AN ANALOGUE OF SERRE’S CONJECTURE FOR GALOIS REPRESENTATIONS AND HECKE EIGENCLASSES IN THE mod p COHOMOLOGY OF GL(n,Z)

1. Introduction. Let p be a prime number and F an algebraic closure of the finite field F p with p elements. Let n and N denote positive integers, N prime to p. We are interested in representations ρ of the Galois group G Q = Gal(¯ Q/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 Frob l in G Q. We also fix a complex conjugation Frob ∞ ∈ G Q. For every prime q, we fix a decomposition group G q with its filtration by its ramification subgroups G q,i. We denote the whole inertia group G q,0 by I q. 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. Set 0 (N) to be the subgroup of SL(n, Z) consisting of those matrices whose first row is congruent to (* , 0,. .. , 0) modulo N. Let S N 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 Ᏼ(N) the F-algebra of double cosets 0 (N)S N 0 (N). It is com-mutative. This algebra acts on the cohomology and homology of 0 (N) with any coefficient FS N-module. When a double coset is acting on cohomology, we call it a Hecke operator. The Hecke algebra Ᏼ(N) contains all double cosets of the form 0 (N)D(l, k)) 0 (N), where D(l, k) is the diagonal matrix with k l's followed by (n − k) l's, and l is a prime not dividing N. We use the notation T (l,k) for the corresponding Hecke operator. Definition 1.1. Let ᐂ be an Ᏼ(pN)-module, and suppose v ∈ ᐂ is an eigenvector for the action of Ᏼ(pN) with T (l,k)v = a(l, k)v for some a(l, k) ∈ F, for all k = 0,. .. , n, and for all l prime to pN. 1 2 ASH AND SINNOTT unramified outside pN such that k (−1) k l k(k−1)/2 …