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Springer New York
p-Adic Automorphic Forms on Shimura Varieties / Edition 1

p-Adic Automorphic Forms on Shimura Varieties / Edition 1

by Haruzo HidaHaruzo Hida
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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).

Product Details

ISBN-13: 9781441919236
Publisher: Springer New York
Publication date: 10/07/2011
Series: Springer Monographs in Mathematics
Edition description: Softcover reprint of the original 1st ed. 2004
Pages: 390
Product dimensions: 6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

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.- 2.1.1 Quadratic Reciprocity Law.- 2.1.2 Cyclotomic Version.- 2.1.3 Geometric Interpretation.- 2.1.4 Kronecker’s Reciprocity Law.- 2.1.5 Reciprocity Law for Elliptic Curves.- 2.2 Cyclotomic Reciprocity Laws and Adeles.- 2.2.1 Cyclotomic Fields.- 2.2.2 Cyclotomic Reciprocity Laws.- 2.2.3 Adelic Reformulation.- 2.3 A Generalization of Galois Theory.- 2.3.1 Infinite Galois Extensions.- 2.3.2 Automorphism Group of a Field.- 2.4 Algebraic Curves over a Field.- 2.4.1 Algebraic Function Fields.- 2.4.2 Zariski Topology.- 2.4.3 Divisors.- 2.4.4 Differentials.- 2.4.5 Adele Rings of Algebraic Function Fields.- 2.5 Elliptic Curves over a Field.- 2.5.1 Dimension Formulas.- 2.5.2 Weierstrass Equations of Elliptic Curves.- 2.5.3 Moduli of Weierstrass Type.- 2.5.4 Group Structure on Elliptic Curves.- 2.5.5 Abel’s Theorem.- 2.5.6 Torsion Points on Elliptic Curves.- 2.5.7 Classical Weierstrass Theory.- 2.6 Elliptic Modular Function Field.- 3 Modular Curves.- 3.1 Basics of Elliptic Curves over a Scheme.- 3.1.1 Definition of Elliptic Curves.- 3.1.2 Cartier Divisors.- 3.1.3 Picard Schemes.- 3.1.4 Invariant Differentials.- 3.1.5 Classification Functors.- 3.1.6 Cartier Duality.- 3.2 Moduli of Elliptic Curves and the Igusa Tower.- 3.2.1 Moduli of Level 1 over ? $$\left[ {1/6} \right]$$.- 3.2.2 Moduli of P?1(N).- 3.2.3 Action of?m.- 3.2.4 Compactification.- 3.2.5 Moduli of ?(N)-Level Structure.- 3.2.6 Hasse Invariant.- 3.2.7 Igusa Curves.- 3.2.8 Irreducibility of Igusa Curves.- 3.2.9 p-Adic Elliptic Modular Forms.- 3.3 p-Ordinary Elliptic Modular Forms.- 3.3.1 Axiomatic Treatment.- 3.3.2 Bounding the p-Ordinary Rank.- 3.3.3 p-Ordinary Projector.- 3.3.4 Families of p-Ordinary Modular Forms.- 3.4 Elliptic ?-Adic Forms and p-Adic L-functions.- 3.4.1 Generality of ?-Adic Forms.- 3.4.2 Some p-Adic L-Functions.- 4 Hilbert Modular Varieties.- 4.1 Hilbert–Blumenthal Moduli.- 4.1.1 Abelian Variety with Real Multiplication.- 4.1.2 Moduli Problems with Level Structure.- 4.1.3 Complex Analytic Hilbert Modular Forms.- 4.1.4 Toroidal Compactification.- 4.1.5 Tate Semi-Abelian Schemes with Real Multiplication.- 4.1.6 Hasse Invariant and Sheaves of Cusp Forms.- 4.1.7 p-Adic Hilbert Modular Forms of Level ?(N).- 4.1.8 Moduli Problem of ?11(N)-Type.- 4.1.9 p-Adic Modular Forms on PGL(2).- 4.1.10 Hecke Operators on Geometrie Modular Forms.- 4.2 Hilbert Modular Shimura Varieties.- 4.2.1 Abelian Varieties up to Isogenies.- 4.2.2 Global Reciprocity Law.- 4.2.3 Local Reciprocity Law.- 4.2.4 Hilbert Modular Igusa Towers.- 4.2.5 Hecke Operators as Algebraic Correspondences.