P-Adic Automorphic Forms on Shimura Varieties

ISBN-13: 9780387207117 / Angielski / Twarda / 2004 / 404 str.

Haruzo Hida

Opis

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.

Recenzje

From the reviews:

"Hida views ... the study of the geometric Galois group of the Shimura tower, as a geometric reciprocity law ... . general goal of the book is to incorporate Shimura's reciprocity law in a broader scheme of integral reciprocity laws which includes Iwasawa theory in its scope. ... a beautiful and very useful reference for anybody interested in the arithmetic theory of automorphic forms." (Jacques Tilouine, Mathematical Reviews, 2005e)

"The first purpose of this book is to supply the base of the construction of the Shimura variety. The second one is to introduce integrality of automorphic forms on such varieties ... . The mathematics discussed here is wonderful but highly nontrivial. ... The book will certainly be useful to graduate students and researchers entering this beautiful and difficult area of research." (Andrzej Dabrowski, Zentralblatt MATH, Vol. 1055, 2005)

"The purpose of this book is twofold: First to establish a p-adic deformation theory of automorphic forms on Shimura varieties; this is recent work of the author. Second, to explain some of the necessary background, in particular the theory of moduli and Shimura varieties of PEL type ... . The book requires some familiarity with algebraic number theory and algebraic geometry (schemes) but is rather complete in the details. Thus, it may also serve as an introduction to Shimura varieties as well as their deformation theory." (J. Mahnkopf, Monatshefte für Mathematik, Vol. 146 (4), 2005)

"The idea is to study the 'p-adic variation' of automorphic forms. ... This book ... is a high-level exposition of the theory for automorphic forms on Shimura Varieties. It includes a discussion of the special cases of elliptic modular forms and Hilbert modular forms, so it will be a useful resource for those wanting to learn the subject. The exposition is very dense, however, and the prerequisites are extensive. Overall, this is a book I am happy to have on my shelves ... ." (Fernando Q. Gouvêa, Math DL, January, 2004)

"Hida ... showed that ordinary p-adic modular forms moved naturally in p-adic families. ... In the book under review ... Hida has returned to the geometric construction of p-adic families of ordinary forms. ... Hida's theory has had many applications in the theory of classical modular forms, and as mathematics continues to mature, this more general theory will no doubt have similarly striking applications in the theory of automorphic forms." (K. Buzzard, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 109 (4), 2007)

Spis treści

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.

Blurb

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).