Smith theory and Hecke operators

Of the connections between the cohomology of arithmetic groups and representations of Galois groups, some are known and more are conjectured. This paper proves one of these conjectures for certain monomial Galois representations. Let p be a prime. We show that the representation of the absolute Galois group of induced from an -valued ray class character of the cyclotomic field is attached to a Hecke eigenclass in the mod p cohomology of a torsion-free congruence subgroup Γ(M) of . The method involves constructing an action of the Hecke algebra on the fundamental exact sequence of Smith theory arising from the action by conjugation of an element of order p in on Γ(M).