Eyal Zvi Goren's Books

Lectures on Hilbert Modular Varieties and Modular Forms

By: Eyal Zvi Goren

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Lectures on Hilbert Modular Varieties and Modular Forms

By: Eyal Zvi Goren

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 abelianvarieties 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 ofmoduli 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.

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Page Count

282

Publisher

American Mathematical Soc.

Published Date

ISBN 10

082181995X

ISBN 13

9780821819951

Number Theory

By: H. Kisilevsky,Eyal Zvi Goren,Canadian Number Theory Association. Conference

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Number Theory

By: H. Kisilevsky,Eyal Zvi Goren,Canadian Number Theory Association. Conference

Collects articles from the meeting of the Canadian Number Theory Association held at the Centre de Recherches Mathematiques (CRM) at the University of Montreal. This book covers topics such as algebraic number theory, analytic number theory, arithmetic algebraic geometry, computational number theory, and Diophantine analysis and approximation.

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Page Count

329

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821833316

ISBN 13

9780821833315

Hilbert Modular Forms: mod $p$ and $p$-Adic Aspects

Mod P and P-adic Aspects

By: Fabrizio Andreatta,Eyal Zvi Goren

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Hilbert Modular Forms: mod $p$ and $p$-Adic Aspects

Mod P and P-adic Aspects

By: Fabrizio Andreatta,Eyal Zvi Goren

We study Hilbert modular forms in characteristic $p$ and over $p$-adic rings. In the characteristic $p$ theory we describe the kernel and image of the $q$-expansion map and prove the existence of filtration for Hilbert modular forms; we define operators $U$, $V$ and $\Theta_\chi$ and study the variation of the filtration under these operators. Our methods are geometric - comparing holomorphic Hilbert modular forms with rational functions on a moduli scheme with level-$p$ structure, whose poles are supported on the non-ordinary locus.In the $p$-adic theory we study congruences between Hilbert modular forms. This applies to the study of congruences between special values of zeta functions of totally real fields. It also allows us to define $p$-adic Hilbert modular forms 'a la Serre' as $p$-adic uniform limit of classical modular forms, and compare them with $p$-adic modular forms 'a la Katz' that are regular functions on a certain formal moduli scheme. We show that the two notions agree for cusp forms and for a suitable class of weights containing all the classical ones. We extend the operators $V$ and $\Theta_\chi$ to the $p$-adic setting.

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