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An Introduction to the Langlands Program

J. Bernstein and S. Gelbart, editors
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
, on

Any graduate student in number theory quickly picks up two important bits of information about the "Langlands Program". The first is that Langlands' amazing vision of where number theory should go is one of the driving ideas in subject. In other words, this is important stuff. The second bit of information is that this material is hard. The pre-requisites themselves are daunting: Langlands' conjectures bring together algebraic number theory, the theory of automorphic forms, representation theory, algebraic groups, and probably some stuff that I've forgotten. However important, this is hard stuff to learn.

As a result, while most number theorists have at least a general feeling for what the Langlands Program is all about, few have the persistence to really learn about it, and even fewer penetrate far enough into the subject to be able to contribute to it. The goal of this book is to make entering into the subject a little easier.

The essays collected here are not broad surveys, aimed at a general mathematical public. Instead, they try to open up the basic issues for students and mathematicians who know some number theory. The editors highlight the fact that the book does not attempt a "complete formal introduction" to the subject; that would have taken a much bigger book. Instead, they offer the authors' personal "take" on the subject, an informal introduction. That, of course, makes them more rather than less useful. I suspect this book will find its way into the hands of many graduate students. Perhaps it will also motivate a few of them to learn more, get involved, and make their own contributions.

Fernando Q. Gouvêa learned a little bit about the Langlands Program as a graduate student, but not enough.

E. Kowalski - Elementary Theory of L-Functions I
E. Kowalski - Elementary Theory of L-Functions II
E. Kowalski - Classical Automorphic Forms
E. DeShalit - Artin L-Functions
E. DeShalit - L-Functions of Elliptic Curves and Modular Forms
S. Kudla - Tate's Thesis
S. Kudla - From Modular Forms to Automorphic Representations
D. Bump - Spectral Theory and the Trace Formula
J. Cogdell - Analytic Theory of L-Functions for GLn
J. Cogdell - Langlands Conjectures for GLn
J. Cogdell - Dual Groups and Langlands Functoriality
D. Gaitsgory - Informal Introduction to Geometric Langlands