Hardcover ISBN:  9780821819951 
Product Code:  CRMM/14 
List Price:  $90.00 
MAA Member Price:  $81.00 
AMS Member Price:  $72.00 
Electronic ISBN:  9781470438593 
Product Code:  CRMM/14.E 
List Price:  $84.00 
MAA Member Price:  $75.60 
AMS Member Price:  $67.20 

Book DetailsCRM Monograph SeriesVolume: 14; 2002; 270 ppMSC: Primary 11; 14;
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 reinterpreted 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 nonexperts. It attempts to provide the necessary background to all concepts exposed in it. It may serve as a textbook for an advanced graduate course.ReadershipGraduate students and research mathematicians interested in number theory and algebraic geometry.

Table of Contents

Chapters

Introduction

Tori and abelian varieties

Complex abelian varieties with real multiplication and Hilbert modular forms

Abelian varieties with real multiplication over general fields

$p$adic elliptic modular forms

$p$adic Hilbert modular forms

Deformation theory of abelian varieties

Appendix A. Group schemes

Appendix B. Calculating with cusps


Reviews

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


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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 reinterpreted 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 nonexperts. It attempts to provide the necessary background to all concepts exposed in it. It may serve as a textbook for an advanced graduate course.
Graduate students and research mathematicians interested in number theory and algebraic geometry.

Chapters

Introduction

Tori and abelian varieties

Complex abelian varieties with real multiplication and Hilbert modular forms

Abelian varieties with real multiplication over general fields

$p$adic elliptic modular forms

$p$adic Hilbert modular forms

Deformation theory of abelian varieties

Appendix A. Group schemes

Appendix B. Calculating with cusps

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