Superelliptic degree sets over Henselian fields

TL;DR

This paper characterizes the degree sets of certain superelliptic curves over Henselian fields, revealing when they can omit infinitely many multiples of the curve's index, and provides a method for their computation.

Contribution

It offers a complete characterization and computational method for degree sets of superelliptic curves over Henselian fields, extending understanding beyond finite fields.

Findings

Degree sets can omit infinitely many multiples of the index over Henselian fields.

Complete characterization of when this omission occurs for cyclic covers of prime degree.

Provides a practical method for computing degree sets of such curves.

Abstract

Let $K$ be a discretely valued Henselian field. Creutz and Viray show that the degree set of a curve $C$ over a $p$-adic field can miss infinitely many multiples of the index of $C$, a phenomenon that cannot occur over finitely generated fields. For curves $C/K$ with a cyclic cover of $\mathbb{P}^1$ of prime degree, under mild assumptions, we completely characterize how and when this behavior can occur, and give a method for computing degree sets of curves of this type.