**CRM Monograph Series**

Volume: 14;
2002;
270 pp;
Hardcover

MSC: Primary 11; 14;

Print ISBN: 978-0-8218-1995-1

Product Code: CRMM/14

List Price: $90.00

AMS Member Price: $72.00

MAA Member Price: $81.00

**Electronic ISBN: 978-1-4704-3859-3
Product Code: CRMM/14.E**

List Price: $84.00

AMS Member Price: $67.20

MAA Member Price: $75.60

# Lectures on Hilbert Modular Varieties and Modular Forms

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*Eyal Z. Goren*

A co-publication of the AMS and Centre de Recherches Mathématiques

This book is devoted to certain aspects of the theory of \(p\)-adic
Hilbert modular forms and moduli spaces of abelian varieties with real
multiplication.

The theory of \(p\)-adic modular forms is presented first in the
elliptic case, introducing the reader to key ideas of N. M. Katz and J.-P.
Serre. It is re-interpreted from a geometric point of view, which is developed
to present the rudiments of a similar theory for Hilbert modular forms.

The theory of moduli spaces of abelian varieties with real multiplication is
presented first very explicitly over the complex numbers. Aspects of the
general theory are then exposed, in particular, local deformation theory of
abelian varieties in positive characteristic.

The arithmetic of \(p\)-adic Hilbert modular forms and the geometry
of moduli spaces of abelian varieties are related. This relation is used to
study \(q\)-expansions of Hilbert modular forms, on the one hand, and
stratifications of moduli spaces on the other hand.

The book is addressed to graduate students and non-experts. It attempts to
provide the necessary background to all concepts exposed in it. It may serve as
a textbook for an advanced graduate course.

Titles in this series are co-published with the Centre de Recherches Mathématiques.

#### Readership

Graduate students and research mathematicians interested in number theory and algebraic geometry.

#### Reviews & Endorsements

It is very nice to have these important topics brought together in a book that could be used as a textbook for a graduate course.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Lectures on Hilbert Modular Varieties and Modular Forms

- Cover Cover11
- Title page v6
- Contents vii8
- Introduction 112
- Tori and abelian varieties 516
- Complex abelian varieties with real multiplication and Hilbert modular forms 4960
- Abelian varieties with real multiplication over general fields 7788
- 𝑝-adic elliptic modular forms 109120
- 𝑝-adic Hilbert modular forms 167178
- Deformation theory of abelian varieties 193204
- Appendix A. Group schemes 235246
- Appendix B. Calculating with cusps 251262
- Bibliography 255266
- Notation index 263274
- Index 267278
- Back Cover Back Cover1282