P-Adic Automorphic Forms On Shimura Varieties

This book covers the following three topics in a manner accessible to graduatestudents who have an understanding of algebraic number theory and schemetheoretic ... » 

This book covers the following three topics in a manner accessible to graduatestudents who have an understanding of algebraic number theory and schemetheoretic algebraic geometry 1. An elementary construction of Shimuravarieties as moduli of abelian schemes 2. padic deformation theory ofautomorphic forms on Shimura varieties 3. A simple proof of irreducibility ofthe generalized Igusa tower over the Shimura variety The book starts with adetailed study of elliptic and Hilbert modular forms and reaches to theforefront of research of Shimura varieties associated with general classicalgroups. The method of constructing padic analytic families and the proof ofirreducibility was recently discovered by the author. The area covered in thisbook is now a focal point of research worldwide with many farreachingapplications that have led to solutions of longstanding problems andconjectures. Specifically the use of padic elliptic and Hilbert modular formshave proven essential in recent breakthroughs in number theory for examplethe proof of Fermats Last Theorem and the ShimuraTaniyama conjecture by A.Wiles and others. Haruzo Hida is Professor of Mathematics at University ofCalifornia Los Angeles. His previous books include Modular Forms and GaloisCohomology Cambridge University Press 2000 and Geometric Modular Forms andElliptic Curves World Scientific Publishing Company 2000. TOCIntroduction.Geometric Reciprocity Laws. Modular Curves. Hilbert Modular Varieties.Generalized EichlerShimura Map. Moduli Schemes. Shimura Varieties. pAdicAutomorphic Forms. Bibliography. «