Moduli of Abelian Varieties

524 Pages·2001·41.88 MB·English

by  Valery Alexeev (auth.)

Progress in Mathematics
Volume 195
Series Editors
H. Bass
1. Oesterle
A. Weinstein
Moduli of
Abelian Varieties
CareI Faber
Gerard van der Geer
Frans Oort
Editors
Springer Basel AG
Editors:
Carei Faber Gerard van der Geer
Institutionen for Matematik Korteweg de Vries Instituut
Kungliga Tekniska H6gskolan (KTH) Universiteit van Amsterdam
10044 Stockholm Plantage Muidergracht 24
Sweden 10 18 TV Amsterdam
The Netherlands
Frans Oort
Mathematisch Instituut
Universiteit Utrecht
Budapestlaan 6
3508 TA Utrecht
The Netherlands
2000 Mathematics Subject Classification 14J60, 14D20, 11 G 15
A CIP catalogue record for this book is available from the Library of Congress,
Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Moduli of Abelian varieties / Carei Faber ... ed. - Basel ; Boston; Berlin: Birkhăuser,
2001
(Progress in mathematics ; VoI. 195)
ISBN 978-3-0348-9509-5 ISBN 978-3-0348-8303-0 (eBook)
DOI 10.1007/978-3-0348-8303-0
This work is subject to copyright. AH rights are reserved, whether the whole or
part of the material is concerned, specifically the rights of translation, reprinting, re-use
of illustrations, broadcasting, reproduction on microfilms or in other ways, and
storage in data banks. For any kind ofuse whatsoever, permission from the copyright
owner must be obtained.
© 2001 Springer Basel AG
Originally published by Birkhăuser Verlag in 2001
Softcover reprint of the hardcover 1s t edition 2001
Printed on acid-free paper produced of chlorine-free pulp. TCF
00
ISBN 978-3-0348-9509-5
987654321
CONTENTS
Participants vii
Contributors IX
Introduction xi
On extra components in the functorial compactification ofA 1
g
V. Alexeev
On Mumford's uniformization and Neron models ofJacobians of
semistable curves over complete rings 11
F. Andreatta
Torelli theorem via Fourier-Mukai transform 127
A. Beilinson and A. Polishchuk
On the Andre-Oort conjecture for Hilbert modular surfaces 133
B. Edixhoven
Toroidal resolutions for some matrix singularities 157
G. Faltings
Formal Brauer groups and moduli ofabelian surfaces 185
G. van der Geer and T. Katsura
Isogeny classes ofabelian varieties with no principal polarizations 203
E. Howe
Igusa's modular form and the classification ofSiegel modular threefolds 217
K. Hulek
Mirror symmetry and quantization ofabelian varieties 231
Yu.I. Manin
Group schemes with additional structures and Weyl group cosets 255
B. Moonen
Moduli space ofelliptic curves with Heisenberg level structure 299
I. Nakamura and T. Terasoma
Singularities ofthe height strata in the moduli ofK3 surfaces 325
A.Ogus
v
Astratification ofa moduli space ofabelian varieties 345
F. Dort
Newton polygon strata in the moduli space ofabelian varieties 417
F. Dort
The dimension ofOort strata ofShimura varieties ofPEL-type 441
T. Wedhorn
Hyperelliptic Jacobians and modular representations 473
Yu.G. Zarhin
Windows for displays ofp-divisible groups 491
Th. Zink
vi
PARTICIPANTS
The following is the list of participants of the conference Moduli ofAbelian
Varieties, Texel '99, which was held on Texel Island during the last week of
April 1999.
Dan Abramovich (Boston), Valery Alexeev (Athens),
Fabrizio Andreatta (Utrecht), Christina Birkenhake (Erlangen),
Michael Bogouslavsky (Amsterdam), Brad Brock (Malibu),
Oliver Biiltel (Kaln), Ching-Li Chai (Philadelphia),
Ciro Ciliberto (Roma), Elisabetta Colombo (Milano),
Tim Dokshitzer (Utrecht), Bas Edixhoven (Rennes),
Torsten Ekedahl (Stockholm), Hel~me Esnault (Essen),
Carel Faber (Stillwater, Stockholm), Gerd Faltings (MPI, Bonn),
Gavril Farkas (Amsterdam), Gerard van der Geer (Amsterdam),
Eyal Goren (Montreal), Ulrich Gartz (Kaln),
Joost van Hamel (Utrecht), Gunter Harder (Bonn),
Everett Howe (San Diego), Johan Huisman (Rennes),
Klaus Hulek (Hannover), Elham Izadi (Athens),
Johan de Jong (MIT, Cambridge), T. Katsura (Tokyo),
David Lehavi (Jerusalem), Ron Livne (Jerusalem),
Andrea Miller (Bonn), A. Mokrane (Paris),
Ben Moonen (Utrecht), Laurent Moret-Bailly (Rennes),
Stefan Muller-Stach (Essen), I. Nakamura (Sapporo),
Wieslawa Niziol (Salt Lake City), Rutger Noot (Rennes),
K. Oguiso (Tokyo), Arthur Ogus (Berkeley),
Cathy O'Neil (MIT, Cambridge), Frans Oort (Utrecht),
Sascha Orlik (Kaln), Pierre Parent (Rennes),
Gian-Pietro Pirola (Pavia), Gregory Sankaran (Bath),
Utpal Sarkar (Amsterdam), Rene Schoof (Roma),
Vasily Shabat (Amsterdam), Bart de Smit (Leiden),
Tomasz Szemberg (MPI, Bonn), Valerio Talamanca (Amsterdam),
Halszka Tutaj-Gasinska (Krakow), Eckart Viehweg (Essen),
Maja Volkov (Kaln), Vitaly Vologodsky (Athens),
Mike Zieve (Los Angeles), Thomas Zink (Bielefeld).
Vll
CONTRIBUTORS
V. Alexeev Department of Mathematics, University of Georgia, Athens, GA
30602, USA.
E-mail: [email protected]
F. Andreatta Department of Mathematics and Statistics, McGill University,
805 Sherbrooke St. West, Montreal, Quebec, Canada H3A 2K6.
E-mail: [email protected]
A. Beilinson Department ofMathematics, UniversityofChicago, 5734S. Uni
versity Avenue, Chicago, IL 60637, USA.
