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p-Adic Automorphic Forms on Shimura Varieties

Haruzo Hida
Publication Date: 
Number of Pages: 
Monographs in Mathematics
[Reviewed by
Fernando Q. Gouvêa
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In his introduction to this book, the author tells of how in 1980 he had the insight that the p-adic L-functions attached to automorphic forms on various different algebraic groups should be functions of several variables. For almost 25 years now, Hida has been working out that insight. This book is one of the fruits of that effort.

The idea is to study the "p-adic variation" of automorphic forms. Figuring out exactly what it means to vary an automorphic form (what exactly would the parameter be?) is part of the program. Hida's first breakthrough dealt with the case of elliptic modular forms, which correspond to the algebraic group GL2 over Q. He has since developed a general theory that (at least conjecturally) describes what happens over general algebraic groups.

This book, which developed from a course taught in France, 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, though I wouldn't want to read it from cover to cover. My only complaint is the funny system of bibliographic references that mixes codes that refer to titles (e.g., [CFT] for Artin and Tate on Class Field Theory), codes that refer to authors (e.g., [Co] for Coleman's fundamental paper on families of modular forms), and codes that are just strange (e.g., [Mt] for one of Miyake's papers). But that is only an annoyance in a book that will be a valuable resource for specialists in the field.

Fernando Q. Gouvêa is a number theorist who has done work related to p-adic modular forms.

Introduction.- Geometric Reciprocity Laws.- Modular Curves.- Hilbert Modular Varieties.- Generalized Eichler-Shimura Map.- Moduli Schemes.- Shimura Varieties.- p-Adic Automorphic Forms.- Bibliography.