Smith theory and Hecke operators. (Q1867298)

Let \(p\) be a prime. In this paper it is shown that the representation of the absolute Galois group of \(\mathbb{Q}\) induced from an \(\overline{\mathbb{F}}^\times_p\)-valued ray class character of the cyclotomic field \(\mathbb{Q}(\zeta_p)\) is attached to a Hecke eigenclass in the \(\text{mod\,}p\) cohomology of a torsion-free congruence subgroup of \(\text{GL}_{p-1}(\mathbb{Z})\). Attachment means that the characteristic polynomial of any Frobenius element is given by Hecke eigenvalues in an explicit way. The proof is an application of the key idea of Smith in his study of the action of a \(p\)-adic group on a topological space.