2 ) —^ r ( £ i ( g ) £ 2 ) 0 O j ^ . BIG LEMMA 10.8.
The map M induced by M:
M:r(£i(8)£2r
—^ r ( X , ( > V + ( £ i ) 0 > V - ( £ 2 ) ) ^ )
is an isomorphism. PROOF:
We introduce the sheaf J^ on X x X x X defined by: T = TTIC];^ <S> TTiaP"^ (g) TT^C^^ 0 n^^V
where TTij (resp. TT,) is the projection onto the (i,i)^^ factors (resp. i^^ factor). The proof goes by calculating the cohomology of J* in two diff'erent ways: via the restriction of T to the fibres of TTS and via the restriction of J' to the fibres of 7ri2. CLAIM I:
PROOF:
r0
if i ^ 2^
U>V+(A)0>V-(£2)r
ifi = 2g
The restriction of .^ to X x X x {a} is Trt(£r^ ® V-a) 0 7r;(£-^ 0 Va),
THE HOMOGENEOUS COORDINATE RING
195
so its k^^ cohomology is the tensor product of terms:
0
[H\X,C-'
^V.a)0
H^{X,C-^
^Va)l
But the Ci are ample, hence H\X^ £,• (g) "P^) = (0) if f > 0, hence by Serre duality H^(X^ CJ^ <^Vh) = (0) if i < p. Thus the only non-zero group here is H^^iX xXx
[«}, ^1;,^;,^^,}) = H^{C:[' (8) V^a) 0 H^{jC^' 0 Va)
by Serre duality. Therefore all WTT^^ are zero except for the 2g^^ one, which is the dual of the bundle
[j[H\Ci®Va)®H\C2^V.a)l i.e, the dual of >V+(/:i) 0 ^ ( £ 2 ) . CLAIM
II:
fO
if2 9«^^
I ( £ r 0Oxxx ^2)
^ ^A
if i = S'
where O^ is the structure sheaf of the diagonal in X x X. PROOF:
Here's where we need the result (0),
i# g i=g
ioxV on X X X d.ni'K : X X X -^ X. (See Ahelian Varieties, §13). To reduce the claim to this, note that
(here (7r2 — iri, ^3) is the map (x, y, z) 1—• (y — a;, z) from X x X x X to X X X). This follows from the theorem of the cube {Abelian Varieties,
196
TATA LECTURES ON THETA III
§10) since the bundles are isomorphic on {0} x X x X, X x {0} x X and X X X X {0}. Therefore substituting the expression for T^ we get:
(the second is flat base change for the diagram: XxXxX
^J^rilljld
XxX
^^"^V-^ X
XxX
.)
This gives Claim II immediately. We now finish the proof of the big lemma. Apply the Leray spectral sequence for 7ri2: HP{X xX,R^iri2,4y='))
= > HP-^^{X xX
xX,T).
By Claim II, this sequence degenerates to: H^{X xXxX.T)^
H^-\X
X
X,R^Ti2,^{T))
-//^-^(A,(£i0/:2r') ^ H^^-^iX, £1 (g) £2)''
by Serre duality.
Thus (*) ^ ^
HHXXXXX.:F) ^
^
= {^'^ ,, lr(£i(g)£2)^
^"^^^ ifk = 2g.
On the other hand, by Claim I the Leray Spectral sequence for TTS degenerates to: (**)
H^{X xXxX.T)^
H^-^'iX,
(W+(£i) ®
^{£2))^).
