Shintani function and its application to automorphic \(L\)-functions for classical groups. I: The case of orthogonal groups
type: Item
instance of: scholarly article
title: Shintani function and its application to automorphic \(L\)-functions for classical groups. I: The case of orthogonal groups
zbMATH Open document ID: 0813.11032
DOI: 10.1007/BF01459771
author: Atsushi Murase
author: Takashi Sugano
published in: Mathematische Annalen
publication date: 1994-07-11T00:00:00Z
full work available at URL: https://eudml.org/doc/165197
review text: Let \(H\) be the orthogonal group of a quadratic form over \(\mathbb{Q}\) which is positive definite over \(\mathbb{R}\) and \(G\) the isotropy subgroup of some nonzero vector. If \(F\) (resp. \(f)\) is an automorphic form on \(H(\mathbb{Q}) \backslash H (\mathbb{A})\) (resp. \(G(\mathbb{Q}) \backslash G (\mathbb{A}))\), the inner product of \(f\) with the restriction of \(\rho (h)F\) to \(G(\mathbb{Q}) \backslash G (\mathbb{A})\) is called Shintani function by the authors. It appears in the ``basic identity'' for a certain Rankin-Selberg type zeta- integral. Assume \(F\) and \(f\) are invariant under \(H(\mathbb{R})\) resp. \(G(\mathbb{R})\) and are eigenfunctions of the Hecke algebras. Then the zeta- integral in question is shown to be (essentially) the quotient of two standard \(L\)-functions for \(H\) and \(G\). This is dense by explicitly computing the local zeta-integrals. The result is proved under cerrtain assumptions on the quadratic form. Conjectures concerning the functional equation and the poles of the \(L\)-functions considered here are formulated and partially proved.
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11F55
zbMATH DE Number: 590001
zbMATH Keywords: automorphic \(L\)-functions
zbMATH Keywords: orthogonal group
zbMATH Keywords: automorphic form
zbMATH Keywords: Shintani function
zbMATH Keywords: Rankin-Selberg type zeta-integral
zbMATH Keywords: functional equation
zbMATH Keywords: poles
reviewed by: J. G. M. Mars
MaRDI profile type: Publication
cites work: Ein Rationalitätssatz für formale Heckereihen zur Siegelschen Modulgruppe. (A rationality theorem for formal Hecke series related to the Siegel modular group)
cites work: Q5816776
cites work: On L-functions for GSp(4) X GL(2) and their special values.
cites work: Q4027534
cites work: Explicit constructions of automorphic \(L\)-functions
cites work: Q5638492
cites work: On the functional equations satisfied by Eisenstein series
cites work: Q3356365
cites work: Q4388148
cites work: ε factor of representations of classical groups
cites work: 𝐿 functions for the group 𝐺₂
cites work: Theory of spherical functions on reductive algebraic groups over p-adic fields
cites work: Q4859137
OpenAlex ID: W1250622352
description: scientific article; zbMATH DE number 590001