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arxiv: 2605.14583 · v1 · submitted 2026-05-14 · 🧮 math.AG

Recognition: no theorem link

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

Huachao Zhang , Chang-An Zhao
Authors on Pith no claims yet

Pith reviewed 2026-05-15 01:30 UTC · model grok-4.3

classification 🧮 math.AG
keywords Weierstrass semigroupKummer extensiontotally ramified placegapssymmetric semigroupfunction fieldminimal generating setalgebraic curve
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The pith

The Weierstrass semigroup at totally ramified degree-one places on Kummer extensions y^m = f(x) admits an explicit unified description.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper provides an explicit description of the gaps and the full Weierstrass semigroup for any totally ramified place of degree one in a Kummer extension given by y^m = f(x). The same formulas apply across different such places without extra conditions. A necessary and sufficient condition is derived for when this semigroup is symmetric. The minimal set of generators is identified and functions are constructed whose pole orders match those generators. The results are illustrated on function fields of GGS curves and subcovers of the BM curve.

Core claim

We explicitly describe the set of gaps and the Weierstrass semigroup at a totally ramified place of degree one on a Kummer extension defined by the affine equation y^m = f(x) over K, an algebraic extension of F_q, where f(x) in K(x). Our description takes a unified form for distinct totally ramified places of degree one. We then provide a necessary and sufficient condition for the Weierstrass semigroup at a totally ramified place of degree one to be symmetric. Furthermore, we investigate the minimal generating set of the Weierstrass semigroups at many totally ramified places of degree one. We not only explicitly describe the minimal generating set, but also construct functions whose pole div

What carries the argument

The uniform explicit description of the gap set for the Weierstrass semigroup at totally ramified places of degree one using the parameters m and the valuations induced by the equation y^m = f(x).

Load-bearing premise

The place must be totally ramified of degree one in the Kummer extension y^m = f(x), with the formulas holding uniformly for any such place and any admissible f.

What would settle it

For a concrete Kummer extension such as y^2 = x^3 + 1 over a finite field, compute the actual gaps at a chosen totally ramified place of degree one by determining the orders of all functions with poles only there, and check whether they match the gaps predicted by the unified description.

read the original abstract

We explicitly describe the set of gaps and the Weierstrass semigroup at a totally ramified place of degree one on a Kummer extension defined by the affine equation $y^m = f(x)$ over $K$, an algebraic extension of $\mathbb{F}_q$, where $f(x)\in K(x)$. Our description takes a unified form for distinct totally ramified places of degree one. We then provide a necessary and sufficient condition for the Weierstrass semigroup at a totally ramified place of degree one to be symmetric. Furthermore, we investigate the minimal generating set of the Weierstrass semigroups at many totally ramified places of degree one. We not only explicitly describe the minimal generating set, but also construct functions whose pole divisors have coefficients lying in the set. Finally, we apply our results to specific Kummer extensions, including function fields of GGS curves and subcovers of the BM curve.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript claims to explicitly describe the set of gaps and the Weierstrass semigroup at totally ramified places of degree one on Kummer extensions defined by the affine equation y^m = f(x) over an algebraic extension K of F_q. It provides a unified form for distinct such places, a necessary and sufficient condition for the semigroup to be symmetric, explicit descriptions and constructions of minimal generating sets via functions with prescribed pole divisors, and applications to GGS curves and subcovers of the BM curve.

Significance. If the derivations hold, the work supplies concrete, unified tools for computing Weierstrass semigroups and their symmetry in Kummer covers, which are standard in the study of algebraic function fields. The explicit constructions of realizing functions and the applications to GGS and BM curves are concrete strengths that could support further computations in coding theory or Hurwitz class number problems.

major comments (2)
  1. [§3] §3 (main description theorem): the unified expression for the gaps via v_P(y) = -v_P(f)/m with gcd(v_P(f), m) = 1 is presented as holding for any such place, but the proof sketch does not explicitly verify that the resulting numerical semigroup has genus exactly matching the Riemann-Hurwitz formula for the Kummer extension; this verification is load-bearing for the claim that the description is complete.
  2. [§5] §5 (symmetry criterion): the necessary and sufficient condition is stated in terms of the genus and the minimal generators, but it reduces to checking whether the largest gap equals 2g-2; the paper should confirm this does not introduce circularity with the genus computation already used in the gap description.
minor comments (3)
  1. [Introduction] Notation for the place P and the valuation v_P is introduced without a dedicated preliminary subsection; a short paragraph recalling the standard ramification data for Kummer extensions would improve readability.
  2. [§6] In the applications to GGS curves, the explicit functions realizing the generators are constructed only in the general case; one or two concrete coefficient examples would help readers verify the construction.
  3. [Abstract] The abstract claims the description holds 'for distinct totally ramified places of degree one,' but the main theorems appear to treat a fixed place; a clarifying sentence on uniformity across places would remove ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive comments. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [§3] §3 (main description theorem): the unified expression for the gaps via v_P(y) = -v_P(f)/m with gcd(v_P(f), m) = 1 is presented as holding for any such place, but the proof sketch does not explicitly verify that the resulting numerical semigroup has genus exactly matching the Riemann-Hurwitz formula for the Kummer extension; this verification is load-bearing for the claim that the description is complete.

    Authors: We agree that an explicit verification strengthens the argument. The gap set is constructed directly from the valuation conditions at P, and the number of gaps is designed to equal the genus g obtained from the Riemann-Hurwitz formula for the degree-m Kummer extension. In the revised manuscript we will add a short paragraph in §3 that counts the gaps produced by the unified expression and confirms that this count equals the genus given by Riemann-Hurwitz (specifically g = (m-1)(deg(f)-1)/2 adjusted for the ramification data). This makes the completeness of the description fully self-contained. revision: yes

  2. Referee: [§5] §5 (symmetry criterion): the necessary and sufficient condition is stated in terms of the genus and the minimal generators, but it reduces to checking whether the largest gap equals 2g-2; the paper should confirm this does not introduce circularity with the genus computation already used in the gap description.

    Authors: The genus g is computed globally and independently from the Riemann-Hurwitz formula applied to the full extension (using only the degree m and the ramification indices at all places), before any local gap set at a specific place P is described. The gap description in §3 then produces a concrete set of size g, and the symmetry test in §5 simply checks whether the largest element of that set equals 2g-2. There is therefore no circularity. We will insert a clarifying sentence in §5 stating that g is fixed by the global Riemann-Hurwitz computation prior to the local description. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives the gaps and Weierstrass semigroup directly from the ramification data of the Kummer equation y^m = f(x) using the standard valuation v_P(y) = -v_P(f)/m (under gcd(v_P(f), m) = 1). The semigroup is expressed as the numerical semigroup generated by m and the adjusted pole orders, with symmetry and minimal generators obtained by explicit comparison to the genus formula and construction of realizing functions. No load-bearing step reduces by definition or self-citation to its own inputs; the central claims remain independent of fitted parameters or prior self-referential results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard properties of algebraic function fields, ramification in Kummer extensions, and the definition of Weierstrass semigroups; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)
  • standard math Standard ramification theory for Kummer extensions of function fields over finite fields
    Invoked to classify totally ramified places of degree one
  • standard math Existence and basic properties of Weierstrass semigroups at places of function fields
    Background from algebraic geometry of curves

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