Representations of quaternion algebras over local fields and trace formulas of Hecke operators
type: Item
instance of: scholarly article
title: Representations of quaternion algebras over local fields and trace formulas of Hecke operators
zbMATH Open document ID: 0819.11018
author: Hiroaki Hijikata
author: Hiroshi Saito
author: Masatoshi Yamauchi
published in: Journal of Number Theory
publication date: 1993-05-16T00:00:00Z
review text: For a nonarchimedean local field \(F\) and the division quaternion algebra \(B\) over \(F\) the authors investigate the decomposition of irreducible admissible representations of \(B^ \times\) upon restriction to the multiplicative group of a quadratic extension of \(F\) contained in \(B\). They give explicit formulae for the decomposition of these restrictions into characters. A similar result of Tunnell describes the decomposition of representations of \(\text{GL}_ 2\) upon restriction to Cartan subgroups using the \(\varepsilon\)-factors of base change lifts of the representations; the explicit formulae given here allow with the help of the correspondence of Jacquet/Langlands to calculate these factors. The authors then calculate the traces of the Hecke operators on the representation spaces and use their results to decompose spaces of cusp forms of prime power level. They give several explicit examples, among them the decomposition of the 1176-dimensional space of newforms of weight 2, level \(7^ 5\) and trivial character into subspaces of dimensions 13, 14, 15 (which are not necessarily irreducible).
reviewed by: Rainer Schulze-Pillot
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11F12
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 166064
zbMATH Keywords: automorphic forms on quaternion algebras
zbMATH Keywords: decomposition of spaces of cusp forms
zbMATH Keywords: traces of the Hecke operators
MaRDI profile type: Publication
full work available at URL: jnth.1993.1013
OpenAlex ID: W2093158518
DOI: 10.1006/JNTH.1993.1013
description: scientific article; zbMATH DE number 166064