An introduction to the Langlands program

de Daniel Bump, Joseph Bernstein, Stephen S. Gelbart

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.