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An introduction to the Langlands program. Lectures presented at the Hebrew University of Jerusalem, Jerusalem, Israel, March 12–16, 2001. (English) Zbl 1112.11028
Boston, MA: Birkhäuser. viii, 281 p. (2003).

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Contents: E. Kowalski, Elementary theory of \(L\)-functions. I (1–20); E. Kowalski, Elementary theory of \(L\)-functions. II (21–37); E. Kowalski, Classical automorphic forms (39–71); Ehud de Shalit, Artin \(L\) functions (73–87); Ehud de Shalit, \(L\)-functions of elliptic curves and modular forms (89–108); Stephen S. Kudla, Tate’s thesis (109–131); Stephen S. Kudla, From modular forms to automorphic representations (133–151); Daniel Bump, Spectral theory and the trace formula (153–196); J. W. Cogdell, Analytic theory of \(L\)-functions for \(\text{GL}_n\) (197–228); J. W. Cogdell, Langlands conjectures for \(\text{GL}_n\) (229–249); J. W. Cogdell, Dual groups and Langlands functoriality (251–268); D. Gaitsgory, Informal introduction to geometric Langlands (269–281).
The 12 chapters presented in this book are based on lectures that were given in the School of Mathematics of the Institute for Advanced Studies at the Hebrew University of Jerusalem, March 12–16, 2001. They 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.
The contribution of each of the lecturers and their articles may now be briefly described as follows.
The first eight chapters are devoted to the case of \(\text{GL}(1)\) and \(\text{GL}(2)\): E. Kowalski classically focuses on the basic zeta-function of Riemann and its generalizations to Dirichlet and Hecke \(L\)-functions, class field theory, and a selection of topics devoted to classical automorphic functions; E. de Shalit carefully surveys the conjectures of Artin and Shimura-Taniyama-Weil.
After discussing Hecke’s \(L\)-functions, S. Kudla examines classical modular (automorphic) \(L\)-functions as \(\text{GL}(2)\) ones, thereby bringing into play the theory of representations. One way to study those representations which are “automorphic” is via Selberg’s theory of the “trace formula”; this is introduced in B. Bump’s chapter.
The last four chapters, by J. Cogdell and D. Gaitsgory, are more abstract. After starting with discussion of cuspidal automorphic representations of \(\text{GL}(2, \mathbb A)\), Cogdell quickly gets to Langlands’ theory for \(\text{GL}(2, \mathbb A)\); then he explains why one needs the Langlands’ dual group in order to formulate the general conjectures for a reductive group \(G\) different from \(\text{GL}(n)\).
Gaitsgory gives an informal introduction to the geometric Langlands program. This is a new and very active area of research which grew out of the theory of automorphic forms and is closely related to it. Roughly speaking, in this theory we everywhere replace functions – like automorphic forms – by sheaves on algebraic varieties; this allows us to use powerful methods of algebraic geometry in order to construct “automorphic sheaves”.

MSC:
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11-06 Proceedings, conferences, collections, etc. pertaining to number theory
00B25 Proceedings of conferences of miscellaneous specific interest
11F12 Automorphic forms, one variable
11G09 Drinfel’d modules; higher-dimensional motives, etc.
11R39 Langlands-Weil conjectures, nonabelian class field theory
11R58 Arithmetic theory of algebraic function fields
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