p-Adic Automorphic Forms on Shimura Varieties

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.

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.

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 deformationtheory 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