ISBN-10:
1441919236
ISBN-13:
9781441919236
Pub. Date:
10/07/2011
Publisher:
Springer New York
p-Adic Automorphic Forms on Shimura Varieties / Edition 1

p-Adic Automorphic Forms on Shimura Varieties / Edition 1

by Haruzo Hida
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Overview

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.

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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