Comparing (*) and (**), we get the lemma. To prove the theorem, it is helpful to have a notation for "relations" in more general contexts: for all sheaves ^F^S on X^ we denote the kernel of multiplication
hyR{T,Gy
THE HOMOGENEOUS COORDINATE RING
197
LEMMA
10.9. 1/^1,^2 > 2 and £1 -f ^2 > 5, then
PROOF:
Let R be the left hand side of this equality. By definition, we have
an exact sequence 0 ^-
R{Ci, £ 2 , 0 ^ 3 ) ^ ^— ( r ( £ i ) 0 r(£2 0 ^^3))"^ ^— r ( £ i 0 £2 0 ^^3)^
so to prove the lemma, we just have to show that if A is a linear functional on r ( £ i ) 0 r(£2 0 £3) that vanishes on R, then A is induced by a linear functional on r ( £ i 0 £2 0 £3). Since
Y, r(£i) 0 r(£2 0 Va) 0 r(£3 0 v.a) - ^ r(£i) 0 r(£2 0 £3) is surjective by Lemma 10.2, A is determined by its restrictions X^ to r ( £ i ) 0 r ( £ 2 0 Va) 0 r ( £ 3 0 V-a). We have the diagram: r ( £ i ) 0 r ( £ 2 0:Pa)
R{^l.C2^Va)
r(£3 0 V-a)
r(£3 0 V-a)
r(£i
xC2^Va)
r(£3 0 v.a)
Surjectivity in the top line follows from Theorem 10.1. Since A^ vanishes on 7^(£i, £2 0 Va) • r ( £ 3 0 V^a), Aa iuduces ^^ as in the diagram. Now let's put all these maps together into maps of bundles over X. the kernel of the map r ( £ i ) 0 7r*(£2 0:P) —>
%{Ci^C2<^V).
This map of bundles is surjective, so <S is a bundle too.
Let S be
198
TATA LECTURES ON THETA III Consider the exact sequence of sheaves on X:
O-^S0%{C^^V-^)-^
r ( £ i ) 0 7f*(£2 0 V) (8) %{C3^V-^)
^
7f*(£l 0 £2 0 ^ ) 0 ^ 7f*(£3 0 ^ " ^ )
0
Here A is just the globalization of the A^'s, i.e., it arises from r ( £ i ) 0 7 f * ( £ 2 0 : P ) 0 7 r , ( £ 3 0 ^ " ' ) — T{Ci)^T{C2^Cs)^0^
^
O^.
A restricts to zero on each fibre of the bundle S 0 7r*(>C3 0 V~^)j hence it is zero on the whole bundle. Consequently // exists. We apply the Big Lemma: Since the multiplication 0 7r*(Z:3 0 : p - ^ ) f )
T{X,{9^{Ci^C2®V)
—^ r ( £ i 0 £ 2 0 £ 3 r
is an isomorphism, Jl corresponds to an element of r ( £ i 0 ^ 2 0 £3)^- QED THEOREM
10.10. Ifii
>3J>4
(or ei> 2,^2 > 5), and £3 > 2, tiien
/^(£l, £2 0 £3) = i?(£i, £2)r(£3). PROOF:
Write £2 = £4 0 £5 with ^5 = 2, so ^4 > 2. By the above lemma R{Ci, £2 0 £3) = X ] ^ ( ^ 1 ' ^^ ^ ^ « ) r ( £ 5 0 £3 0 ^ - a ) . a
By our surj activity results
r(£5 ® P-a)r(£3) = r(£5 0 -Ca ® V-a); therefore
a ^
p.
|n
'
THE HOMOGENEOUS COORDINATE RING
199
This gives R{Ci, £2 0 £3) = R{Cu C2)T(C3).
QED
In particular, if £1 > 4, i2(£i,/:^+i) = jR(£i,£i)r(£5*)
if n > 0
and if £1 > 3
This proves Theorem 10.6. Let us recapitulate the results so far: let A^ = £", n > 4, £ ample of degree one. Let Hn denote ^Z^/Z^. Then the basis {i?
{nz,nT)}aeHn
of r ( X , M) defines an embedding ie:X
^
P Hn
into the fixed projective space whose coordinates are indexed by Hn- Here we have taken the complex case for simplicity, but the same construction can be made using algebraic theta functions so long as char (A:) \ n. We have shown a) the ideal of io{X) is generated by quadrics, b) if n is even, the quadrics can be given explicitly by (10.7) in terms of the coordinates of ie{0). We can go further and give a geometric construction of the quadrics containing i${X) using only the point ie{0) which is valid for any n > 4, even or odd. To do this, let Kn = ^Z^^/Z^^ and let Kn act on P ^ - by the maps:
for all (61,62) G Kn- This is obviously the projective version of the irreducible action of the finite Heisenberg group Heis(2y, (Z/nZ)) and the usual formulae for ^-functions show that:
200
TATA LECTURES ON THETA III
PROPOSITION
10.11. Kn maps i9{X) to iself and restricts on ie{X)
to
translating by the n-torsion subgroup X^. In particular, ie{Xn) = Kn{U{^)) and we claim: PROPOSITION
10.12. Ifn > 3, a quadric Q C P^** contains io{X) if and
only if it contains ie {Xn). PROOF:
Note that d
(nz, T) is a basis of r ( £ " ). Choose some y E X
where none of these functions are zero. Assume Q is a quadric which contains f^(Xn). Then Q defines a section s G r(£^") which is zero on XnThen Tly{s) G T{TlyC'^'') is zero on y -f X„. Since n > 3, £ " ' 0
TlyC'^''
is ample so it has a non-zero section t. Then Ty{s) (g) t is a section of £ " zero on t/ + X„. Now Xn is an isotropic subgroup of K{C"' ) = X^^ so translation by Xn lifts to an action of Xn on £ "
and on r ( X , £ " ).
