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Author: Baskar Balasubramanyam
Publisher:
ISBN: 9781109959567
Category : Hilbert modular surfaces
Languages : en
Pages : 61
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Book Description
We construct a measure-valued cohomology class that interpolates the modular symbols attached to a nearly ordinary Hida family of Hilbert modular forms over a totally real field F. We call such a class an overconvergent modular symbol. Our construction is a generalization to totally real fields of results obtained in [7] by Greenberg and Stevens for F = Q . Under the assumption that F has strict class number one, the overconvergent modular symbol is used to define a two variable p-adic L-function that interpolates special values of classical L-functions.
Author: Baskar Balasubramanyam
Publisher:
ISBN: 9781109959567
Category : Hilbert modular surfaces
Languages : en
Pages : 61
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Book Description
We construct a measure-valued cohomology class that interpolates the modular symbols attached to a nearly ordinary Hida family of Hilbert modular forms over a totally real field F. We call such a class an overconvergent modular symbol. Our construction is a generalization to totally real fields of results obtained in [7] by Greenberg and Stevens for F = Q . Under the assumption that F has strict class number one, the overconvergent modular symbol is used to define a two variable p-adic L-function that interpolates special values of classical L-functions.
Author: Baskar Balasubramanyam
Publisher: World Scientific
ISBN: 9814719242
Category : Mathematics
Languages : en
Pages : 342
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Book Description
The aim of this book is to give a systematic exposition of results in some important cases where p-adic families and p-adic L-functions are studied. We first look at p-adic families in the following cases: general linear groups, symplectic groups and definite unitary groups. We also look at applications of this theory to modularity lifting problems. We finally consider p-adic L-functions for GL(2), the p-adic adjoint L-functions and some cases of higher GL(n).
Author: John Bergdall
Publisher: American Mathematical Society
ISBN: 1470470314
Category : Mathematics
Languages : en
Pages : 138
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Book Description
View the abstract.
Author: Haruzo Hida
Publisher: Oxford University Press
ISBN: 019857102X
Category : Mathematics
Languages : en
Pages : 417
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Book Description
Describing the applications found for the Wiles and Taylor technique, this book generalizes the deformation theoretic techniques of Wiles-Taylor to Hilbert modular forms (following Fujiwara's treatment), and also discusses applications found by the author.
Author: Fabrizio Andreatta
Publisher: American Mathematical Soc.
ISBN: 0821836099
Category : Mathematics
Languages : en
Pages : 114
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Book Description
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: Haruzo Hida
Publisher: Springer Science & Business Media
ISBN: 9780387207117
Category : Mathematics
Languages : en
Pages : 414
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Book Description
This book covers the following three topics in a manner accessible to graduate students who have an understanding of algebraic number theory and scheme theoretic algebraic geometry: 1. An elementary construction of Shimura varieties as moduli of abelian schemes. 2. p-adic deformation theory of automorphic forms on Shimura varieties. 3. A simple proof of irreducibility of the generalized Igusa tower over the Shimura variety. The book starts with a detailed study of elliptic and Hilbert modular forms and reaches to the forefront of research of Shimura varieties associated with general classical groups. The method of constructing p-adic analytic families and the proof of irreducibility was recently discovered by the author. The area covered in this book is now a focal point of research worldwide with many far-reaching applications that have led to solutions of longstanding problems and conjectures. Specifically, the use of p-adic elliptic and Hilbert modular forms have proven essential in recent breakthroughs in number theory (for example, the proof of Fermat's Last Theorem and the Shimura-Taniyama conjecture by A. Wiles and others). Haruzo Hida is Professor of Mathematics at University of California, Los Angeles. His previous books include Modular Forms and Galois Cohomology (Cambridge University Press 2000) and Geometric Modular Forms and Elliptic Curves (World Scientific Publishing Company 2000).
Author: Haruzo Hida
Publisher: Springer Science & Business Media
ISBN: 1468493906
Category : Mathematics
Languages : en
Pages : 397
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Book Description
In the early years of the 1980s, while I was visiting the Institute for Ad vanced Study (lAS) at Princeton as a postdoctoral member, I got a fascinating view, studying congruence modulo a prime among elliptic modular forms, that an automorphic L-function of a given algebraic group G should have a canon ical p-adic counterpart of several variables. I immediately decided to find out the reason behind this phenomenon and to develop the theory of ordinary p-adic automorphic forms, allocating 10 to 15 years from that point, putting off the intended arithmetic study of Shimura varieties via L-functions and Eisenstein series (for which I visited lAS). Although it took more than 15 years, we now know (at least conjecturally) the exact number of variables for a given G, and it has been shown that this is a universal phenomenon valid for holomorphic automorphic forms on Shimura varieties and also for more general (nonholomorphic) cohomological automorphic forms on automorphic manifolds (in a markedly different way). When I was asked to give a series of lectures in the Automorphic Semester in the year 2000 at the Emile Borel Center (Centre Emile Borel) at the Poincare Institute in Paris, I chose to give an exposition of the theory of p-adic (ordinary) families of such automorphic forms p-adic analytically de pending on their weights, and this book is the outgrowth of the lectures given there.
Author: A. A. Panchishkin
Publisher:
ISBN:
Category :
Languages : en
Pages : 29
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Book Description
Author: A. A. Pančiškin
Publisher:
ISBN:
Category :
Languages : de
Pages : 29
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Book Description
Author: Haruzo Hida
Publisher: Clarendon Press
ISBN: 0191513873
Category : Mathematics
Languages : en
Pages : 420
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Book Description
The 1995 work of Wiles and Taylor-Wiles opened up a whole new technique in algebraic number theory and, a decade on, the waves caused by this incredibly important work are still being felt. This book, authored by a leading researcher, describes the striking applications that have been found for this technique. In the book, the deformation theoretic techniques of Wiles-Taylor are first generalized to Hilbert modular forms (following Fujiwara's treatment), and some applications found by the author are then discussed. With many exercises and open questions given, this text is ideal for researchers and graduate students entering this research area.
