On the eigenvalues of the Hecke operator \(T_ 2\)
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 \(q\).
scientific article; zbMATH DE number 874235