P-Adic automorphic forms on Shimura varieties

Includes index

Includes bibliography (p. 375-382) and references

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

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: an elementary construction of Shimura varieties as moduli of abelian schemes; p-adic deformation theory of automorphic forms on Shimura varieties; and a simple proof of irreducibility of the generalized Igusa tower over the Shimura variety.