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

Author Shinya Harada

(Received August 10, 2005, revised March 24, 2006)
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 $K$.

2000 Mathematics Subject Classification. Primary 11F80; Secondary 11S15.

Key words and phrases. Galois representations, local fields.

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