Wild conductor exponents of curves

TL;DR

This paper provides an explicit formula for wild conductor exponents of plane curves over p-adic fields, generalizing previous results for hyperelliptic curves and relating them to invariants of explicit extensions.

Contribution

It introduces a general relation between wild conductor exponents of branched covers and their discriminant covers, extending known formulas to broader classes of curves.

Findings

Derived an explicit formula for wild conductor exponents in terms of invariants.

Proved a general relation connecting wild conductor exponents with discriminant covers.

Addressed a minor issue in the literature on 3-torsion of genus 2 curves.

Abstract

We give an explicit formula for wild conductor exponents of plane curves over $\mathbb{Q}_p$ in terms of standard invariants of explicit extensions of $\mathbb{Q}_p$, generalising a formula for hyperelliptic curves. To do so, we prove a general result relating the wild conductor exponent of a simply branched cover of the projective line with its associated discriminant cover. In an appendix, we resolve a minor issue in the literature on the $3$-torsion of genus 2 curves.