# An Introduction to the Langlands Program

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For the past several decades the theory of automorphic forms has become a major focal point of development in number theory and algebraic geometry, with applications in many diverse areas, including combinatorics and mathematical physics.

The twelve chapters of this monograph present a broad, user-friendly introduction to the Langlands program, that is, the theory of automorphic forms and its connection with the theory of L-functions and other fields of mathematics.

*Key features of this self-contained presentation:*

A variety of areas in number theory from the classical zeta function up to the Langlands program are covered.

The exposition is systematic, with each chapter focusing on a particular topic devoted to special cases of the program:

• Basic zeta function of Riemann and its generalizations to Dirichlet and Hecke L-functions, class field theory and some topics on classical automorphic functions** (E. Kowalski)**

• A study of the conjectures of Artin and Shimura–Taniyama–Weil **(E. de Shalit)**

• An examination of classical modular (automorphic) L-functions as GL(2) functions, bringing into play the theory of representations **(S.S. Kudla)**

• Selberg's theory of the trace formula, which is a way to study automorphic representations **(D. Bump)**

• Discussion of cuspidal automorphic representations of GL(2,(A)) leads to Langlands theory for GL(n) and the importance of the Langlands dual group **(J.W. Cogdell)**

• An introduction to the geometric Langlands program, a new and active area of research that permits using powerful methods of algebraic geometry to construct automorphic sheaves **(D. Gaitsgory)**

Graduate students and researchers will benefit from this beautiful text.

Grad algebra algebraic geometry elliptic curve modular form number theory zeta function

- DOI https://doi.org/10.1007/978-0-8176-8226-2
- Copyright Information Birkhäuser Boston 2004
- Publisher Name Birkhäuser, Boston, MA
- eBook Packages Springer Book Archive
- Print ISBN 978-0-8176-3211-3
- Online ISBN 978-0-8176-8226-2
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