Conductor-Discriminant Inequality for Tamely Ramified Cyclic Covers I

We prove conductor-discriminant inequalities for all $\mathbb{Z}\mathrm{/}n$-covers of ${\mathbb{P}}^1$ defined over discretely valued fields $K$ with excellent valuation ring ${\mathcal{O}}_K$ and perfect residue field of characteristic not dividing $n$, modulo some calculations appearing in work of the third author. Specifically, when such a curve $X$ is given by $y^n=f(x)$ with $f(x)\in {\mathcal{O}}_K[x]$ and $n\mid \text{deg}(f)$, and if $\mathcal{X}$ is its minimal regular model over ${\mathcal{O}}_K$, then the negative of the Artin conductor of $\mathcal{X}$ is bounded above by $(n-1)v_K(\text{disc}(\text{rad}(f)))$. This is a direct generalization of previous work of the first two authors on hyperelliptic curves, which in turn generalized work of Ogg, Saito, Liu, and the second author. When $f$ is monic, this strengthens a result of Kohls stating that the conductor exponent of the Jacobian of such a curve is bounded above by $(n-1)v_K(\text{disc}(\text{rad}(f)))$.