On Shimura's correspondence
type: Item
instance of: scholarly article
title: On Shimura's correspondence
zbMATH Open document ID: 0631.10015
DOI: 10.1215/S0012-7094-87-05533-5
author: Daniel Bump
author: Jeff Hoffstein
published in: Duke Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: The authors make an important contribution to the theory of the generalized Shimura correspondence for GL(2) which was proved using the Selberg trace formula by Y. Z. Flicker. They show that if \(\phi\) is an automorphic form on an n-fold cover of GL(2) over a number field then this corresponds to an automorphic form \(\phi\) on GL(2). To \(\phi\) one can associate an L-function. The crux of the author's method is that they show that this L-function can also be represented essentially as a global integral of a type introduced by H. Jacquet, and that the analytic properties of the L-function can be deduced from this representation. The integral involves the metaplectic theta function on the n-fold cover of GL(n). Central to this approach is a very interesting calculation at the unramified places. The results are almost, but not quite enough to give an ``L-function'' proof of the Shimura correspondence in the case of GL(2).
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11R42
zbMATH DE Number: 4025512
zbMATH Keywords: generalized Shimura correspondence for GL(2)
zbMATH Keywords: automorphic form
zbMATH Keywords: L- function
zbMATH Keywords: metaplectic theta function
zbMATH Keywords: unramified places
reviewed by: Samuel James Patterson
MaRDI profile type: Publication
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full work available at URL: s0012-7094-87-05533-5
OpenAlex ID: W2084425420
description: scientific article; zbMATH DE number 4025512