On the eigenvalues of the Hecke operator \(T_ 2\)
type: Item
instance of: scholarly article
title: On the eigenvalues of the Hecke operator \(T_ 2\)
zbMATH Open document ID: 0847.11024
author: Kevin Buzzard
published in: Journal of Number Theory
publication date: 1996-05-06T00:00:00Z
review text: Let \(K\) be the splitting field of the characteristic polynomial \({\mathcal X}_k\) of the Hecke operator \(T_2\) on the space \(S_k\) of cusp forms of weight \(k\) on the modular group \(\text{SL}_2(\mathbb{Z})\). The dimension of \(S_k\) is \(\ell\) for \(k= 12\ell\), and the author shows that for all primes \(\ell\leq 19\) the Galois group of \(K\) over \(\mathbb{Q}\) is the full symmetric group on \(\ell\) symbols. The proof requires machine calculations to find primes \(p\) and \(q\) such that the reduction of \({\mathcal X}_k\) is irreducible modulo \(p\) and decomposes in a particular way modulo \(q\).
reviewed by: Günter Köhler
Mathematics Subject Classification ID: 11F25
Mathematics Subject Classification ID: 11F11
zbMATH DE Number: 874235
zbMATH Keywords: eigenvalues
zbMATH Keywords: Hecke operator
zbMATH Keywords: cusp forms
zbMATH Keywords: Galois group
MaRDI profile type: Publication
full work available at URL: jnth.1996.0039
OpenAlex ID: W2052624439
DOI: 10.1006/JNTH.1996.0039
description: scientific article; zbMATH DE number 874235