Pith Number

pith:H466YKTR

pith:2026:H466YKTRKSOSYVLOANOV6JQI4J

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

Chang-An Zhao, Huachao Zhang

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

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

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Claims

C1strongest 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.

C2weakest assumption

The assumption that the place is totally ramified of degree one on the Kummer extension y^m = f(x), with the description holding uniformly without additional restrictions on the ramification or the polynomial f beyond the Kummer setup.

C3one line summary

Explicit descriptions of Weierstrass semigroups and gaps at totally ramified places on Kummer extensions, plus symmetry conditions and minimal generators, with applications to GGS and BM curves.

References

44 extracted · 44 resolved · 0 Pith anchors

[1] A. S. Castellanos, E. A. R. Mendoza, L. Quoos, Weierstrass semigroups, pure gaps and codes on function fields, Designs, Codes and Cryptography 92 (5) (2024) 1219–1242.doi:10.1007/s10623-023-01339-w 2024 · doi:10.1007/s10623-023-01339-w

[2] E. A. Mendoza, On Kummer extensions with one place at infinity, Finite Fields and Their Applications 89 (2023) 102209.doi:10.1016/j.ffa.2023.102209 2023 · doi:10.1016/j.ffa.2023.102209

[3] M. Abdón, H. Borges, L. Quoos, Weierstrass points on Kummer extensions, Ad- vances in Geometry 19 (3) (2019) 323–333.doi:10.1515/advgeom-2018-0021 2019 · doi:10.1515/advgeom-2018-0021

[4] P. Beelen, L. Landi, M. Montanucci, Weierstrass semigroups on the Skabelund maximal curve, Finite Fields and Their Applications 72 (2021) 101811.doi:10. 1016/j.ffa.2021.101811 2021

[5] D. Bartoli, M. Montanucci, G. Zini, Weierstrass semigroups at every point of the Suzuki curve, Acta Arithmetica 197 (1) (2021) 1–20.doi:10.4064/ aa181203-24-2 2021

Receipt and verification

First computed	2026-05-17T23:39:05.338853Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

3f3dec2a71549d2c556e035d5f2608e2509da17997f8cfa8663b086080a1903c

Aliases

arxiv: 2605.14583 · arxiv_version: 2605.14583v1 · doi: 10.48550/arxiv.2605.14583 · pith_short_12: H466YKTRKSOS · pith_short_16: H466YKTRKSOSYVLO · pith_short_8: H466YKTR

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/H466YKTRKSOSYVLOANOV6JQI4J \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 3f3dec2a71549d2c556e035d5f2608e2509da17997f8cfa8663b086080a1903c

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "cac63ac7a91b745b7cfb97e4b6af66d117d7aefcc4738f986a3ae738a221c3c4",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.AG",
    "submitted_at": "2026-05-14T08:54:10Z",
    "title_canon_sha256": "bc14ba4a47a28998215f4728d2214ca4fac25bd095986b0d7d9e30a5ed316556"
  },
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  "source": {
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    "kind": "arxiv",
    "version": 1
  }
}