On the image of \(\Lambda\)-adic Galois representations
type: Item
instance of: scholarly article
title: On the image of \(\Lambda\)-adic Galois representations
zbMATH Open document ID: 1020.11037
author: Ami Fischman
publication date: 2002-09-03T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/115983
review text: Let \(f\) be a cuspidal Hecke-eigenform without complex multiplication, \(E\) the number field generated by its Hecke-eigenvalues, \(\mathcal O\) the ring of integers in \(E\). Following \textit{F.\ Momose} [J.\ Fac.\ Sci., Univ.\ Tokyo, Sect.\ I A 28, 89-109 (1981; Zbl 0482.10023)], the author attaches to \(f\) a certain subfield \(\tilde E\) of \(E\) and a subgroup \(H\) of finite index inside the absolute Galois group \(G\) of the rationals. The author shows that there exists a set \(\Sigma\) of prime numbers which has density 1 and only contains primes \(l\) where \(f\) is ordinary such that for its family of \(\Lambda\)-adic forms (where \(\Lambda = \mathbb{Z}_l T\)) in the sense of \textit{H.\ Hida} [Invent. Math. 85, 545-613 (1986; Zbl 0612.10021)] the following holds: If \(f\) is \(\ell\)-ordinary for a prime \(\ell\) of \(E\) above \(l\in\Sigma\), then the \(\Lambda\)-adic ordinary Hecke algebra above \(\ell\) is isomorphic to \({\mathcal O}_\ellT\) and the image of \(H\) under the \(\Lambda\)-adic Galois-representation attached to \(f\) contains SL\(_2(\tilde{\mathcal O}_{\tilde\ell}T)\), where \(\tilde\ell = \ell\cap \tilde {\mathcal O}.\) This interesting result is additionally spiced up by a specialization result: if a \(\Lambda\)-adic modular form does not admit complex multiplication, then neither do its ``classical''specializations of weight at least 2.
reviewed by: Stefan Kühnlein
Mathematics Subject Classification ID: 11F85
Mathematics Subject Classification ID: 11F80
zbMATH DE Number: 1794802
zbMATH Keywords: modular form
zbMATH Keywords: \(p\)-adic family
zbMATH Keywords: Galois representation
zbMATH Keywords: \(p\)-adic modular form
MaRDI profile type: Publication
OpenAlex ID: W655913820
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DOI: 10.5802/AIF.1890
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published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1794802