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Field name	Details
Dewey Class	512.7
Title	p-Adic Automorphic Forms on Shimura Varieties ([EBook] /) / by Haruzo Hida.
Author	Hida, Haruzo
Other name(s)	SpringerLink (Online service)
Publication	New York, NY : : Springer New York, , 2004.
Physical Details	XI, 390 p. : online resource.
Series	Springer monographs in mathematics 1439-7382
ISBN	9781468493900
Summary Note	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).:
Contents note	1 Introduction -- 1.1 Automorphic Forms on Classical Groups -- 1.2 p-Adic Interpolation of Automorphic Forms -- 1.3 p-Adic Automorphic L-functions -- 1.4 Galois Representations -- 1.5 Plan of the Book -- 1.6 Notation -- 2 Geometric Reciprocity Laws -- 2.1 Sketch of Classical Reciprocity Laws -- 2.2 Cyclotomic Reciprocity Laws and Adeles -- 2.3 A Generalization of Galois Theory -- 2.4 Algebraic Curves over a Field -- 2.5 Elliptic Curves over a Field -- 2.6 Elliptic Modular Function Field -- 3 Modular Curves -- 3.1 Basics of Elliptic Curves over a Scheme -- 3.2 Moduli of Elliptic Curves and the Igusa Tower -- 3.3 p-Ordinary Elliptic Modular Forms -- 3.4 Elliptic ?-Adic Forms and p-Adic L-functions -- 4 Hilbert Modular Varieties -- 4.1 Hilbert–Blumenthal Moduli -- 4.2 Hilbert Modular Shimura Varieties -- 4.3 Rank of p-Ordinary Cohomology Groups -- 4.4 Appendix: Fundamental Groups -- 5 Generalized Eichler–Shimura Map -- 5.1 Semi-Simplicity of Hecke Algebras -- 5.2 Explicit Symmetric Domains -- 5.3 The Eichler–Shimura Map -- 6 Moduli Schemes -- 6.1 Hilbert Schemes -- 6.2 Quotients by PGL(n) -- 6.3 Mumford Moduli -- 6.4 Siegel Modular Variety -- 7 Shimura Varieties -- 7.1 PEL Moduli Varieties -- 7.2 General Shimura Varieties -- 8 Ordinary p-Adic Automorphic Forms -- 8.1 True and False Automorphic Forms -- 8.2 Deformation Theory of Serre and Tate -- 8.3 Vertical Control Theorem -- 8.4 Irreducibility of Igusa Towers -- References -- Symbol Index -- Statement Index.
System details note	Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users)
Internet Site	http://dx.doi.org/10.1007/978-1-4684-9390-0
Links to Related Works	Subject References: Algebraic Geometry . Mathematics . Number Theory . Authors: Hida, Haruzo . Corporate Authors: SpringerLink (Online service) . Series: Springer monographs in mathematics . Classification: 512.7 . 512.7 (DDC 23) .

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p-Adic Automorphic Forms on Shimura Varieties

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