Hilbert Modular Forms Mod P And P Adic Aspects

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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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Genre : Mathematics
Author : Fabrizio Andreatta
Publisher : American Mathematical Soc.
Release : 2005
File : 114 Pages
ISBN-13 : 9780821836095


P Adic Aspects Of Modular Forms

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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).

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Genre : Mathematics
Author : Baskar Balasubramanyam
Publisher : World Scientific
Release : 2016-06-14
File : 342 Pages
ISBN-13 : 9789814719247


Hilbert Modular Forms And Iwasawa Theory

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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.

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Genre : Mathematics
Author : Haruzo Hida
Publisher : Clarendon Press
Release : 2006-06-15
File : 420 Pages
ISBN-13 : 9780191513879


Geometric Aspects Of Dwork Theory

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This two-volume book collects the lectures given during the three months cycle of lectures held in Northern Italy between May and July of 2001 to commemorate Professor Bernard Dwork (1923 - 1998). It presents a wide-ranging overview of some of the most active areas of contemporary research in arithmetic algebraic geometry, with special emphasis on the geometric applications of the p-adic analytic techniques originating in Dwork's work, their connection to various recent cohomology theories and to modular forms. The two volumes contain both important new research and illuminating survey articles written by leading experts in the field. The book will provide an indispensable resource for all those wishing to approach the frontiers of research in arithmetic algebraic geometry.

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Genre : Mathematics
Author : Alan Adolphson
Publisher : Walter de Gruyter
Release : 2008-08-22
File : 1150 Pages
ISBN-13 : 9783110198133


Automorphic Forms And Related Geometry Assessing The Legacy Of I I Piatetski Shapiro

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This volume contains the proceedings of the conference Automorphic Forms and Related Geometry: Assessing the Legacy of I.I. Piatetski-Shapiro, held from April 23-27, 2012, at Yale University, New Haven, CT. Ilya I. Piatetski-Shapiro, who passed away on 21 February 2009, was a leading figure in the theory of automorphic forms. The conference attempted both to summarize and consolidate the progress that was made during Piatetski-Shapiro's lifetime by him and a substantial group of his co-workers, and to promote future work by identifying fruitful directions of further investigation. It was organized around several themes that reflected Piatetski-Shapiro's main foci of work and that have promise for future development: functoriality and converse theorems; local and global -functions and their periods; -adic -functions and arithmetic geometry; complex geometry; and analytic number theory. In each area, there were talks to review the current state of affairs with special attention to Piatetski-Shapiro's contributions, and other talks to report on current work and to outline promising avenues for continued progress. The contents of this volume reflect most of the talks that were presented at the conference as well as a few additional contributions. They all represent various aspects of the legacy of Piatetski-Shapiro.

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Genre : Mathematics
Author : James W. Cogdell
Publisher : American Mathematical Soc.
Release : 2014-04-01
File : 454 Pages
ISBN-13 : 9780821893944


Flat Level Set Regularity Of P Laplace Phase Transitions

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We prove a Harnack inequality for level sets of $p$-Laplace phase transition minimizers. In particular, if a level set is included in a flat cylinder, then, in the interior, it is included in a flatter one. The extension of a result conjectured by De Giorgi and recently proven by the third author for $p=2$ follows.

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Genre : Mathematics
Author : Enrico Valdinoci
Publisher : American Mathematical Soc.
Release : 2006
File : 158 Pages
ISBN-13 : 9780821839102


The Calculus Of One Sided M Ideals And Multipliers In Operator Spaces

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The theory of one-sided $M$-ideals and multipliers of operator spaces is simultaneously a generalization of classical $M$-ideals, ideals in operator algebras, and aspects of the theory of Hilbert $C*$-modules and their maps. Here we give a systematic exposition of this theory. The main part of this memoir consists of a 'calculus' for one-sided $M$-ideals and multipliers, i.e. a collection of the properties of one-sided $M$-ideals and multipliers with respect to the basic constructions met in functional analysis. This is intended to be a reference tool for 'noncommutative functional analysts' who may encounter a one-sided $M$-ideal or multiplier in their work.

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Genre : Mathematics
Author : David P. Blecher
Publisher : American Mathematical Soc.
Release : 2006
File : 102 Pages
ISBN-13 : 9780821838235


Measure Theoretic Laws For Lim Sup Sets

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Given a compact metric space $(\Omega,d)$ equipped with a non-atomic, probability measure $m$ and a positive decreasing function $\psi$, we consider a natural class of lim sup subsets $\Lambda(\psi)$ of $\Omega$. The classical lim sup set $W(\psi)$ of `$\psi$-approximable' numbers in the theory of metric Diophantine approximation fall within this class. We establish sufficient conditions (which are also necessary under some natural assumptions) for the $m$-measure of $\Lambda(\psi)$ to be either positive or full in $\Omega$ and for the Hausdorff $f$-measure to be infinite. The classical theorems of Khintchine-Groshev and Jarnik concerning $W(\psi)$ fall into our general framework. The main results provide a unifying treatment of numerous problems in metric Diophantine approximation including those for real, complex and $p$-adic fields associated with both independent and dependent quantities. Applications also include those to Kleinian groups and rational maps. Compared to previous works our framework allows us to successfully remove many unnecessary conditions and strengthen fundamental results such as Jarnik's theorem and the Baker-Schmidt theorem. In particular, the strengthening of Jarnik's theorem opens up the Duffin-Schaeffer conjecture for Hausdorff measures.

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Genre : Diophantine approximation
Author : Victor Beresnevich Detta Dickinson Sanju Velani
Publisher : American Mathematical Soc.
Release : 2005-12-01
File : 116 Pages
ISBN-13 : 0821865684


Relatively Hyperbolic Groups Intrinsic Geometry Algebraic Properties And Algorithmic Problems

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In this the authors obtain an isoperimetric characterization of relatively hyperbolicity of a groups with respect to a collection of subgroups. This allows them to apply classical combinatorial methods related to van Kampen diagrams to obtain relative analogues of some well-known algebraic and geometric properties of ordinary hyperbolic groups. There is also an introduction and study of the notion of a relatively quasi-convex subgroup of a relatively hyperbolic group and solve somenatural algorithmic problems.

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Genre : Mathematics
Author : Denis V. Osin
Publisher : American Mathematical Soc.
Release : 2006
File : 114 Pages
ISBN-13 : 9780821838211


An Algebraic Structure For Moufang Quadrangles

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Features an article that intends to present a uniform algebraic structure for Moufang quadrangles, and to classify these structures without referring back to the original Moufang quadrangles from which they arise, thereby also giving a new proof for the classification of Moufang quadrangles, which does consist of the division into these 2 parts.

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Genre : Mathematics
Author : Tom de Medts
Publisher : American Mathematical Soc.
Release : 2005
File : 114 Pages
ISBN-13 : 9780821836088