Godement–Jacquet

Godement–Jacquet
Name	Godement–Jacquet
Authors	Roger Godement; Hervé Jacquet
Field	Number theory; Representation theory; Automorphic forms
First published	1972
Notable for	Zeta integrals for GL_n; standard L-function; functional equation; Rankin–Selberg convolution

Godement–Jacquet

Godement–Jacquet refers to the foundational work of Roger Godement and Hervé Jacquet on zeta integrals and L-functions for GL_n representations, establishing analytic continuation and functional equations that connect to the theory of adelic automorphic representations. Their methods bridge harmonic analysis on adèles, representation theory of GL_n, and analytic properties of L-functions, influencing later developments such as the Langlands program, Rankin–Selberg theory, and converse theorems. The approach introduced zeta integrals built from matrix coefficients and Schwartz–Bruhat functions, yielding local factors at places of global fields and global functional equations for standard L-functions.

Introduction

Godement–Jacquet arose from work by Roger Godement and Hervé Jacquet on analytic properties of L-functions attached to automorphic representations of GL_n(A), where A denotes the adèle ring of a global field; this work directly influenced research by Jacquet–Langlands, Langlands, Shahidi, Piatetski-Shapiro, and Bump. The monograph developed zeta integrals using spherical functions and matrix coefficients from representations of GL_n(F), with local theory at nonarchimedean places tied to Casselman and Bernstein–Zelevinsky results and archimedean analysis related to Harish-Chandra and Wallach. The framework provided tools later used by Gelbart, Jacquet, Shalika, Cogdell, Piatetski-Shapiro, and Mœglin in converse theorems and classification problems.

Godement–Jacquet Zeta Integral and L-function

The Godement–Jacquet zeta integral pairs a matrix coefficient of an admissible representation of GL_n(F) with a Schwartz–Bruhat function on the space of n×n matrices, generalizing Tate's thesis on Hecke L-function and connecting with the Rankin–Selberg integrals studied by Rankin and Selberg. For a global automorphic representation of GL_n(A), the product of local zeta integrals defines the standard L-function, compatible with local factors studied by Deligne, Weil, Tate, and Iwasawa. Their construction produces an Euler product whose unramified local integrals match the characteristic polynomial of Frobenius in the Satake parameterization appearing in works by Satake, Kottwitz, and Langlands.

Local Theory and Factors

Godement–Jacquet developed a local theory at each place of a global field, producing epsilon and gamma factors paralleling the local constants of Tate and the local Langlands correspondence formalized by Harris–Taylor and Henniart. At nonarchimedean places the zeta integral uses compactly supported matrix coefficients and Bruhat–Schwartz functions on M_n(F_v), invoking results of Bernstein and Zelevinsky on admissible representations and the theory of spherical functions by Macdonald. At archimedean places analytic continuation and meromorphicity rely on techniques from Harish-Chandra’s Plancherel theory and the asymptotic expansions used by Langlands and Shalika (1970s). The identification of the unramified local factor with the reciprocal of the characteristic polynomial determinant follows the Satake isomorphism and computations akin to their original calculations.

Global Theory and Automorphic Representations

Globally, the Godement–Jacquet zeta integral attaches a completed L-function to a cuspidal automorphic representation of GL_n(A), functioning within the adelic framework popularized by Weil and used extensively by Langlands. The global integral is constructed from automorphic forms in the space of a cuspidal representation and Schwartz–Bruhat functions on M_n(A), with convergence and analytic continuation proved via Eisenstein series techniques related to Piatetski-Shapiro and constant term computations akin to Langlands' Eisenstein series. This theory interfaces with the classification of automorphic representations by Moeglin–Waldspurger and the trace formula developed by Selberg and advanced by Arthur.

Functional Equation and Converse Theorem

Godement–Jacquet established a functional equation for the completed standard L-function, paralleling the functional equations in Tate and later used in converse theorems by Jacquet–Piatetski-Shapiro–Shalika and Cogdell–Piatetski-Shapiro. The functional equation involves a global epsilon factor factoring as a product of local epsilon factors, a phenomenon central to the Langlands functoriality conjectures and to the compatibility of the global Langlands correspondence with local reciprocity results proven by Harris, Taylor, and Henniart. Converse theorems utilizing Godement–Jacquet L-functions have been instrumental in proving automorphy lifting results and cases of functorial transfer examined by Kim–Shahidi and Cogdell–Piatetski-Shapiro.

Examples and Applications

Applications include recovery of classical Hecke L-functions via the n=1 case following Tate (1950s), the Rankin–Selberg L-function comparisons for GL_2 by Jacquet–Langlands, and input to symmetric power L-functions studied by Gelbart–Jacquet and Kim. The method gives explicit local factors used in numerical computations influenced by Odlyzko and in nonvanishing results related to central value conjectures of Birch–Swinnerton-Dyer in special cases via periods studied by Waldspurger and Gross–Zagier. Godement–Jacquet techniques inform the analysis of newforms and local newvector theory linked to Atkin–Lehner and Casselman as well as the study of cohomology of arithmetic manifolds pursued by Harder, Borel, and Franke.

Historical Development and Impact

Published in the early 1970s, the Godement–Jacquet monograph consolidated analytic and representation-theoretic methods, influencing the development of the modern Langlands program articulated by Langlands and subsequent work by Shahidi, Arthur, Mœglin–Waldspurger, and others. Its ideas seeded research on converse theorems by Piatetski-Shapiro, Cogdell, and Soudry, and on local-global compatibility culminating in results by Harris–Taylor and Henniart. The framework continues to underpin advances in automorphic L-functions, functoriality conjectures, and arithmetic applications pursued by contemporary researchers including Buzzard, Taylor, Clozel, Flicker, and Venkatesh.

Category:Automorphic forms