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Modular Forms and Galois Cohomology

Haruzo Hida
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Studies in Advanced Mathematics 69
[Reviewed by
Fernando Q. Gouvêa
, on

By now, everyone knows that Wiles' proof of Fermat's Last Theorem is based on the theories of elliptic curves and of modular forms. What most people don't know is that after Ribet's 1991 work which established the connection between the modularity conjecture and Fermat's Last Theorem, most of the work that remained to be done (and that Wiles and Taylor did) had to do with the connection between modular forms and Galois representations. In this book, Hida explores exactly this fundamental connection between modular forms and Galois representations (including an account of Wiles' crucial theorem right in the middle of the book). Along the way, he explores both the theory of Galois representations and the cohomology theory of Galois groups. The final chapter, on "Modular L-values and Selmer Groups," includes a number of Hida's own results. The book is based on a course taught by Hida at UCLA, and it looks to be a good source for people with the appropriate background who want to learn this material. 

Fernando Q. Gouvêa ( is the editor of MAA Online. He teaches both "History of Mathematics" and "Number Theory", among others, at Colby College. He is a number theorist whose main research focus is on p-adic modular forms and Galois representations.

Preface; 1. Overview of modular forms; 2. Representations of a group; 3. Representations and modular forms; 4. Galois cohomology; 5. Modular L-values and Selmer groups; Bibliography; Subject index; List of statements; List of symbols.