Introduction

Summary

The theory of L-functions is an old subject with a long history. In the 1940s Hecke and Maass rewrote the classical theory in the setting of automorphic forms, and it seemed as if the theory of L-functions had settled into a fairly final form. This view was effectively overturned with the publication of two major books: [Gelfand-Graev-Piatetski-Shapiro, 1969], [Jacquet-Langlands, 1970], where it was shown that the theory of L-functions could be recast in the language of infinite dimensional complex representations of reductive groups.

Another milestone in the recent theory of L-functions was the book by Roger Godement and Hervé Jacquet, [Godement-Jacquet, 1972], which defined for the first time the standard L-functions attached to automorphic representations of the general linear group, and proved their key properties by generalizing the seminal ideas of [Tate, 1950], [Iwasawa, 1952, 1992]. The proofs in [Godement-Jacquet, 1972] made fundamental use of matrix coefficients associated to automorphic representations. The standard L-functions of the general linear group are often called Godement-Jacquet L-functions. Although several other techniques have since been discovered to obtain the main analytic properties of such L-functions, none is more beautiful and elegant than the method of matrix coefficients, originally devised by Godement and Jacquet, which is a major theme of this book.

Modern research in the theory of automorphic representations and L-functions is largely focused in the direction of the Langlands program.