Lectures On Hilbert Modular Varieties And Modular Forms

Author : Eyal Zvi Goren,Zvi Goren
Publisher : American Mathematical Soc.
Page : 282 pages
File Size : 50,7 Mb
Release : 2002
Category : Abelian varieties
ISBN : 9780821819951

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Lectures on Hilbert Modular Varieties and Modular Forms by Eyal Zvi Goren,Zvi Goren Pdf

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.

Author : Friedrich Hirzebruch,Gerard van der Geer
Publisher : Unknown
Page : 200 pages
File Size : 43,8 Mb
Release : 1981
Category : Discontinuous groups
ISBN : UOM:39015015619466

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Lectures on Hilbert Modular Surfaces by Friedrich Hirzebruch,Gerard van der Geer Pdf

Author : Gerard van der Geer
Publisher : Springer Science & Business Media
Page : 301 pages
File Size : 40,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642615535

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Hilbert Modular Surfaces by Gerard van der Geer Pdf

Over the last 15 years important results have been achieved in the field of Hilbert Modular Varieties. Though the main emphasis of this book is on the geometry of Hilbert modular surfaces, both geometric and arithmetic aspects are treated. An abundance of examples - in fact a whole chapter - completes this competent presentation of the subject. This Ergebnisbericht will soon become an indispensible tool for graduate students and researchers in this field.

Author : Robert C. Gunning
Publisher : Princeton University Press
Page : 116 pages
File Size : 47,9 Mb
Release : 1962-03-21
Category : Education
ISBN : 0691079951

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Lectures on Modular Forms by Robert C. Gunning Pdf

New interest in modular forms of one complex variable has been caused chiefly by the work of Selberg and of Eichler. But there has been no introductory work covering the background of these developments. H. C. Gunning's book surveys techniques and problems; only the simpler cases are treated-modular forms of even weights without multipliers, the principal congruence subgroups, and the Hecke operators for the full modular group alone.

Author : Laurent Berger,Gebhard Böckle,Lassina Dembélé,Mladen Dimitrov,Tim Dokchitser,John Voight
Publisher : Springer Science & Business Media
Page : 257 pages
File Size : 45,9 Mb
Release : 2013-06-13
Category : Mathematics
ISBN : 9783034806183

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Elliptic Curves, Hilbert Modular Forms and Galois Deformations by Laurent Berger,Gebhard Böckle,Lassina Dembélé,Mladen Dimitrov,Tim Dokchitser,John Voight Pdf

The notes in this volume correspond to advanced courses held at the Centre de Recerca Matemàtica as part of the research program in Arithmetic Geometry in the 2009-2010 academic year. The notes by Laurent Berger provide an introduction to p-adic Galois representations and Fontaine rings, which are especially useful for describing many local deformation rings at p that arise naturally in Galois deformation theory. The notes by Gebhard Böckle offer a comprehensive course on Galois deformation theory, starting from the foundational results of Mazur and discussing in detail the theory of pseudo-representations and their deformations, local deformations at places l ≠ p and local deformations at p which are flat. In the last section,the results of Böckle and Kisin on presentations of global deformation rings over local ones are discussed. The notes by Mladen Dimitrov present the basics of the arithmetic theory of Hilbert modular forms and varieties, with an emphasis on the study of the images of the attached Galois representations, on modularity lifting theorems over totally real number fields, and on the cohomology of Hilbert modular varieties with integral coefficients. The notes by Lassina Dembélé and John Voight describe methods for performing explicit computations in spaces of Hilbert modular forms. These methods depend on the Jacquet-Langlands correspondence and on computations in spaces of quaternionic modular forms, both for the case of definite and indefinite quaternion algebras. Several examples are given, and applications to modularity of Galois representations are discussed. The notes by Tim Dokchitser describe the proof, obtained by the author in a joint project with Vladimir Dokchitser, of the parity conjecture for elliptic curves over number fields under the assumption of finiteness of the Tate-Shafarevich group. The statement of the Birch and Swinnerton-Dyer conjecture is included, as well as a detailed study of local and global root numbers of elliptic curves and their classification.

Author : Jan Hendrik Bruinier,Gerard van der Geer,Günter Harder,Don Zagier
Publisher : Springer Science & Business Media
Page : 273 pages
File Size : 49,6 Mb
Release : 2008-02-10
Category : Mathematics
ISBN : 9783540741190

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The 1-2-3 of Modular Forms by Jan Hendrik Bruinier,Gerard van der Geer,Günter Harder,Don Zagier Pdf

This book grew out of three series of lectures given at the summer school on "Modular Forms and their Applications" at the Sophus Lie Conference Center in Nordfjordeid in June 2004. The first series treats the classical one-variable theory of elliptic modular forms. The second series presents the theory of Hilbert modular forms in two variables and Hilbert modular surfaces. The third series gives an introduction to Siegel modular forms and discusses a conjecture by Harder. It also contains Harder's original manuscript with the conjecture. Each part treats a number of beautiful applications.

Author : Fabrizio Andreatta,Eyal Zvi Goren
Publisher : American Mathematical Soc.
Page : 100 pages
File Size : 44,8 Mb
Release : 2005
Category : Mathematics
ISBN : 9780821836095

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Hilbert Modular Forms: mod $p$ and $p$-Adic Aspects by Fabrizio Andreatta,Eyal Zvi Goren Pdf

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.

