A refinement of the Artin conductor and the base change conductor

TL;DR

This paper introduces a refined version of the Artin conductor that provides higher resolution on tame ramification and relates it to the base change conductor of K-tori and semiabelian varieties, generalizing existing formulas.

Contribution

It defines a new refinement of the Artin distribution, bAr_K, and establishes its connection to the base change conductor for K-tori and semiabelian varieties.

Findings

bAr_K refines Ar_K with higher tame ramification resolution

bAr_K equals the base change conductor when evaluated on Galois representations

Generalizes formulas for base change conductors of algebraic K-tori and semiabelian varieties

Abstract

For a local field K with positive residue characteristic p, we introduce, in the first part of this paper, a refinement bAr_K of the classical Artin distribution Ar_K. It takes values in cyclotomic extensions of Q which are unramified at p, and it bisects Ar_K in the sense that Ar_K is equal to the sum of bAr_K and its conjugate distribution. Compared with 1/2 Ar_K, the bisection bAr_K provides a higher resolution on the level of tame ramification. In the second part of this article, we prove that the base change conductor c(T) of an analytic K-torus T is equal to the value of bAr_{K} on the Q_p-rational Galois representation X^*(T)_{Q_p} that is given by the character module X^*(T) of T. We hereby generalize a formula for the base change conductor of an algebraic K-torus, and we obtain a formula for the base change conductor of a semiabelian K-variety with potentially ordinary reduction.