| Author | : Pierre Deligne |
| Publisher | : Springer |
| Total Pages | : 423 |
| Release | : 2009-03-20 |
| Genre | : Mathematics |
| ISBN | : 3540389555 |
Shimura Varieties
| Author | : Thomas Haines |
| Publisher | : Cambridge University Press |
| Total Pages | : 341 |
| Release | : 2020-02-20 |
| Genre | : Mathematics |
| ISBN | : 1108704867 |
This volume forms the sequel to "On the stabilization of the trace formula", published by International Press of Boston, Inc., 2011
Shimura Varieties
| Author | : Thomas Haines |
| Publisher | : Cambridge University Press |
| Total Pages | : 341 |
| Release | : 2020-02-20 |
| Genre | : Mathematics |
| ISBN | : 1108632068 |
This is the second volume of a series of mainly expository articles on the arithmetic theory of automorphic forms. It forms a sequel to On the Stabilization of the Trace Formula published in 2011. The books are intended primarily for two groups of readers: those interested in the structure of automorphic forms on reductive groups over number fields, and specifically in qualitative information on multiplicities of automorphic representations; and those interested in the classification of I-adic representations of Galois groups of number fields. Langlands' conjectures elaborate on the notion that these two problems overlap considerably. These volumes present convincing evidence supporting this, clearly and succinctly enough that readers can pass with minimal effort between the two points of view. Over a decade's worth of progress toward the stabilization of the Arthur-Selberg trace formula, culminating in Ngo Bau Chau's proof of the Fundamental Lemma, makes this series timely.
The Geometry and Cohomology of Some Simple Shimura Varieties. (AM-151)
| Author | : Michael Harris |
| Publisher | : Princeton University Press |
| Total Pages | : 287 |
| Release | : 2001-11-04 |
| Genre | : Mathematics |
| ISBN | : 0691090920 |
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.
p-Adic Automorphic Forms on Shimura Varieties
| Author | : Haruzo Hida |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 397 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 1468493906 |
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.
p-Adic Automorphic Forms on Shimura Varieties
| Author | : Haruzo Hida |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 414 |
| Release | : 2004-05-10 |
| Genre | : Mathematics |
| ISBN | : 9780387207117 |
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).
Harmonic Analysis, the Trace Formula, and Shimura Varieties
| Author | : Clay Mathematics Institute. Summer School |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 708 |
| Release | : 2005 |
| Genre | : Mathematics |
| ISBN | : 9780821838440 |
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.
Arithmetic Compactifications of PEL-type Shimura Varieties
| Author | : Kai-Wen Lan |
| Publisher | : Princeton University Press |
| Total Pages | : 587 |
| Release | : 2013-03-24 |
| Genre | : Mathematics |
| ISBN | : 0691156549 |
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).
Galois Representations in Arithmetic Algebraic Geometry
| Author | : A. J. Scholl |
| Publisher | : Cambridge University Press |
| Total Pages | : 506 |
| Release | : 1998-11-26 |
| Genre | : Mathematics |
| ISBN | : 0521644194 |
Conference proceedings based on the 1996 LMS Durham Symposium 'Galois representations in arithmetic algebraic geometry'.
