Shimura Varieties

Author : Haruzo Hida
Publisher : Springer Science & Business Media
Page : 397 pages
File Size : 54,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781468493900

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p-Adic Automorphic Forms on Shimura Varieties by Haruzo Hida Pdf

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 : Thomas Haines,Michael Harris
Publisher : Cambridge University Press
Page : 341 pages
File Size : 48,6 Mb
Release : 2020-02-20
Category : Mathematics
ISBN : 9781108704861

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Shimura Varieties by Thomas Haines,Michael Harris Pdf

This volume forms the sequel to "On the stabilization of the trace formula", published by International Press of Boston, Inc., 2011

Author : Pierre Deligne,James S. Milne,Arthur Ogus,Kuang-yen Shih
Publisher : Springer
Page : 423 pages
File Size : 43,8 Mb
Release : 2009-03-20
Category : Mathematics
ISBN : 9783540389552

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Hodge Cycles, Motives, and Shimura Varieties by Pierre Deligne,James S. Milne,Arthur Ogus,Kuang-yen Shih Pdf

Author : Clay Mathematics Institute. Summer School
Publisher : American Mathematical Soc.
Page : 708 pages
File Size : 47,7 Mb
Release : 2005
Category : Mathematics
ISBN : 082183844X

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Harmonic Analysis, the Trace Formula, and Shimura Varieties by Clay Mathematics Institute. Summer School Pdf

Langlands program proposes fundamental relations that tie arithmetic information from number theory and algebraic geometry with analytic information from harmonic analysis and group representations. This title intends to provide an entry point into this exciting and challenging field.

Author : Atsushi Ichino,Kartik Prasanna
Publisher : American Mathematical Society
Page : 214 pages
File Size : 54,5 Mb
Release : 2021-02-23
Category : Mathematics
ISBN : 9781470448943

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Periods of Quaternionic Shimura Varieties. I. by Atsushi Ichino,Kartik Prasanna Pdf

This book formulates a new conjecture about quadratic periods of automorphic forms on quaternion algebras, which is an integral refinement of Shimura's algebraicity conjectures on these periods. It also provides a strategy to attack this conjecture by reformulating it in terms of integrality properties of the theta correspondence for quaternionic unitary groups. The methods and constructions of the book are expected to have applications to other problems related to periods, such as the Bloch-Beilinson conjecture about special values of $L$-functions and constructing geometric realizations of Langlands functoriality for automorphic forms on quaternion algebras.

Author : Harry Reimann
Publisher : Springer
Page : 152 pages
File Size : 47,9 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540684145

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The semi-simple zeta function of quaternionic Shimura varieties by Harry Reimann Pdf

This monograph is concerned with the Shimura variety attached to a quaternion algebra over a totally real number field. For any place of good (or moderately bad) reduction, the corresponding (semi-simple) local zeta function is expressed in terms of (semi-simple) local L-functions attached to automorphic representations. In an appendix a conjecture of Langlands and Rapoport on the reduction of a Shimura variety in a very general case is restated in a slightly stronger form. The reader is expected to be familiar with the basic concepts of algebraic geometry, algebraic number theory and the theory of automorphic representation.

Author : Kai-Wen Lan
Publisher : Princeton University Press
Page : 584 pages
File Size : 55,9 Mb
Release : 2013-03-21
Category : Mathematics
ISBN : 9781400846016

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Arithmetic Compactifications of PEL-Type Shimura Varieties by Kai-Wen Lan Pdf

By studying the degeneration of abelian varieties with PEL structures, this book explains the compactifications of smooth integral models of all PEL-type Shimura varieties, providing the logical foundation for several exciting recent developments. The book is designed to be accessible to graduate students who have an understanding of schemes and abelian varieties. PEL-type Shimura varieties, which are natural generalizations of modular curves, are useful for studying the arithmetic properties of automorphic forms and automorphic representations, and they have played important roles in the development of the Langlands program. As with modular curves, it is desirable to have integral models of compactifications of PEL-type Shimura varieties that can be described in sufficient detail near the boundary. This book explains in detail the following topics about PEL-type Shimura varieties and their compactifications: A construction of smooth integral models of PEL-type Shimura varieties by defining and representing moduli problems of abelian schemes with PEL structures An analysis of the degeneration of abelian varieties with PEL structures into semiabelian schemes, over noetherian normal complete adic base rings A construction of toroidal and minimal compactifications of smooth integral models of PEL-type Shimura varieties, with detailed descriptions of their structure near the boundary Through these topics, the book generalizes the theory of degenerations of polarized abelian varieties and the application of that theory to the construction of toroidal and minimal compactifications of Siegel moduli schemes over the integers (as developed by Mumford, Faltings, and Chai).

Author : Yuval Zvi Flicker
Publisher : World Scientific
Page : 338 pages
File Size : 52,8 Mb
Release : 2005
Category : Mathematics
ISBN : 9789812564030

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Automorphic Forms and Shimura Varieties of PGSp (2) by Yuval Zvi Flicker Pdf

The area of automorphic representations is a natural continuation of studies in the 19th and 20th centuries on number theory and modular forms. A guiding principle is a reciprocity law relating infinite dimensional automorphic representations with finite dimensional Galois representations. Simple relations on the Galois side reflect deep relations on the automorphic side, called ?liftings.' This in-depth book concentrates on an initial example of the lifting, from a rank 2 symplectic group PGSp(2) to PGL(4), reflecting the natural embedding of Sp(2,ó) in SL(4, ó). It develops the technique of comparing twisted and stabilized trace formulae. It gives a detailed classification of the automorphic and admissible representation of the rank two symplectic PGSp(2) by means of a definition of packets and quasi-packets, using character relations and trace formulae identities. It also shows multiplicity one and rigidity theorems for the discrete spectrum.Applications include the study of the decomposition of the cohomology of an associated Shimura variety, thereby linking Galois representations to geometric automorphic representations.To put these results in a general context, the book concludes with a technical introduction to Langlands' program in the area of automorphic representations. It includes a proof of known cases of Artin's conjecture.