Author : Joseph Lehner
Publisher : Unknown
Page : 88 pages
File Size : 41,9 Mb
Release : 1969
Category : Finite fields (Algebra)
ISBN : UCSD:31822014374235

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Lectures on Modular Forms by Joseph Lehner Pdf

Author : Joseph J. Lehner
Publisher : Courier Dover Publications
Page : 96 pages
File Size : 49,5 Mb
Release : 2017-05-17
Category : Mathematics
ISBN : 9780486821405

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Lectures on Modular Forms by Joseph J. Lehner Pdf

Concise book offers expository account of theory of modular forms and its application to number theory and analysis. Substantial notes at the end of each chapter amplify the more difficult subjects. 1969 edition.

Author : Jayce Getz,Mark Goresky
Publisher : Springer Science & Business Media
Page : 264 pages
File Size : 43,8 Mb
Release : 2012-03-28
Category : Mathematics
ISBN : 9783034803519

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Hilbert Modular Forms with Coefficients in Intersection Homology and Quadratic Base Change by Jayce Getz,Mark Goresky Pdf

In the 1970s Hirzebruch and Zagier produced elliptic modular forms with coefficients in the homology of a Hilbert modular surface. They then computed the Fourier coefficients of these forms in terms of period integrals and L-functions. In this book the authors take an alternate approach to these theorems and generalize them to the setting of Hilbert modular varieties of arbitrary dimension. The approach is conceptual and uses tools that were not available to Hirzebruch and Zagier, including intersection homology theory, properties of modular cycles, and base change. Automorphic vector bundles, Hecke operators and Fourier coefficients of modular forms are presented both in the classical and adèlic settings. The book should provide a foundation for approaching similar questions for other locally symmetric spaces.

Author : Eberhard Freitag
Publisher : Springer Science & Business Media
Page : 255 pages
File Size : 52,7 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9783662026380

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Hilbert Modular Forms by Eberhard Freitag Pdf

Important results on the Hilbert modular group and Hilbert modular forms are introduced and described in this book. In recent times, this branch of number theory has been given more and more attention and thus the need for a comprehensive presentation of these results, previously scattered in research journal papers, has become obvious. The main aim of this book is to give a description of the singular cohomology and its Hodge decomposition including explicit formulae. The author has succeeded in giving proofs which are both elementary and complete. The book contains an introduction to Hilbert modular forms, reduction theory, the trace formula and Shimizu's formulae, the work of Matsushima and Shimura, analytic continuation of Eisenstein series, the cohomology and its Hodge decomposition. Basic facts about algebraic numbers, integration, alternating differential forms and Hodge theory are included in convenient appendices so that the book can be used by students with a knowledge of complex analysis (one variable) and algebra.

Author : Fabrizio Andreatta,Eyal Zvi Goren
Publisher : American Mathematical Soc.
Page : 100 pages
File Size : 41,7 Mb
Release : 2005
Category : Mathematics
ISBN : 1470404206

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Hilbert Modular Forms by Fabrizio Andreatta,Eyal Zvi Goren Pdf

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.

Author : T. Oda
Publisher : Springer Science & Business Media
Page : 141 pages
File Size : 52,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781468492019

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Periods of Hilbert Modular Surfaces by T. Oda Pdf

Author : John Cremona,John Jones,Jennifer Paulhus,Andrew V. Sutherlan,John Voight
Publisher : American Mathematical Soc.
Page : 386 pages
File Size : 45,9 Mb
Release : 2024-03-22
Category : Mathematics
ISBN : 9781470472603

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LuCaNT: LMFDB, Computation, and Number Theory by John Cremona,John Jones,Jennifer Paulhus,Andrew V. Sutherlan,John Voight Pdf

This book will be published Open Access with a Creative Commons Attribution 4.0 International License (CC BY 4.0). The eBook can be downloaded electronically for free. This volume contains the proceedings of the LuCaNT (LMFDB, Computation, and Number Theory) conference held from July 10–14, 2023, at the Institute for Computational and Experimental Research in Mathematics (ICERM), Providence, Rhode Island and affiliated with Brown University. This conference provided an opportunity for researchers, scholars, and practitioners to exchange ideas, share advances, and collaborate in the fields of computation, mathematical databases, number theory, and arithmetic geometry. The papers that appear in this volume record recent advances in these areas, with special focus on the LMFDB (the L-Functions and Modular Forms Database), an online resource for mathematical objects arising in the Langlands program and the connections between them.

Author : Robert C. Gunning
Publisher : Princeton University Press
Page : 96 pages
File Size : 48,8 Mb
Release : 2016-03-02
Category : Mathematics
ISBN : 9781400881666

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Lectures on Modular Forms. (AM-48), Volume 48 by Robert C. Gunning Pdf

New interest in modular forms of one complex variable has been caused chiefly by the work of Selberg and of Eichler. But there has been no introductory work covering the background of these developments. H. C. Gunning's book surveys techniques and problems; only the simpler cases are treated-modular forms of even weights without multipliers, the principal congruence subgroups, and the Hecke operators for the full modular group alone.