mod \(p\) descent for Hilbert modular forms
type: Item
instance of: scholarly article
title: mod \(p\) descent for Hilbert modular forms
zbMATH Open document ID: 0983.11028
published in: Mathematical Research Letters
publication date: 2001-11-21T00:00:00Z
review text: Let \(F\) be a totally real number field, and \(k\) a finite field of characteristic \(p>2\). Let \(\rho_F: \text{Gal} (\overline{F}/F)\to \text{GL}_2(k)\) be an absolutely irreducible, continuous and odd representation. Assume that \(\rho_F\) is modular (i.e. arises from a holomorphic Hilbert modular form of some weight and level) and extends to a representation \(\rho_{\mathbb{Q}}: \text{Gal} (\overline{\mathbb{Q}}/\mathbb{Q})\to \text{GL}_2(k')\) with \((k':k)< \infty\). Question (Khare). Is the extension \(\rho_{\mathbb{Q}}\) modular? Any such extension is odd and absolutely irreducible, hence, Serre's conjecture implies the answer to this question is ``yes''. The author provides an answer to this question under some assumptions (including the solvability of \(F/\mathbb{Q}\)).
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11F41
Mathematics Subject Classification ID: 11F33
zbMATH DE Number: 1539311
zbMATH Keywords: mod \(p\) descent
zbMATH Keywords: modular representation
zbMATH Keywords: Serre's conjecture
zbMATH Keywords: Hilbert modular form
author: Chandrashekhar Khare
reviewed by: Andrzej Dąbrowski
MaRDI profile type: Publication
full work available at URL: mrl.2000.v7.n4.a11
OpenAlex ID: W17635663
DOI: 10.4310/MRL.2000.V7.N4.A11
description: scientific article; zbMATH DE number 1539311