Therefore, write T;{s)®t=
^
cx'sx
xeXn
where sx are eigenfunctions for Xn- There sx are just 'd immediate by calculating i^ i
(nz), as is
(^(^ -f pt of order n)). Since each cxsx is
a linear combination of translates of T*{s) <S) t hy Xn, cxSx is zero at y. But all I? , r s are non-zero at y, so CA = 0 all A. Therefore T^^yS (S)t = 0, hence s = 0, hence Q contains id{X) COROLLARY
QED
10.13. For all n > 4,
iff{X) - {r\Q\all quadrics Q such that Q D Kn{id{0))}. To end this section, we want to rephrase the fact that the coordinates of 2^(0) determine X and Ai by saying instead that {i9
{0,nT)}aeHn
are
homogeneous coordinates on a suitable moduli space of abelian varieties. The precise statement is a little technical because, before the algebraic analogs of i)
(0, nT) can be defined on an arbitrary abelian variety X,
THE HOMOGENEOUS COORDINATE RING
201
some labelling of its points of finite order must be done. We will sketch the result. Whenever char(fc) f n, the moduli space An is defined to be the variety which classifies up to isomorphism triples (X, £,<^„), where X is an abelian variety, £ is a degree one, ample line bundle on X and (f>n is an isomorphism of Xn with (Z/nZ)^^ carrying the skew-symmetric form e\i on Xn to the standard form on (Z/nZ)^^ (a primitive n^^ root of 1 in ^ must be fixed to define this). n is called a "level n structure". When n|m, there is a map Am —^ An because a level m structure determines a level n structure, i.e., we have the usual tower {An} of moduli spaces. To work out the meaning of Cor. 10.13, we need a moduli space intermediate beween An and ^2nj which Igusa named ^n,2n- We assume n is even. An,2n is the variety which classifies up to isomorphism triples (X, Ai, a ) , where X is an abelian variety, Ai is an ample symmetric line bundle on X with e;|^ = 1 and Of is a symmetric^ isomorphism g{M)
-^Heis(2y,Z/nZ).
Then a induces an isomorphism (unique up to scalars) between the Heisenberg representation T{XjAi)
o{Q{M) and any of the standard realizations
of this representation for lleis(2g, Z/nZ). This gives us algebraically a basis Sa, a E {Z/nZy,
of T{XjM)
generalizing the basis d
in the complex
case, hence an embedding ie : X ^-^ P ^ " . The map
gives us a map
e„ : An,2n which is itself one-to-one because ie(0) allows us to reconstruct X^M Of. The final result is: ^i.e., the symmetry i ^ o{Q{M) corresponds to the involution (A, a:, ?/) (A,-x,-y).
and
202
TATA LECTURES ON THETA III
THEOREM
10.14. aj For aii n > 4, Bn is an iniTnersion of the scheme ^A^ 2n
h) Ifn is even and n>6j
then Im{Sn) is a Zariski-open subset of the
closed subscheme in P^"" defined by the quartic polynomials given by the equations (10.7) with z = 0: I / ^SrjXa'^d-\-rjXi,'^d-\-ri
1 ' I / , SrjXg^f) Xb^rj 1 =
( 2^Sr)Xa-\.d-^r}Xb^d-{-T]
where Srj = (-1)*2^'2^, a,h,a\h\d and the sum being over rj G PROOF:
1 * I 2^ SrjXa'+fjXh'+r}
1
G (^Z^/Z^) satisfying a -f 6 = a' + 6'
\V/J.^.
See Mumford, Equations Defining Abelian Varieties 11, §6 for the
case S\n and Kempf, Linear systems on abelian varieties, Am. J. Math., (1989), for the general case. It would be really nice if this were also true for n = 4, but this is open.
