### An analogue of Serre's conjecture for Galois representations and Hecke eigenclasses in the mod-p cohomology of GL(n,Z), by Avner Ash and Warren Sinnott

The conjecture of Serre referred in the title is the one about
modularity of odd Galois representations into GL(2,F) where
F is a finite field of characteristic p. We present an analogous
conjecture where GL(2) is replaced by GL(n). We explain the analogue
of "oddness." We then present some theoretical and experimental
evidence for the conjecture, primarily when n = 3. Our conjecture
does not require the Galois representation to be irreducible. Our
most interesting examples involve the sum of an irreducible even
2-dimensional representation and a 1-dimensional representation.

Avner Ash and Warren Sinnott <ash@math.ohio-state.edu, sinnott@math.ohio-state.edu>