Superelliptic degree sets over Henselian fields

TL;DR

This paper studies the degree set $\mathcal{D}(C/K)$ for curves $C/K$ over discretely valued Henselian fields, with a focus on superelliptic curves of prime degree. It introduces a cluster-diagram framework that expresses degree-set obstructions and densities through root clusters, their Galois orbits $\mathcal{O}(\mathfrak{s})$, and valuations of $F(x)$ via the slope formula, plus the computable quantities $c_{\mathfrak{s}}$ and $\gamma_{\mathfrak{s}}$. The authors prove criteria for when $\mathcal{D}(C/K)$ is not cofinite and construct explicit examples showing $\mathcal{D}(C/K)$ can be a union of $q\mathbb{N}$ with additional arithmetic progressions, thereby achieving controlled non-cofiniteness. These results connect to the special fiber of regular models and the Degrees-on-Varieties framework, enabling explicit computations and broadening understanding of rational points on curves over local fields.

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.

Theorems & Definitions (47)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof