p-Adic Automorphic Forms on Shimura Varieties

The book starts with a detailed study of elliptic and Hilbert modular forms and reaches to the forefront of research. 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.

Includes supplementary material: sn.pub/extras

Texte du rabat

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).

Résumé
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.

Contenu
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 HilbertBlumenthal Moduli.- 4.2 Hilbert Modular Shimura Varieties.- 4.3 Rank of p-Ordinary Cohomology Groups.- 4.4 Appendix: Fundamental Groups.- 5 Generalized EichlerShimura Map.- 5.1 Semi-Simplicity of Hecke Algebras.- 5.2 Explicit Symmetric Domains.- 5.3 The EichlerShimura 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.