On mod p modular representations which are defined over \F_p

In this paper, we use techniques of Conrey, Farmer and Wallace to find spaces of modular forms $S_k(\Gamma_0(N))$ where all of the eigenspaces have Hecke eigenvalues defined over \F_p, and give a heuristic indicating that these are all such spaces.Comment: This is the final version, accepted and to appear in Glasnik Matematick

Irreducibility of automorphic Galois representations of GL(n), n at most 5

Let pi be a regular, algebraic, essentially self-dual cuspidal automorphic representation of GL_n(A_F), where F is a totally real field and n is at most 5. We show that for all primes l, the l-adic Galois representations associated to pi are irreducible, and for all but finitely many primes l, the mod l Galois representations associated to pi are also irreducible. We also show that the Lie algebras of the Zariski closures of the l-adic representations are independent of l.Comment: Erratum: there is a gap in the proof of the main theorem for n=4,