One-point covers of elliptic curves and good reduction

TL;DR

This paper extends Raynaud's criterion for good reduction of certain covers of elliptic curves over mixed-characteristic fields, allowing for larger cyclic Sylow p-subgroups, thus broadening the understanding of reduction behavior.

Contribution

It generalizes Raynaud's criterion to cases where the Galois group has larger cyclic Sylow p-subgroups in elliptic curve covers.

Findings

Generalization of good reduction criterion to larger cyclic Sylow p-subgroups.

Applicable to elliptic curve covers with more complex Galois groups.

Provides conditions under which such covers have potentially good reduction.

Abstract

Raynaud gave a criterion for a branched $G$-cover of curves defined over a mixed-characteristic discretely valued field $K$ with residue characteristic $p$ to have good reduction in the case of either a three-point cover of $\mathbb{P}^1$ or a one-point cover of an elliptic curve. Specifically, such a cover has potentially good reduction whenever $G$ has a Sylow $p$-subgroup of order $p$ and the absolute ramification index of $K$ is less than the number of conjugacy classes of order $p$ in $G$. In the case of an elliptic curve, we generalize this to the case in which $G$ has an arbitrarily large cyclic Sylow $p$-subgroup.