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Title:P-Adic Automorphic Forms on Shimura Varieties
Format Type:Ebook
Author:
Publisher:Springer
ISBN:0387207112
ISBN 13:
Number of Pages:390
Category:Manga

P-Adic Automorphic Forms on Shimura Varieties by Haruzo Hida

PDF, EPUB, MOBI, TXT, DOC P-Adic Automorphic Forms on Shimura Varieties In the early years of the s 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 to 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 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 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

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Modular Forms and Galois Cohomology, P-Adic Automorphic Forms on Shimura Varieties, p-Adic Automorphic Forms on Shimura Varieties (Springer Monographs in Mathematics), Geometric Modular Forms and Elliptic Cur, Hilbert Modular Forms and Iwasawa Theory, Modular Forms and Galois Cohomology, Geometric Modular Forms and Elliptic Curves, Elementary Theory of L-Functions and Eisenstein Series, Elliptic Curves and Arithmetic Invariants, Contributions to Automorphic Forms, Geometry, and Number Theory: A Volume in Honor of Joseph Shalika
This book provides a comprehensive account of the theory of moduli spaces of elliptic curves over integer rings and its application to modular forms The construction of Galois representations which play a fundamental role in Wiles proof of the Shimura Taniyama conjecture is given In addition the book presents an outline of the proof of diverse modularity results of two dimensional Galois representations including that of Wiles as well as some of the author s new results in that direction In this new second edition a detailed description of Barsotti Tate groups including formal Lie groups is added to Chapter As an application a down to earth description of formal deformation theory of elliptic curves is incorporated at the end of Chapter in order to make the proof of regularity of the moduli of elliptic curve more conceptual and in Chapter though limited to ordinary cases newly incorporated are Ribet s theorem of full image of modular p adic Galois representation and its generalization to big adic Galois representations under mild assumptions a new result of the author Though some of the striking developments described above is out of the scope of this introductory book the author gives a taste of present day research in the area of Number Theory at the very end of the book giving a good account of modularity theory of abelian varieties and curves, No description available, This book provides a comprehensive account of a key perhaps the most important theory that forms the basis of Taylor Wiles proof of Fermat s last theorem Hida begins with an overview of the theory of automorphic forms on linear algebraic groups and then covers the basic theory and recent results on elliptic modular forms including a substantial simplification of the Taylor Wiles proof by Fujiwara and Diamond He offers a detailed exposition of the representation theory of profinite groups including deformation theory as well as the Euler characteristic formulas of Galois cohomology groups The final chapter presents a proof of a non abelian class number formula, This book contains a detailed account of the result of the author s recent Annals paper and JAMS paper on arithmetic invariant including u invariant L invariant and similar topics This book can be regarded as an introductory text to the author s previous book p Adic Automorphic Forms on Shimura Varieties Written as a down to earth introduction to Shimura varieties this text includes many examples and applications of the theory that provide motivation for the reader Since it is limited to modular curves and the corresponding Shimura varieties this book is not only a great resource for experts in the field but it is also accessible to advanced graduate students studying number theory Key topics include non triviality of arithmetic invariants and special values of L functions elliptic curves over complex and p adic fields Hecke algebras scheme theory elliptic and modular curves over rings and Shimura curves