Conductor-discriminant inequality for hyperelliptic curves in odd residue characteristic

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.