AN ANALOGUE OF SERRE’S CONJECTURE FOR GALOIS REPRESENTATIONS AND HECKE EIGENCLASSES IN THE MOD-p COHOMOLOGY OF GL(n,Z) AVNER ASH AND WARREN SINNOTT

Let p be a prime number and F an algebraic closure of the finite field Fp with p elements. Let n and N denote positive integers, N prime to p. We are interested in representations of the Galois group GQ = Gal(Q̄/Q) into GL(n,F), unramified at all finite primes not dividing pN . (We shall say “ρ is unramified outside pN .”) In this paper, “representation” will always mean “continuous, semisimple representation.” We choose for each prime l not dividing pN a Frobenius element Frobl in GQ. We also fix a complex conjugation Frob∞ ∈ GQ. For every prime q we fix a decomposition group Gq with its filtration by its ramification subgroups Gq,i. We denote the whole inertia group Gq,0 by Tq. Our aim is to make a conjecture about when such a representation should be attached to a cohomology class of a congruence subgroup of level N of GL(n,Z). Then we exhibit such evidence for the conjecture as we are able to. Set Γ0(N) to be the subgroup of SL(n,Z) consisting of those matrices whose first row is congruent to (∗, 0, . . . , 0) modulo N . Let SN be the subsemigroup of the integral matrices in GL(n,Q) whose first row is congruent to (∗, 0, . . . , 0) modulo N and with determinant positive and prime to N . We denote by H(N) the F-algebra of double cosets Γ0(N)SNΓ0(N). It is commutative. This algebra acts on the cohomology and homology of Γ0(N) with any coefficient FSN module. When a double coset is acting on cohomology, we call it a Hecke operator. H(N) contains all double cosets of the form Γ0(N)D(l, k)Γ0(N), where D(l, k) is the diagonal matrix with