Author : Michael Harris,Richard Lawrence Taylor,Richard Taylor
Publisher : Princeton University Press
Page : 287 pages
File Size : 47,9 Mb
Release : 2001-11-04
Category : Mathematics
ISBN : 9780691090924

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The Geometry and Cohomology of Some Simple Shimura Varieties. (AM-151) by Michael Harris,Richard Lawrence Taylor,Richard Taylor Pdf

This book aims first to prove the local Langlands conjecture for GLn over a p-adic field and, second, to identify the action of the decomposition group at a prime of bad reduction on the l-adic cohomology of the "simple" Shimura varieties. These two problems go hand in hand. The results represent a major advance in algebraic number theory, finally proving the conjecture first proposed in Langlands's 1969 Washington lecture as a non-abelian generalization of local class field theory. The local Langlands conjecture for GLn(K), where K is a p-adic field, asserts the existence of a correspondence, with certain formal properties, relating n-dimensional representations of the Galois group of K with the representation theory of the locally compact group GLn(K). This book constructs a candidate for such a local Langlands correspondence on the vanishing cycles attached to the bad reduction over the integer ring of K of a certain family of Shimura varieties. And it proves that this is roughly compatible with the global Galois correspondence realized on the cohomology of the same Shimura varieties. The local Langlands conjecture is obtained as a corollary. Certain techniques developed in this book should extend to more general Shimura varieties, providing new instances of the local Langlands conjecture. Moreover, the geometry of the special fibers is strictly analogous to that of Shimura curves and can be expected to have applications to a variety of questions in number theory.

Author : Harry Reimann
Publisher : Lecture Notes in Mathematics
Page : 166 pages
File Size : 43,8 Mb
Release : 1997-04-14
Category : Mathematics
ISBN : STANFORD:36105020404443

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The Semi-simple Zeta Function of Quaternionic Shimura Varieties by Harry Reimann Pdf

This monograph is concerned with the Shimura variety attached to a quaternion algebra over a totally real number field. For any place of good (or moderately bad) reduction, the corresponding (semi-simple) local zeta function is expressed in terms of (semi-simple) local L-functions attached to automorphic representations. In an appendix a conjecture of Langlands and Rapoport on the reduction of a Shimura variety in a very general case is restated in a slightly stronger form. The reader is expected to be familiar with the basic concepts of algebraic geometry, algebraic number theory and the theory of automorphic representation.

Author : Haruzo Hida
Publisher : Springer Science & Business Media
Page : 414 pages
File Size : 44,7 Mb
Release : 2004-05-10
Category : Mathematics
ISBN : 0387207112

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p-Adic Automorphic Forms on Shimura Varieties by Haruzo Hida Pdf

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 : A. J. Scholl,Richard Lawrence Taylor
Publisher : Cambridge University Press
Page : 506 pages
File Size : 50,6 Mb
Release : 1998-11-26
Category : Mathematics
ISBN : 9780521644198

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Galois Representations in Arithmetic Algebraic Geometry by A. J. Scholl,Richard Lawrence Taylor Pdf

Conference proceedings based on the 1996 LMS Durham Symposium 'Galois representations in arithmetic algebraic geometry'.

Author : Richard Pink
Publisher : Unknown
Page : 368 pages
File Size : 51,5 Mb
Release : 1989
Category : Algebraic varieties
ISBN : UOM:39015049381372

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Arithmetical Compactification of Mixed Shimura Varieties by Richard Pink Pdf

Author : Michael Harris,Steven Zucker
Publisher : Unknown
Page : 132 pages
File Size : 44,6 Mb
Release : 2001
Category : Hodge theory
ISBN : STANFORD:36105112168005

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Boundary Cohomology of Shimura Varieties, III by Michael Harris,Steven Zucker Pdf

In this book, the authors complete the verification of the following fact: The nerve spectral sequence for the cohomology of the Borel-Serre boundary of a Shimura variety $\mathrm{Sh}$ is a spectral sequence of mixed Hodge-de Rham structures over the field of definition of its canonical model. To achieve that, they develop the machinery of automorphic vector bundles on mixed Shimura varieties, for the latter enter in the boundary of the toroidal compactifications of $\mathrm{Sh}$; and study the nerve spectral sequence for the automorphic vector bundles and the toroidal boundary. They also extend the technique of averting issues of base-change by taking cohomology with growth conditions. They give and apply formulas for the Hodge gradation of the cohomology of both $\mathrm{Sh}$ and its Borel-Serre boundary.

Author : Gorō Shimura
Publisher : Princeton University Press
Page : 292 pages
File Size : 46,8 Mb
Release : 1971-08-21
Category : Mathematics
ISBN : 0691080925

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Introduction to the Arithmetic Theory of Automorphic Functions by Gorō Shimura Pdf

The theory of automorphic forms is playing increasingly important roles in several branches of mathematics, even in physics, and is almost ubiquitous in number theory. This book introduces the reader to the subject and in particular to elliptic modular forms with emphasis on their number-theoretical aspects. After two chapters geared toward elementary levels, there follows a detailed treatment of the theory of Hecke operators, which associate zeta functions to modular forms. At a more advanced level, complex multiplication of elliptic curves and abelian varieties is discussed. The main question is the construction of abelian extensions of certain algebraic number fields, which is traditionally called "Hilbert's twelfth problem." Another advanced topic is the determination of the zeta function of an algebraic curve uniformized by modular functions, which supplies an indispensable background for the recent proof of Fermat's last theorem by Wiles.