A local analysis of congruences in the \((p,p)\) case. II
type: Item
instance of: scholarly article
title: A local analysis of congruences in the \((p,p)\) case. II
zbMATH Open document ID: 0971.11028
published in: Inventiones Mathematicae
publication date: 2001-05-07T00:00:00Z
review text: Fix an odd rational prime \(p\). Let \(\rho:G_\mathbb{Q} \to\text{GL}_2 (\overline \mathbb{F}_p)\) be a continuous, irreducible representation of the absolute Galois group \(G_\mathbb{Q}= \text{Gal} (\overline\mathbb{Q}/ \mathbb{Q})\). We say that \(\rho\) arises from a newform \(f\in S_k(\Gamma_1 (M))\) \((k\geq 2)\) if \(\rho\) is the reduction modulo the maximal ideal of an integral model of the irreducible \(p\)-adic representation \(\rho_f: G_\mathbb{Q} \to\text{GL}_2(K)\) attached to \(f\) by Eichler, Shimura and Deligne. The author studies the local components at \(p\) of newforms \(f\) that give rise to \(\rho\) (Theorems 1 to 5). His proofs of these theorems (given in sections 3, 4 and 5) are along the lines outlined in the introduction of his paper [Compos. Math. 112, 363-376 (1998; Zbl 1072.11506)].
Mathematics Subject Classification ID: 11F33
Mathematics Subject Classification ID: 11F80
zbMATH DE Number: 1553216
zbMATH Keywords: \(p\)-adic Galois representation
zbMATH Keywords: modular curve
zbMATH Keywords: newform
zbMATH Keywords: Jacquet-Langlands correspondence
zbMATH Keywords: Steinberg lift
author: Chandrashekhar Khare
reviewed by: Andrzej Dąbrowski
MaRDI profile type: Publication
full work available at URL: s002220000103
OpenAlex ID: W2092774145
DOI: 10.1007/S002220000103
description: scientific article; zbMATH DE number 1553216