On Artin representations and nearly ordinary Hecke algebras over totally real fields
type: Item
instance of: scholarly article
title: On Artin representations and nearly ordinary Hecke algebras over totally real fields
zbMATH Open document ID: 1293.11073
author: Shu Sasaki
published in: Documenta Mathematica
publication date: 2013-10-25T00:00:00Z
full work available at URL: 31.html
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11F33
Mathematics Subject Classification ID: 11F41
Mathematics Subject Classification ID: 14G22
Mathematics Subject Classification ID: 14G35
zbMATH DE Number: 6220362
zbMATH Keywords: Galois representations
zbMATH Keywords: p-adic modular forms
zbMATH Keywords: Hilbert modular varieties
zbMATH Keywords: strong Artin conjecture
MaRDI profile type: Publication
review text: The main theme of this paper is to prove several modularity theorems for Hilbert cusp forms over totally real fields.NEWLINENEWLINELet \(F\) be a totally real number field and \(\rho : \text{Gal}(\overline F/F)\longrightarrow \mathrm{GL}_2(\mathbb C)\) a totally odd, irreducible continuous Galois representation. The author considers the case of icosahedral representations, namely, those having projective image in \(\mathrm{GL}_2(\mathbb C)/\mathbb C^\times\) is isomorphic to \(A_5\). The case when \(F=\mathbb Q\) is treated in [\textit{K. Buzzard} et al., Duke Math. J. 109, No. 2, 283--318 (2001; Zbl 1015.11021)] and the author follows basically the approach of [loc. cit.] Under some technical assumptions, the author shows that such a representation is modular, namely, comes form a Hilbert cusp eigenform of weight 1, and therefore the \(L\)-function \(L(\rho,s)\) of \(\rho\) is entire, proving a conjecture of Artin.NEWLINENEWLINEThe above results are obtained as consequences of the main result of the paper, which can be stated as follows. Let \(F\) be a totally real field, \(K\) be a finite extension of \(\mathbb Q_p\) with ring of integers \(\mathcal O\) and maximal ideal \(\mathfrak m\), \(p\) a rational prime which splits completely in \(F\) and \(\rho:\text{Gal}(\overline F/F)\longrightarrow \mathrm{GL}_2(\mathcal O)\) a continuous representation which is ramified only at finitely many primes. One supposes that \(\bar\rho=\rho\mod\mathfrak m\) admits a modular lifting, which is potentially ordinary and potentially Barsotti-Tate at every prime of \(F\) over \(p\) and that the restriction of \(\bar\rho\) to \(\text{Gal}(\overline F/F(\zeta_p))\) is absolutely irreducible. One finally assumes that the restriction of \(\bar\rho\) at the decomposition group at every prime of \(F\) above \(p\) is a direct sum of characters, distinct modulo \(\mathfrak m\), such that the images of the inertia subgroups via each of these characters are finite. (If \(p=2\) one also needs…
description: scientific article; zbMATH DE number 6220362