## Tohoku Mathematical Journal 2007 March SECOND SERIES VOL. 59, NO. 1

 Tohoku Math. J. 59 (2007), 67-77

Title ON THE FINITENESS OF MOD $p$ GALOIS REPRESENTATIONS OF A LOCAL FIELD

Abstract. Let $K$ be a local field and $k$ an algebraically closed field. We prove the finiteness of isomorphism classes of semisimple Galois representations of $K$ into $\mathrm{GL}_d(k)$ with bounded Artin conductor and residue degree. We calculate explicitly the number of totally ramified finite abelian extensions of $K$ with bounded conductor. Using this result, we give an upper bound for the number of certain Galois extensions of $K$.