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Modern Birkhauser Classics Many of the original research and survey monographs in pure and applied mathematics published by Birkhauser in recent decades have been groundbreaking and have come to be regarded as foundational to the subject. Through the MBC Series, a select number of these modern classics, entirely uncorrected, are being re-released in paperback (and as eBooks) to ensure that these treasures remain accessible to new generations of students, scholars, and researchers.
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 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{ 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, '^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 \.\ 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 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„, n) G ^(Li) and v = {xn.tpn) G ^(Ls). Then (Ti:„,