Good reduction of three-point Galois covers

TL;DR

This paper provides a new proof of Raynaud's criterion for good reduction of three-point Galois covers over p-adic fields, extending it to larger cyclic p-Sylow subgroups and illustrating with examples.

Contribution

It offers a different proof of the criterion and extends it to cases with large cyclic p-Sylow subgroups, answering Raynaud's question.

Findings

Extended the criterion to larger cyclic p-Sylow subgroups.

Provided examples of covers with good reduction and large p-Sylow subgroups.

Abstract

Michel Raynaud gave a criterion for a three-point G-cover f : Y \rightarrow X = P^1, defined over a p-adic field K, to have good reduction. In particular, if the order of a p-Sylow subgroup of G is p, and the number of conjugacy classes of elements of order p is greater than the absolute ramification index e of K, then f has potentially good reduction. We give a different proof of this criterion, which extends to the case where G has an arbitrarily large cyclic p-Sylow subgroup, answering a question of Raynaud. We then use the criterion to give a family of examples of three-point covers with good reduction to characteristic p and arbitrarily large p-Sylow subgroups.