Modular representations of \(GL_ 2\) of a local field: The ordinary, unramified case
type: Item
instance of: scholarly article
title: Modular representations of \(GL_ 2\) of a local field: The ordinary, unramified case
zbMATH Open document ID: 0841.11026
published in: Journal of Number Theory
publication date: 1995-11-29T00:00:00Z
review text: Let \(F\) be a local field of residue characteristic \(p\), and let \(G= PGL_2 (F)\). Let \(E\) be an algebraically closed field of characteristic \(p\), and let \(V\) be an \(E\)-vector space on which \(G\) acts with \(V^K\neq 0\). If \(V\) has a trivial central character, the authors prove that there exists an eigenvector in \(V^K\) for an appropriate Hecke operator \(T\) and classify them by the eigenvalues \(\lambda\) of \(T\) to one-dimensional \((\lambda= \pm 1)\), principal series \((\lambda\neq 0,\pm1)\), or supersingular \((\lambda =0)\). They also show that any \(\lambda\) can appear and prove the uniqueness for \(\lambda \neq 0\).
reviewed by: Min Ho Lee
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11F85
zbMATH DE Number: 821860
zbMATH Keywords: modular representations
zbMATH Keywords: principal series
zbMATH Keywords: local field
zbMATH Keywords: Hecke operator
zbMATH Keywords: eigenvalues
author: Laure Barthel
author: Ron Livné
MaRDI profile type: Publication
full work available at URL: jnth.1995.1124
OpenAlex ID: W2052550983
DOI: 10.1006/JNTH.1995.1124
description: scientific article; zbMATH DE number 821860