My current projects involve studying the behavior of Galois covers of curves defined over discretely valued fields.

Current projects

Good reduction of covers of elliptic curves. We start with a Galois cover \(f\) of an elliptic curve \(E\) defined over a "small" mixed-characteristic discretely valued field \(K\) (where "small" involves some technical conditions) with Galois group \(G\) branched over one point. The aim is to prove that \(f\) has potentially good reduction when \(G\) has a cyclic Sylow \(p\)-subgroup; that is, that there is a finite of \(K\) over which \(f\) has a smooth model.

Conductors of one-point covers in characteristic \(p\). Abhyankar's conjecture gives conditions under which a finite group \(G\) can be realized as the Galois group of a branched cover \(f\) of a curve \(X\) in positive characteristic \(p\). It does not, however, provide a description of the ramification at the branch points. In this project, I try to place bounds on the ramification jumps for covers of \(\mathbb{P}^1\) branched over one point in the case where \(G\) has a cyclic Sylow \(p\)-subgroup. The strategy involves looking at deformations of \(f\) to generate new covers with "smaller" ramification data.

Examples of one-point covers of elliptic curves. Roughly, the rigidity criterion provides a way of determining when a Galois cover \(X \to \mathbb{P}^1\) branched at three points is defined over a "small" field. There is no known analogous method for Galois covers of genus 1 curves branched at one point, though these two cases are often seen as natural analogues. My goal in this project is to try to generalize this criterion to the genus 1 case which would, in particular, give a systematic way of finding examples of covers that arise in my other project involving covers of elliptic curves.

Papers and publications.

Deformations of wildly ramified one-point covers, 2021. Preprint.

One-point covers of elliptic curves and good reduction, 2018. Submitted for publication.