Conductor-discriminant inequality for hyperelliptic curves in odd residue characteristic

TL;DR

This paper establishes an inequality relating the conductor and discriminant of hyperelliptic curves over discretely valued fields with odd residue characteristic, generalizing previous results without restrictions on genus or ramification.

Contribution

It proves a conductor-discriminant inequality for all hyperelliptic curves over fields with odd residue characteristic, extending prior work to all genera and ramification cases.

Findings

The negative Artin conductor is bounded by the valuation of the discriminant.

No restrictions on the genus or ramification of the splitting field.

Generalizes earlier inequalities by Ogg, Saito, Liu, and others.

Abstract

We prove an inequality between the conductor and the discriminant for all hyperelliptic curves defined over discretely valued fields $K$ with perfect residue field of characteristic not 2. Specifically, if such a curve is given by $y^2 = f(x)$ with $f(x) \in \mathcal{O}_K[x]$, and if $X$ is its minimal regular model over $\mathcal{O}_K$, then the negative of the Artin conductor of $X$ (and thus also the number of irreducible components of the special fiber of $X$) is bounded above by the valuation of disc$(f)$. There are no restrictions on genus of the curve or on the ramification of the splitting field of $f$. This generalizes earlier work of Ogg, Saito, Liu, and the second author.