A Remark on a Finiteness Conjecture on mod p by C. Khare

The following conjecture on finiteness of mod p Galois representations was formulated by C. Khare in a recent article: Let F denote the algebraic closure of a finite field. Then for each number field K , each integer n , and each ideal n of the ring of integers of K there are only finitely many isomorphism classes of continuous semisimple n -dimensional representations of the absolute Galois group G_K of K over F_p whose prime to-p conductor is bounded by n. We show, as was conjectured by Khare, that the above is implied by the seemingly weaker conjecture where the prime-to- p conductor is assumed to be trivial, provided one considers all number fields simultaneously.