Simply branched covers of curves and wild conductor exponents

Abstract

We use deformation theory to show that a cover of smooth, projective, geometrically connected curves $\pi :C\to D$ over a $p$-adic field can be $p$-adically perturbed to obtain a nearby simply branched cover ${\pi }^{\prime }:C^{\prime }\to D$ and that, if $D={\mathbb{P}}^1$ and ${\pi }^{\ast }{\mathcal{O}}_{{\mathbb{P}}^1}(1)$ is very ample, then we may take $C^{\prime }=C$. As an application, we give a formula for the wild conductor exponent of a curve at a prime $p>d$ in terms of the ramification data of any degree $d$ cover $C\to {\mathbb{P}}^1$.

14 pages