› › Convolutions Of Hilbert Modular Forms Motives And P Adic L Functions




Motives

Author : Uwe Jannsen

Publisher : American Mathematical Soc.

Page : 696 pages

File Size : 16,70 MB

Release : 1994-02-28

Category : Mathematics

ISBN : 9780821827994

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Book Description

Motives were introduced in the mid-1960s by Grothendieck to explain the analogies among the various cohomology theories for algebraic varieties, to play the role of the missing rational cohomology, and to provide a blueprint for proving Weil's conjectures about the zeta function of a variety over a finite field. Over the last ten years or so, researchers in various areas--Hodge theory, algebraic $K$-theory, polylogarithms, automorphic forms, $L$-functions, $ell$-adic representations, trigonometric sums, and algebraic cycles--have discovered that an enlarged (and in part conjectural) theory of ``mixed'' motives indicates and explains phenomena appearing in each area. Thus the theory holds the potential of enriching and unifying these areas. These two volumes contain the revised texts of nearly all the lectures presented at the AMS-IMS-SIAM Joint Summer Research Conference on Motives, held in Seattle, in 1991. A number of related works are also included, making for a total of forty-seven papers, from general introductions to specialized surveys to research papers.



$p$-adic $L$-Functions and $p$-adic Representations

Author : Bernadette Perrin-Riou

Publisher : American Mathematical Soc.

Page : 176 pages

File Size : 29,7 MB

Release : 2000

Category : Mathematics

ISBN : 9780821819463

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Book Description

Traditionally, p-adic L-functions have been constructed from complex L-functions via special values and Iwasawa theory. In this volume, Perrin-Riou presents a theory of p-adic L-functions coming directly from p-adic Galois representations (or, more generally, from motives). This theory encompasses, in particular, a construction of the module of p-adic L-functions via the arithmetic theory and a conjectural definition of the p-adic L-function via its special values. Since the original publication of this book in French (see Astérisque 229, 1995), the field has undergone significant progress. These advances are noted in this English edition. Also, some minor improvements have been made to the text.



Non-Archimedean L-Functions and Arithmetical Siegel Modular Forms

Author : Michel Courtieu

Publisher : Springer

Page : 202 pages

File Size : 50,8 MB

Release : 2003-12-09

Category : Mathematics

ISBN : 3540451781

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Book Description

This book, now in its 2nd edition, is devoted to the arithmetical theory of Siegel modular forms and their L-functions. The central object are L-functions of classical Siegel modular forms whose special values are studied using the Rankin-Selberg method and the action of certain differential operators on modular forms which have nice arithmetical properties. A new method of p-adic interpolation of these critical values is presented. An important class of p-adic L-functions treated in the present book are p-adic L-functions of Siegel modular forms having logarithmic growth. The given construction of these p-adic L-functions uses precise algebraic properties of the arithmetical Shimura differential operator. The book will be very useful for postgraduate students and for non-experts looking for a quick approach to a rapidly developing domain of algebraic number theory. This new edition is substantially revised to account for the new explanations that have emerged in the past 10 years of the main formulas for special L-values in terms of arithmetical theory of nearly holomorphic modular forms.



Hida Families of Hilbert Modular Forms and P-adic L-functions

Author : Baskar Balasubramanyam

Publisher :

Page : 61 pages

File Size : 15,11 MB

Release : 2007

Category : Hilbert modular surfaces

ISBN : 9781109959567

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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.



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

Author : Fabrizio Andreatta

Publisher : American Mathematical Soc.

Page : 114 pages

File Size : 50,78 MB

Release : 2005

Category : Mathematics

ISBN : 0821836099

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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.





Lectures on Hilbert Modular Varieties and Modular Forms

Author : Eyal Zvi Goren

Publisher : American Mathematical Soc.

Page : 282 pages

File Size : 41,26 MB

Release : 2002

Category : Mathematics

ISBN : 082181995X

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Book Description

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.