Residually reducible representations and modular forms
type: Item
instance of: scholarly article
title: Residually reducible representations and modular forms
zbMATH Open document ID: 1005.11030
DOI: 10.1007/BF02698855
published in: Publications Mathématiques
publication date: 2003-02-25T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104160
review text: We cite from the excellent introduction of this beautiful paper: In this paper we give criteria for the modularity of certain two-dimensional Galois representations. Originally conjectural criteria were formulated for compatible systems of \(\lambda\)-adic representations, but a more suitable formulation for our work was given by \textit{J.-M. Fontaine} and \textit{B. Mazur} [Elliptic curves, modular forms, and Fermat's Last Theorem (Hong Kong, 1993), Internatational Press, Ser. Number Theory 1, 41-78 (1995; Zbl 0839.14011)]. Throughout this paper \(p\) will denote an odd prime. Conjecture (Fontaine-Mazur): Suppose that \(\rho: \text{Gal} (\overline{\mathbb{Q}}/\mathbb{Q})\to \text{GL}_2(E)\) is a continuous representation, irreducible and unramified outside a finite set of primes, where \(E\) is a finite extension of \(\mathbb{Q}_p\). Suppose also that (i) \(\rho|_{I_p} \simeq \binom {*\;*}{0\;1}\), where \(I_p\) is an inertia group at \(p\), (ii) \(\det\rho= \psi\varepsilon^{k-1}\) for some \(k\geq 2\) and is odd, where \(\varepsilon\) is the cyclotomic character and \(\psi\) is of finite order. Then \(\rho\) comes from a modular form. To say that \(\rho\) comes from a modular form is to mean that there exists a modular form \(f\) with the property that \(T(\ell)f= \operatorname {trace}\rho (\text{Frob}_\ell)f\) for all \(\ell\) at which \(\rho\) is unramified. Here \(T(\ell)\) is the \(\ell\)th Hecke operator, and an arbitrary embedding of \(E\) into \(\mathbb{C}\) is chosen so that \(\operatorname {trace}\rho (\text{Frob}_\ell)\) can be viewed in \(\mathbb{C}\). Fontaine and Mazur actually state a more general conjecture where condition (i) is replaced by a more general, but more technical, hypothesis. The condition which we use, which we refer to as the condition that \(\rho\) be ordinary, is essential to the methods of this paper. If we pick a stable lattice in \(E^2\), and reduce \(\rho\) modulo a uniformizer \(\lambda\) of \(O_E\), the ring of integers of \(…
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11F11
zbMATH DE Number: 1560381
zbMATH Keywords: modularity
zbMATH Keywords: two-dimensional Galois representations
zbMATH Keywords: universal deformation rings
zbMATH Keywords: Hecke rings
author: Christopher M. Skinner
author: Andrew Wiles
reviewed by: Olaf Ninnemann
MaRDI profile type: Publication
Wikidata QID: Q102397270
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OpenAlex ID: W2050988815
description: scientific article; zbMATH DE number 1560381