The twisted endoscopy of \(GL(4)\) and \(GL(5)\): Transfer of Shalika germs
type: Item
instance of: scholarly article
title: The twisted endoscopy of \(GL(4)\) and \(GL(5)\): Transfer of Shalika germs
zbMATH Open document ID: 0853.22013
DOI: 10.1215/S0012-7094-94-07621-7
author: Thomas C. Hales
published in: Duke Mathematical Journal
publication date: 1997-01-13T00:00:00Z
review text: Let \(F\) be a \(p\)-adic field of characteristic zero and of odd residual characteristic. Over the field \(F\) the author obtains a necessary statement about a matching of Shalika germs at the identity for the groups \(GL(4)\) and \(GL(5)\): each stable Shalika germ on \({\mathfrak s} o(5)\) is a linear combination of Shalika germs on \({\mathfrak s} p(4)\), and, conversely, each stable Shalika germ on \({\mathfrak s} p(4)\) is a linear combination of Shalika germs on \({\mathfrak s} o(5)\). This matching of stable Shalika germs at the identity implies the transfer of twisted stable Shalika germs from \(GL(4)\) and \(GL(5)\) to the endoscopic groups \(SO(5)\) and \(Sp(4)\), and, coupled with an expected theory of descent, it would imply the transfer of twisted orbital integrals on the groups to stable orbital integrals on the endoscopic groups. In order to obtain the matching the author finds that the stable Shalika germs of the twisted orbital integrals on \(GL(4)\) and \(GL(5)\) and the stable orbital integrals on \(SO(5)\) and \(Sp(4)\) are expressed by the number of points on a family of elliptic curves, and then he obtains the relations between the families of elliptic curves, that implies the matching. These calculations give a complete formula for the Shalika germs on \(Sp(4)\), especially, the analysis of two-regular conjugacy class of \(Sp(4)\) gives a proof of the transfer of orbital integrals on \(Sp(4)\) to its endoscopic groups.
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 727837
zbMATH Keywords: \(p\)-adic field
zbMATH Keywords: endoscopic groups
zbMATH Keywords: orbital integrals
zbMATH Keywords: elliptic curves
reviewed by: Takeshi Kawazoe
MaRDI profile type: Publication
cites work: The local behaviour of weighted orbital integrals
cites work: Q3682663
cites work: On the Degree of Igusa's Local Zeta Function
cites work: Q3475501
cites work: The subregular germ of orbital integrals
cites work: Q3871104
cites work: Q3741682
cites work: Fixed point varieties on affine flag manifolds
cites work: Cuspidal geometry of p-adic groups
cites work: Stable trace formula: Elliptic singular terms
cites work: Q4243095
cites work: Orbital Integrals on Forms of SL(3), I
cites work: Q3322323
cites work: Q3972743
cites work: Q3924412
cites work: Profinite Groups, Arithmetic, and Geometry. (AM-67)
cites work: Q3202317
cites work: Q3329671
full work available at URL: s0012-7094-94-07621-7
OpenAlex ID: W1513280919
description: scientific article; zbMATH DE number 727837