On the modularity of certain \(\text{GL}_2(\mathbb{F}_7)\) Galois representations
type: Item
instance of: scholarly article
title: On the modularity of certain \(\text{GL}_2(\mathbb{F}_7)\) Galois representations
zbMATH Open document ID: 1001.11023
published in: Mathematical Research Letters
publication date: 2002-12-13T00:00:00Z
review text: Let \(\overline\rho\) be an absolutely irreducible continuous representation of the absolute Galois group of the rationals to \(\text{GL}_2(\mathbb{F}_7)\). Assume that any complex conjugation has image under \(\rho\) with eigenvalues 1 and \(-1\). Then, according to a conjecture of Serre, \(\overline\rho\) should be modular. Here, the author imposes further conditions on \(\overline\rho\) concerning the decomposition groups at 3 and 7. Then he can show modularity of \(\overline\rho\). The proof goes along the lines of \textit{R. Taylor}'s paper [On icosahedral Artin representations. II, Am. J. Math. 125, No. 3, 549--566 (2003; Zbl 1031.11031)]. It uses a concrete description of the moduli space \(X(7)\) of elliptic curves with prescribed level 7 structure in order to find an elliptic curve (defined over some totally real field \(F\)) whose \(G_F\)-representation on \(E[7]\) is equivalent to \(\overline\rho|_{G_F}\) and which thereby assists in finishing the proof.
reviewed by: Stefan Kühnlein
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 11F33
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 1744228
zbMATH Keywords: Galois representation
zbMATH Keywords: modularity
zbMATH Keywords: elliptic curve
author: Jayanta Manoharmayum
MaRDI profile type: Publication
full work available at URL: mrl.2001.v8.n6.a2
OpenAlex ID: W2312463804
DOI: 10.4310/MRL.2001.V8.N6.A2
Recommended article: On the modularity of certain 2-adic Galois representations
Recommended article: Modularity of \(\mathrm{GL}_2 (\mathbb{F}_p)\)-representations over CM fields
Recommended article: Q3579568
Recommended article: On the modular representations of degree two of \(\text{Gal}({\overline {\mathbb Q}}/{\mathbb Q})\)
Recommended article: Modular multiplicities and representations of \(\text{GL}_2(\mathbb Z_p)\) and \(\text{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)\) at \(\ell=p\)
Recommended article: On the modularity of reducible \(\mathrm{mod}\;l\) Galois representations
Recommended article: Q2996976
Recommended article: On modular representations of \(\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\) arising from modular forms
Recommended article: SUR QUELQUES REPRÉSENTATIONS MODULAIRES ET $p$-ADIQUES DE $\mathrm{GL}_2(\bm{Q}_{p})$. II
Recommended article: Q4672842
description: scientific article; zbMATH DE number 1744228