Conductor-Discriminant Inequality for Hyperelliptic Curves in Odd Residue Characteristic

Abstract

AbstractWe 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 $\mathcal{X}$ is its minimal regular model over $\mathcal{O}_{K}$, then the negative of the Artin conductor of $\mathcal{X}$ (and thus also the number of irreducible components of the special fiber of $\mathcal{X}$) is bounded above by the valuation of $\operatorname{disc}(f)$. 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.