  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.