E-mail: [email protected]
B. Edixhoven IRMAR, Campus de Beaulieu, 35042 Rennes cedex, France.
E-mail: [email protected]
G. Faltings Max-Planck-Institut fur Mathematik, Postfach 7280, D-53072
Bonn, Germany.
E-mail: [email protected]
G. van der Geer Korteweg-de Vries Instituut, Universiteit van Amsterdam,
Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands.
E-mail: [email protected]
E. Howe Center for Communications Research, 4320 Westerra Court, San
Diego, CA 92121-1967, USA.
E-mail: [email protected]
K. Hulek Institut fur Mathematik (C), Universitat Hannover, Postfach 6009,
D-30060 Hannover, Germany.
E-mail: [email protected]
T. Katsura Graduate School of Mathematical Sciences, University ofTokyo,
3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan.
E-mail: [email protected]
Yu.I. Manin Max-Planck-Institut fur Mathematik, Postfach 7280, D-53072
Bonn, Germany.
E-mail: [email protected]
B. Moonen Mathematisch Instituut, Universiteit Utrecht, Postbus 80010, NL
3508 TA Utrecht, The Netherlands.
E-mail: [email protected]
I. Nakamura Department ofMathematics, Hokkaido University, Sapporo 060
810, Japan.
E-mail: [email protected]
A. Ogus Department ofMathematics, University of California, Berkeley, CA
94720, USA.
E-mail: [email protected]
ix
F. Oart MathematischInstituut, Universiteit Utrecht, Postbus80010, NL-3508
TA Utrecht, The Netherlands.
E-mail: [email protected]
A. Palishchuk Department ofMathematics and Statistics, Boston University,
III Cummington Street, Boston, MA 02215, USA.
E-mail; [email protected]
T. Terasama Graduate SchoolofMathematical Sciences, UniversityofTokyo,
3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan.
E-mail;[email protected]
T. Wedham Mathematisches Institut, Universitiit zu Koln, Weyertal 86-90,
D-50931 Koln, Germany.
E-mail;[email protected]
Yu.G. Zarhin Department of Mathematics, Pennsylvania State University,
University Park, PA 16802, USA.
Email: [email protected]
Th. Zink Fakultiit fill Mathematik, Universitat Bielefeld, Postfach 100131,
D-33501 Bielefeld, Germany.
E-mail: [email protected]
x
INTRODUCTION
That moduli spaces of abelian varieties are intriguing mathematical ob
jects became clear in the 19th century through the work of Kronecker, Klein,
Weber, and many others on the moduli of elliptic curves. Not only did the
modular curves and their function theory illustrate the new concept of Rie
mann surfaces in a beautiful way, the theorem of Kronecker-Weber pointed
to much deeper arithmetic properties ofthese modular curves. It took a long
time before the tools were shaped that made it possible to study the arith
metic properties ofmodular curves and then these modular curves became the
focus of intense study after the early 1970's, which resulted in Mazur's theo
rem on the rational points ofXo(N). The insight that all motives comingfrom
elliptic curves defined over Q are present in the motives defined by modular
curves (Shimura-Taniyama-WeiI) came slowly but developed into a main goal
ofresearch, culminating in the proof by Wiles ofFermat's Last Theorem.
Inthemeantimethemoduliofhigher-dimensionalabelianvarietiesstarted
to attract attention and the ground was prepared by work of Siegel, Satake,
Igusa, Baily and others on the complex analytic case. A definitive advance
was made by the work of Mumford who proved in the 1960's the existence
of moduli spaces for polarized abelian varieties in the sense of Grothendieck.
Moreover, Mumford and his collaboratorsdesigned the tools to construct com
pactifications ofthese moduli spaces over the complex numbers. By the work
ofFaltings and Chai we now have good compactifications of moduli spaces of
abelian varieties over the integers.
This does not mean, however, that everything is now set for an explo
ration ofthe moduli ofabelian varieties ofhigher dimension. We do not doubt
that fascinating facts are waiting to be discovered by those who will venture
into this terrain, but they will face great obstacles and will need endurance be
cause progress will come slowly. The links with such fields as algebraic groups,
representation theory, algebraic geometry, and number theory vote for the im
portanceofthesubjectbutthedifficultiesseemtorepelmanyandtheattention
that these spaces got so far is limited. One of the insights obtained in recent
years is that maybe the moduli spaces in positive characteristic are more ac
cessible than their characteristic zero counterparts.
Since abelian varieties and their moduli constitute in our view one of
the most important subjects in algebraic geometry and possess an enormous
potential for number theory, we decided to devote the 3rd Texel conference
to this topic. It was held during the last week of April 1999. This volume
appears on the occasion of the conference, but the articles presented here are
not necessarily connected to lectures given at the conference.
We would like to take the opportunity to thank the participants and
the speakers who made the conference to a success. We also like to thank
the institutions that financed the conference: NWO, het Korteweg-de Vries
Xl