Superelliptic degree sets over Henselian fields

Technical Summary: Superelliptic Degree Sets Over Henselian Fields

Problem Statement
The paper addresses the determination of the degree set $D(V/K)$ for a variety $V$ over a discretely valued Henselian field $K$. The degree set is defined as the collection of degrees of the residue fields of closed points on $V$. While the index $\text{ind}(V/K)$ (the greatest common divisor of the degree set) is often computable from the special fiber of a model, the full degree set can be more complex. Specifically, Creutz and Viray [CV25] demonstrated that over $p$-adic fields, the degree set of a curve can exclude infinitely many multiples of its index, a phenomenon that does not occur over finitely generated fields. The authors investigate this "non-cofinite" behavior for superelliptic curves $C/K$ of prime degree $q$ (curves admitting a cyclic cover $C \to \mathbb{P}^1$ of degree $q$), which are birational to affine curves of the form $y^q = F(x)$.

Methodology
The authors employ a combination of valuation theory, cluster analysis, and Henselian field properties to characterize the degree set.

1. Cluster Theory: Following [DDMM19], the roots of $F(x)$ are analyzed via "clusters"—subsets of roots contained in open disks. The Galois group $\text{Gal}(K^{\text{sep}}/K)$ acts on these clusters.
2. Valuation Formulas: A central tool is Lemma 3.4, which provides an explicit formula for the valuation $v(F(x_0))$ for any $x_0 \in K$ in terms of the nearest cluster $s$. The formula is $v(F(x_0)) = |s| \cdot v(x_0 - \alpha) + c_s$, where $|s|$ is the size of the cluster and $c_s$ is a computable constant derived from the roots.
3. Galois-Invariant Clusters: Under the assumption that the residue field is algebraically closed and roots are tamely ramified, Lemma 4.2 establishes the existence of a "center" $\gamma_s \in K$ for any Galois-invariant cluster $s$. This allows the authors to define regions $X_s$ where the valuation behavior is uniform.
4. Obstruction and Construction: The proof strategy involves two directions:
   • Obstruction (Lemma 5.1): Using the valuation formula, the authors derive modular arithmetic conditions under which points of a specific degree $r$ (where $q \nmid r$) cannot exist. If certain conditions regarding $|s|$, $c_s$, and $v(F(\gamma_s))$ are met, points of degree $r$ are obstructed.
   • Existence (Lemma 5.5): Conversely, if the obstruction conditions are not met, the authors use Hensel's Lemma and Newton polygon arguments to construct points of arbitrarily large degrees $r$ (with $\gcd(r, q)=1$) within specific regions $X_s$.

Key Contributions and Results
The paper provides a complete characterization of when the degree set of a superelliptic curve is not cofinite (i.e., misses infinitely many multiples of the index).

• Theorem 1.1: Assuming an algebraically closed residue field and tame ramification, the degree set $D(C/K)$ is not cofinite if and only if $v(F(0)) \not\equiv 0 \pmod q$ and for every Galois-invariant cluster $s$:

  1. $|s| \equiv 0 \pmod q$ and $c_s \not\equiv 0 \pmod q$;
  2. If the center $\gamma_s$ lies in the region $X_s$, then $v(F(\gamma_s)) \not\equiv 0 \pmod q$.
     When these conditions hold, the degree set is contained in $q\mathbb{N} \cup \bigcup_{s \text{ not Galois invariant}} |O(s)|\mathbb{N}$.

• Theorem 1.2: The authors demonstrate the flexibility of these degree sets. For any prime $q \neq p$ and any finite sequence of positive integers $n_1, \dots, n_\ell$, there exists a superelliptic curve $C/K$ such that $D(C/K) = q\mathbb{N} \cup n_1\mathbb{N} \cup \dots \cup n_\ell\mathbb{N}$. As a corollary, for any pair of odd primes $p$ and $q$, there exists a curve where $D(C/K) = p\mathbb{N} \cup q\mathbb{N}$, showing that the density of the degree set can be arbitrarily small.

• Computational Method: The paper offers a practical method for computing degree sets using cluster diagrams and the quantities $c_s$ and $\gamma_s$, which are computable by hand from the roots of $F(x)$ (Example 6.1).

Significance and Claims
The authors claim that their work completely characterizes the mechanism behind the non-cofinite degree set phenomenon for superelliptic curves of prime degree over Henselian fields.

• Theoretical Insight: The results link the arithmetic of the degree set directly to the combinatorial structure of the roots (clusters) and their valuations, rather than solely relying on the geometry of the special fiber.
• Independence from Regular Models: The paper notes that while previous results [CV25] rely on regular models with strict normal crossings, the methods here can compute degree sets even when the curve is not $\Delta_v$-regular (Remark 6.2), a case where existing algorithms for special fibers may fail to produce a regular model.
• Independence of Tools: Lemma 3.4, which relates the valuation of a polynomial to the nearest cluster, is noted as potentially of independent interest.

The paper concludes that for the class of curves studied, the degree set is determined entirely by the cluster data of the defining polynomial $F(x)$, providing a concrete algorithm to determine whether the set is cofinite and, if not, exactly which multiples of the index are missing.