Exponential sums for \(GL(n)\) and their applications to base change
type: Item
instance of: scholarly article
title: Exponential sums for \(GL(n)\) and their applications to base change
zbMATH Open document ID: 0922.11066
author: Yangbo Ye
published in: Journal of Number Theory
publication date: 1999-10-03T00:00:00Z
review text: This article gives an identity for \(GL(n)\) Kloosterman sums which is related to the relative trace formula. To put this in context, \textit{H. Jacquet} and \textit{Y. Ye} [C. R. Acad. Sci., Paris, Sér. I 311, No. 11, 671-676 (1990; Zbl 0715.11026)] have conjectured a relative trace formula for \(GL(n)\), which characterizes the automorphic representations of \(GL(n)\) which are quadratic base changes as being distinguished with respect to a quasi-split unitary group. This formula should be proved locally. On the one hand, one has Kloosterman sums or local Kloosterman integrals for \(GL(n)\), as in the work of \textit{D. Bump, S. Friedberg} and \textit{D. Goldfeld} [Acta Arith. 50, No. 1, 31-89 (1988; Zbl 0647.10020)], \textit{S. Friedberg} [Math. Z. 196, 165-188 (1987; Zbl 0612.10020)], and \textit{G. Stevens} [Math. Ann. 277, 25-51 (1987; Zbl 0597.12017)]. On the other, there are certain naturally occuring exponential sums (which should perhaps be called relative Kloosterman sums) associated with \(GL(n)\) and a quadratic extension. The fundamental lemma for the relative trace formula states that these sums should be equal, up to a transfer factor. The author proves such a matching for one set of exponential sums, corresponding to the Weyl group element represented by the permutation matrix with the \((n-1)\times(n-1)\) anti-identity matrix in the upper left corner. (In this case the \(GL(n)\) Kloosterman sum reduces to the classical hyper-Kloosterman sum.) When \(n=3\), this identity was previously established by the author in [J. Number Theory 51, No. 2, 275-287 (1995; Zbl 0822.11057)]. The relative trace formula also requires proving a matching of Shalika germs, formulated for \(GL(n)\) by \textit{H. Jacquet} and the author in [Trans. Am. Math. Soc. 348, No. 3, 913-939 (1996; Zbl 0861.11033)]. As a consequence of the matching of exponential sums established here, the author deduces one case of such a matching of Shalika germs.
Mathematics Subject Classification ID: 11L05
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 1146323
zbMATH Keywords: Kloosterman sum
zbMATH Keywords: base change
zbMATH Keywords: relative trace formula
zbMATH Keywords: Shalika germ
zbMATH Keywords: automorphic representations
zbMATH Keywords: exponential sums
zbMATH Keywords: Weyl group
reviewed by: Solomon Friedberg
MaRDI profile type: Publication
full work available at URL: jnth.1997.2196
OpenAlex ID: W2091481400
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DOI: 10.1006/JNTH.1997.2196
description: scientific article; zbMATH DE number 1146323