Cohomology with Sym^g coefficients for congruence subgroups of SL_4(Z) and Galois representations

Abstract

We extend the computations in our prior work to find the cohomology in degree five of a congruence subgroup Gamma of SL_4(Z) with coefficients in Sym^g(K^4), twisted by a nebentype character eta, along with the action of the Hecke algebra. This is the top cuspidal degree. In this paper we take K to be a finite field of large characteristic, as a proxy for the complex numbers. For each Hecke eigenclass found, we produce the unique Galois representation that appears to be attached to it. The computations require modifications to our previous algorithms to accommodate the fact that the coefficients are not one-dimensional.