Modular Representations of GL(3, p), Symmetric Squares, and Mod-p Cohomology of GL(3, )

We prove certain cases of nonabelian reciprocity between mod p Galois representations and Hecke eigenclasses in the mod p cohomology of GL(3, ), using the symmetric square liftings from GL(2) and congruences on Hecke eigenvalues. We need to develop some modular representation theory for GL(3, /p). In some cases we also need to invoke certain computer calculations to finish the proof.