\(l\)-adic representations associated to modular forms over imaginary quadratic fields. II
type: Item
instance of: scholarly article
title: \(l\)-adic representations associated to modular forms over imaginary quadratic fields. II
zbMATH Open document ID: 0823.11020
DOI: 10.1007/BF01231575
published in: Inventiones Mathematicae
publication date: 1995-11-06T00:00:00Z
full work available at URL: https://eudml.org/doc/144202
review text: This paper is part of a series which establishes an important new link between automorphic representations and arithmetic, by associating a system of \(\lambda\)-adic representations to certain modular forms over an imaginary quadratic field \(M\). [For more background, see the review of Part I: the author, \textit{D. Soudry} and \textit{M. Harris}, Invent. Math. 112, 377-411 (1993; Zbl 0797.11047).] Let \(\pi\) be a cuspidal automorphic representation of \(GL(2, A_ M)\) which is regular algebraic up to twist. Thus, \(\pi\) contributes to the cohomology of the standard local system on a three manifold obtained as the quotient of hyperbolic three-space by a congruence subgroup \(GL(2, {\mathcal O}_ M)\), or equivalently \(\pi_ \infty\) has Langlands parameter \(z\mapsto \left( \begin{smallmatrix} z^{1-k} &0\\ 0 &\overline {z}^{k-1} \end{smallmatrix} \right)\), with \(k\geq 2\) an integer. Suppose further that the central character of \(\pi\) is Galois-invariant, and \(k\) is even. For good places \(v\), let \(\{ \alpha_ v, \beta_ v\}\) denote the Langlands parameters of \(\pi_ v\), and let \(F_ \pi\) denote the field generated by the \(\alpha_ v+ \beta_ v\) and \(\alpha_ v \beta_ v\) for all good places \(v\); \(F_ \pi\) is a number field. Then the main theorem of the author states that there is an extension \(E/ F_ \pi\) of degree at most four and for each prime \(\lambda\) of \(F_ \pi\) there is a continuous irreducible representation \(\rho\) of \(\text{Gal} (\overline {M}/ M)\) to \(GL_ 2 (E_{\lambda'} )\) (\(\lambda'\) a prime of \(E\) over \(\lambda\)) such that on the complement of a set of Dirichlet density zero the characteristic polynomial of \(\rho (\text{Frob}_ v)\) is \((X- \alpha_ v) (X-\beta_ v)\). The proof of this result uses the work of Harris, Soudry, and the author (loc. cit.) to associate to \(\pi\) and its twists a family of holomorphic Siegel modular forms of genus 2. The forms obtained are of low weight and do not contribute to the \(l\)-adic …
reviewed by: Solomon Friedberg
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11F33
Mathematics Subject Classification ID: 11F75
Mathematics Subject Classification ID: 11F46
Mathematics Subject Classification ID: 11F55
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 14G20
zbMATH DE Number: 554031
zbMATH Keywords: cohomology of hyperbolic three-manifolds
zbMATH Keywords: automorphic representations
zbMATH Keywords: holomorphic Siegel modular forms
zbMATH Keywords: \(l\)-adic representations
zbMATH Keywords: elliptic curves over imaginary quadratic fields
zbMATH Keywords: Tate module
zbMATH Keywords: Ramanujan conjecture
zbMATH Keywords: \(L\)-function
MaRDI profile type: Publication
cites work: Q3484924
cites work: Q3337589
cites work: Eisenstein cohomology of arithmetic groups. The case \(GL_ 2\)
cites work: \(l\)-adic representations associated to modular forms over imaginary quadratic fields. I: Lifting to \(GSp_ 4(\mathbb{Q})\)
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cites work: Q3758032
cites work: Appendix: A refinement of the strong multiplicity one theorem for \(GL(2)\)
cites work: On Certain L-Functions
cites work: Galois representations associated to Siegel modular forms of low weight
author: Richard Taylor
description: scientific article; zbMATH DE number 554031