- 4.2.6 Modular Line Bundles.- 4.2.7 Sheaves over the Shimura Variety of PGL(2).- 4.2.8 Hecke Algebra of Finite Level.- 4.2.9 Effect on q-Expansion.- 4.2.10 Adelic q-Expansion.- 4.2.11 Nearly Ordinary Hecke Algebra with Central Character.- 4.2.12 p-Adic Universal Hecke Algebra.- 4.3 Rank of p-Ordinary Cohomology Groups.- 4.3.1 Archimedean Automorphic Forms.- 4.3.2 Jacquet–Langlands–Shimizu Correspondence.- 4.3.3 Integral Correspondence.- 4.3.4 Eichler–Shimura Isomorphisms.- 4.3.5 Constant Dimensionality.- 4.4 Appendix: Fundamental Groups.- 4.4.1 Categorical Galois Theory.- 4.4.2 Algebraic Fundamental Groups.- 4.4.3 Group-Theoretic Results.- 5 Generalized Eichler–Shimura Map.- 5.1 Semi-Simplicity of Hecke Algebras.- 5.1.1 Jacquet Modules.- 5.1.2 Double Coset Algebras.- 5.1.3 Rational Representations of G.- 5.1.4 Nearly p-Ordinary Representations.- 5.1.5 Semi-Simplicity of Interior Cohomology Groups.- 5.2 Explicit Symmetric Domains.- 5.2.1 Hermitian Forms over ?.- 5.2.2 Symmetric Spaces of Unitary Groups.- 5.2.3 Invariant Measure.- 5.3 The Eichler–Shimura Map.- 5.3.1 Unitary Groups.- 5.3.2 Symplectic Groups.- 5.3.3 Hecke Equivariance.- 6 Moduli Schemes.- 6.1 Hilbert Schemes.- 6.1.1 Vector Bundles.- 6.1.2 Grassmannians.- 6.1.3 Flag Varieties.- 6.1.4 Flat Quotient Modules.- 6.1.5 Morphisms Between Schemes.- 6.1.6 Abelian Schemes.- 6.2 Quotients by PGL(n).- 6.2.1 Line Bundles on Projective Spaces.- 6.2.2 Automorphism Group of a Projective Space.- 6.2.3 Quotient of a Product of Projective Spaces.- 6.3 Mumford Moduli.- 6.3.1 Dual Abelian Scheme and Polarization.- 6.3.2 Moduli Problem.- 6.3.3 Abelian Scheme with Linear Rigidification.- 6.3.4 Embedding into the Hilbert Scheme.- 6.3.5 Conclusion.- 6.3.6 Smooth Toroidal Compactification.- 6.4 Siegel Modular Variety.- 6.4.1 Moduli Functors.- 6.4.2 Siegel Modular Reciprocity Law.- 6.4.3 Siegel Modular Igusa Tower.- 7 Shimura Varieties.- 7.1 PEL Moduli Varieties.- 7.1.1 Polarization, Endomorphism, and Lattice.- 7.1.2 Construction of the Moduli.- 7.1.3 Moduli Variety for Similitude Groups.- 7.1.4 Classification of G.- 7.1.5 Generic Fiber of Shk(p).- 7.2 General Shimura Varieties.- 7.2.1 Axioms Defining Shimura Varieties.- 7.2.2 Reciprocity Law at Special Points.- 7.2.3 Shimura’s Reciprocity Law.- 8 Ordinary p-Adic Automorphic Forms.- 8.1 True and False Automorphic Forms.- 8.1.1 An Axiomatic Igusa Tower.- 8.1.2 Rational Representation and Vector Bundles.- 8.1.3 Weight of Automorphic Forms and Representations.- 8.1.4 Density Theorems.- 8.1.5 p-Ordinary Automorphic Forms.- 8.1.6 Construction of the Projector eGL.- 8.1.7 Axiomatic Control Result.- 8.2 Deformation Theory of Serre and Tate.- 8.2.1 A Theorem of Drinfeld.- 8.2.2 A Theorem of Serre–Tate.- 8.2.3 Deformation of an Ordinary Abelian Variety.- 8.2.4 Symplectic Case.- 8.2.5 Unitary Case.- 8.3 Vertical Control Theorem.- 8.3.1 Hecke Operators on Deformation Space.- 8.3.2 Statements and Proof.- 8.4 Irreducibility of Igusa Towers.- 8.4.1 Irreducibility and p-Decomposition Groups.- 8.4.2 Closed Immersion into the Siegel Modular Variety.- 8.4.3 Description of a p-Decomposition Group.- 8.4.4 Irreducibility Theorem in Cases A and C.- References.- Symbol Index.- Statement Index.

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