Log ramification via curves in rank 1

TL;DR

This paper proves that in rank 1, the Abbes-Saito log conductor can be determined by its restriction to curves, confirming a conjecture and connecting to higher class field theory.

Contribution

It establishes the rank 1 case of log ramification being determined by curves and extends the understanding of log conductors in this setting.

Findings

Log conductor in rank 1 is determined by curve restrictions

Confirms an expectation of Esnault and Kerz

Links higher class field theory to log ramification

Abstract

We prove that in rank 1, the Abbes-Saito log conductor is determined by its restriction to curves. This result is essentially established by analyzing Artin-Schreier-Witt extensions. Consequently, we confirm an expectation of H. Esnault and M. Kerz. We also conjecture that this result holds in arbitrary finite rank. As an application, we translate recent results in higher class field theory of M. Kerz and S. Saito to the log-ramification context.