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.