\(l\)-adic representations associated to modular forms over imaginary quadratic fields. I: Lifting to \(GSp_ 4(\mathbb{Q})\)
type: Item
instance of: scholarly article
title: \(l\)-adic representations associated to modular forms over imaginary quadratic fields. I: Lifting to \(GSp_ 4(\mathbb{Q})\)
zbMATH Open document ID: 0797.11047
DOI: 10.1007/BF01232440
author: Michael Harris
author: David Soudry
author: Richard Taylor
published in: Inventiones Mathematicae
publication date: 1994-04-13T00:00:00Z
full work available at URL: https://eudml.org/doc/144109
review text: This paper is part of a series which establishes an important new link between automorphic representations and arithmetic. Namely this paper, together with the subsequent papers of \textit{R. Taylor} [On the \(l\)-adic cohomology of Siegel threefolds, Invent. Math. 114, 289-310 (1993; Zbl 0810.11034); \(l\)- adic representations associated to modular forms over imaginary quadratic fields, II; Invent. Math. 116, 619-643 (1994; Zbl 0823.11020)]shows how one may associate a \(\lambda\)-adic representation to certain modular forms over an imaginary quadratic field. By way of background, recall that if \(f\) is a holomorphic elliptic modular newform then one can associate to \(f\) a compatible system of \(\lambda\)-adic representations, obtained from the cohomology of certain sheaves on modular curves. Similar statements are true for Hilbert modular forms. However, if \(g\) is in the space of an automorphic representation over an imaginary quadratic field, no such arithmetic object had previously been found. Indeed, the corresponding quotient of hyperbolic three-space is not an algebraic variety, and no link to arithmetic is apparent. The authors establish such a link by using the theta correspondence to associate to \(g\) a holomorphic Siegel modular form \(G\) in the same near- equivalence class. This paper is concerned with the details of this correspondence, and the subsequent papers of Taylor with applying the methods of algebraic geometry to \(G\) to attach to it a compatible system of \(\lambda\)-adic representations. In the paper at hand, the authors first observe that \(GL_ 2\) over an imaginary quadratic field \(K\) is closely related to a four-variable group of orthogonal similitudes \(GO\) over \(\mathbb{Q}\). (This similitude group is disconnected, and the disconnectedness plays an important role.) There is a theta lifting of automorphic representations on \(GO\) to automorphic representations on \(GSp_ 4\). After establishing some generalities on theta lift…
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11F46
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11R39
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 22E55
Mathematics Subject Classification ID: 11G40
zbMATH DE Number: 447108
zbMATH Keywords: \(l\)-adic representation
zbMATH Keywords: theta series
zbMATH Keywords: \(l\)-adic cohomology of Siegel three-folds
zbMATH Keywords: automorphic representations
zbMATH Keywords: modular forms
zbMATH Keywords: theta correspondence
zbMATH Keywords: holomorphic Siegel modular form
zbMATH Keywords: theta lifting
reviewed by: Solomon Friedberg
MaRDI profile type: Publication
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OpenAlex ID: W3160345189
description: scientific article; zbMATH DE number 447108