Weierstrass semigroups at totally ramified places of degree one on generalized Kummer extensions
Abstract
We explicitly describe the set of gaps and the Weierstrass semigroup at a totally ramified place of degree one on a generalized Kummer extension defined by the affine equation ym = f(x) over K, an algebraic extension of 𝔽q, where f(x)∈ K(x) and K is not required to contain a primitive m-th root of unity. For the different totally ramified places of degree one, our description takes a unified form. Then we 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 provide functions whose coefficients of pole divisors lie in the minimal generating set. Finally, we apply our results to generalized Kummer extensions with the same multiplicities, as well as function fields of GGS curves and subcovers of the BM curve.
Related
- Non-admissible smooth irreducible representations of GL2(F) for totally ramified extensions F/ℚp
- Harmonic torsion of algebraic generalized Ricci solitons and the structure of solvable extensions
- Generalized Frobenius Structures on Hurwitz Spaces, Their Open Extensions and Associated Principal Hierarchies
- Derived equivalence classification of twisted and generalized multifold extensions of affine type
- Virasoro extensions of generalized Kac-Moody algebras
BibTeX
@misc{zhang2026weierstrass,
title = {{Weierstrass semigroups at totally ramified places of degree one on generalized Kummer extensions}},
author = {Huachao Zhang and Chang-An Zhao},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2605.14583},
eprint = {2605.14583},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.14583},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Weierstrass semigroups at totally ramified places of degree one on generalized Kummer extensions
AU - Huachao Zhang
AU - Chang-An Zhao
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2605.14583
UR - https://arxiv.org/abs/2605.14583
AB - We explicitly describe the set of gaps and the Weierstrass semigroup at a totally ramified place of degree one on a generalized Kummer extension defined by the affine equation y^m = f(x) over K, an algebraic extension of 𝔽_q, where f(x)∈ K(x) and K is not required to contain a primitive m-th root of unity. For the different totally ramified places of degree one, our description takes a unified form. Then we 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 provide functions whose coefficients of pole divisors lie in the minimal generating set. Finally, we apply our results to generalized Kummer extensions with the same multiplicities, as well as function fields of GGS curves and subcovers of the BM curve.
ER -