### Synopses & Reviews

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.

#### Review

From the reviews: "The six chapters of this monograph give 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. First-year graduate students and researchers will benefit from this beautiful text." --Zentralblatt Math ". . . the present volume constitutes the most readable entree into the subject to date, suitable both for serious reading and for browsing, and should attract a new generation to this exciting subject. . . . Recommended." --CHOICE "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." (MAA REVIEWS)

#### Review

From the reviews:

"The six chapters of this monograph give 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. First-year graduate students and researchers will benefit from this beautiful text."

--Zentralblatt Math

". . . the present volume constitutes the most readable entree into the subject to date, suitable both for serious reading and for browsing, and should attract a new generation to this exciting subject. . . . Recommended."

--CHOICE

"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." (MAA REVIEWS)

#### Synopsis

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. This monograph gives 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. First-year graduate students and researchers will benefit from this beautiful presentation.

#### Synopsis

This book presents 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. Each of the twelve chapters focuses on a particular topic devoted to special cases of the program. The book is suitable for graduate students and researchers.

### Table of Contents

Preface * 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