Weierstrass semigroups at totally ramified places of degree one on Kummer extensions

Technical Summary: Weierstrass Semigroups at Totally Ramified Places of Degree One on Kummer Extensions

Problem Statement
The paper investigates the structure of Weierstrass semigroups and their associated gap sets at totally ramified places of degree one within Kummer extensions of algebraic function fields. Specifically, the authors consider the extension $F = K(x, y)/K(x)$ defined by the affine equation $y^m = f(x)$, where $K$ is an algebraic extension of a finite field $\mathbb{F}_q$, $\gcd(m, q) = 1$, and $f(x) \in K(x)$ is given by $f(x) = \alpha \prod_{i=1}^r p_i(x)^{\lambda_i}$.

While previous literature has explicitly described these semigroups for specific cases—often requiring $f(x)$ to completely split over $K$ (i.e., $\deg p_i(x) = 1$) or restricting exponents $\lambda_i$ to the range $[1, m-1]$—this work aims to provide a unified description for arbitrary exponents $\lambda_i \in \mathbb{Z} \setminus \{0\}$ and arbitrary irreducible polynomials $p_i(x)$, without requiring $f(x)$ to split completely over $K$. The study focuses on places $Q_s$ lying over $P_s$ (where $P_s$ corresponds to the zero of $p_s(x)$ or the pole of $x$) such that $\gcd(m, \lambda_s) = 1$, ensuring $Q_s$ is a totally ramified place of degree one.

Methodology
The authors employ the theory of algebraic function fields, specifically utilizing the Riemann-Roch theorem and the relationship between gaps and regular Weil differentials. The methodology proceeds through the following steps:

1. Preliminaries and Lemmas: The paper establishes necessary lemmas regarding floor and ceiling functions and properties of constant field extensions. A key tool is the decomposition of Riemann-Roch spaces in Kummer extensions (based on Theorem 2.9 and Lemma 3.3), which allows the dimension $\ell(D)$ to be expressed as a sum over $t \in \{0, \dots, m-1\}$ involving restrictions to the rational function field $K(x)$.
2. Unified Gap Description: By analyzing the condition $\ell(aQ_s) = \ell((a-1)Q_s)$, the authors derive a unified formula for the set of gaps $G(Q_s)$. This involves characterizing the integers $a$ for which a specific inequality involving floor functions and the degrees of the polynomials $p_i(x)$ holds.
3. Symmetry Analysis: Using the characterization of gaps and the existence of a canonical divisor, the authors derive a necessary and sufficient condition for the Weierstrass semigroup $H(Q_s)$ to be symmetric (i.e., $F_{H(Q_s)} = 2g - 1$). This relies on the existence of a specific integer $u$ satisfying a congruence relation involving the exponents $\lambda_i$ and their greatest common divisors with $m$.
4. Minimal Generating Sets: To determine the minimal generating set $\tilde{\Gamma}(Q_{\ell_1}, \dots, Q_{\ell_s})$ for multiple totally ramified places, the authors adapt the techniques introduced by Matthews. They define auxiliary sets $\Gamma(Q_{\ell_1}, \dots, Q_{\ell_s})$ and prove that these sets correspond to specific tuples of integers derived from the gap structure. Crucially, they construct explicit rational functions whose pole divisors realize these generating tuples.

Key Contributions and Results

• Unified Gap and Semigroup Description: The paper provides an explicit, unified description of the gap set $G(Q_s)$ and the Weierstrass semigroup $H(Q_s)$ for any totally ramified place of degree one. Unlike prior results, this description holds for arbitrary integer exponents $\lambda_i$ and does not require $f(x)$ to split completely over $K$.
  • Result: $G(Q_s)$ is described as a set of integers of the form $mj + i$, where bounds on $j$ are determined by sums of ceiling functions involving the exponents $\lambda_i$ and the degrees $d_i = \deg p_i(x)$.
• Symmetry Condition: A necessary and sufficient condition for $H(Q_s)$ to be symmetric is established.
  • Result: $H(Q_s)$ is symmetric if and only if there exists an integer $u \in \{1, \dots, m-1\}$ such that $\frac{u\lambda_i}{\gcd(m, \lambda_i)} \equiv \frac{u\lambda_j}{\gcd(m, \lambda_j)} \pmod m$ for all $i, j \neq s$. This generalizes previous results by Mendoza and Cotterill et al.
• Minimal Generating Sets: The authors explicitly describe the set $\Gamma(Q_{\ell_1}, \dots, Q_{\ell_s})$ for $s$ distinct totally ramified places of degree one.
  • Result: The set is characterized by tuples $(m j_1 + i, \dots, m j_s + i)$ subject to specific linear constraints on the $j_k$ values. Furthermore, the paper provides explicit functions $z \in F$ such that the pole divisor $(z)_\infty$ has coefficients exactly equal to the elements of $\Gamma(Q_{\ell_1}, \dots, Q_{\ell_s})$.
• Non-emptiness Condition: A necessary and sufficient condition for the set $\Gamma(Q_{\ell_1}, \dots, Q_{\ell_s})$ to be non-empty is derived, relating the number of places $s$ to the maximum value of a specific sum involving the exponents.

Applications and Examples
The theoretical results are applied to three specific classes of Kummer extensions:

1. Kummer extensions with equal multiplicities: Where $f(x) = \prod (x-\alpha_i)^\lambda$. The paper refines previous descriptions of the semigroups at infinite and finite places, correcting imprecisions in prior literature regarding minimal generating sets.
2. GGS Curves: The function fields of GGS curves (generalizations of GK curves) are analyzed. The authors determine the gap sets and semigroups at totally ramified places and confirm that the semigroup is symmetric if and only if $n=3$ (the GK curve case).
3. Subcovers of the BM Curve: The paper analyzes the function fields of subcovers of the Beelen-Montanucci (BM) curve, providing explicit gap sets and minimal generating sets for multiple totally ramified places.

Significance
The paper claims significance by unifying and extending existing results on Weierstrass semigroups in Kummer extensions. By removing the restriction that $f(x)$ must split completely over $K$ and allowing arbitrary integer exponents, the results cover a broader class of function fields. The explicit construction of functions realizing the minimal generating sets is a notable contribution, as it moves beyond abstract existence to concrete realization. These results are intended to facilitate the construction of algebraic geometry codes with improved parameters, as the knowledge of Weierstrass semigroups at multiple places is directly applicable to code design. The authors also note that their symmetry conditions extend and generalize previous theorems by Mendoza and Cotterill et al.