Galois representations with conjectural connections to arithmetic cohomology

Abstract

In this paper we extend a conjecture of A. Ash and W. Sinnott relating niveau 1 Galois representations to the $\mod p$ cohomology of congruence subgroups of ${\rm SL}\sb n(\mathbb {Z})$ to include Galois representations of higher niveau. We then present computational evidence for our conjecture in the case $n=3$ in the form of three-dimensional Galois representations which appear to correspond to cohomology eigenclasses as predicted by the conjecture. Our examples include Galois representations with nontrivial weight and level, as well as irreducible three-dimensional representations that are in no obvious way related to lower-dimensional representations. In addition, we prove that certain symmetric square representations are actually attached to cohomology eigenclasses predicted by the conjecture.