Critical values of automorphic \(L\)-functions for \(\text{GL}(r)\times\text{GL}(r)\)
type: Item
instance of: scholarly article
title: Critical values of automorphic \(L\)-functions for \(\text{GL}(r)\times\text{GL}(r)\)
zbMATH Open document ID: 1027.11032
author: Loïc Grenié
published in: Manuscripta Mathematica
publication date: 2003-07-02T00:00:00Z
review text: Let \(F\) be a totally imaginary number field and let \(G = \text{GL}_r\). Let \(\pi\) and \(\pi'\) be two irreducible cuspidal representations of \(G(\mathbb A)\). There is a well-known Rankin-Selberg integral for the L-function of \(\pi\times\pi'\), which involves an Eisenstein series attached to a maximal parabolic subgroup of \(G\). When \(\pi\) and \(\pi'\) are cohomological (i.e. contribute to the cohomology of \(G(F)\setminus G(\mathbb A)/K\)) and satisfy a condition of compatibility, the special values of the \(L\)-function can be computed by interpreting the Rankin-Selberg integral as a cup product of differential forms. This is done in the paper. The result is an expression for \(L_f(\pi\times\pi',s)\) as a product of periods for the three representations involved, divided by an Archimedean integral, times a factor from a finite extension of \(F\) determined by \(\pi\) and \(\pi'\). Rationality of cohomology was known for the cuspidal representations and is proved here for the Eisenstein cohomology. It is not proved that the Archimedean integral in the final formula does not vanish.
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11F75
zbMATH DE Number: 1945276
reviewed by: J. G. M. Mars
MaRDI profile type: Publication
full work available at URL: s00229-002-0333-5
OpenAlex ID: W2135521445
DOI: 10.1007/S00229-002-0333-5
description: scientific article; zbMATH DE number 1945276