Sharp bounds and gap phenomena for large p-subgroups of automorphism groups of curves
Abstract
Let X be a curve of genus g≥2 over an algebraically closed field of odd characteristic p, and let G=Aut(X). A theorem of Giulietti and Korchmaros shows that if G fixes no point and a p-subgroup S≤ GP satisfies |S|>p/(p−1)g, then X is one of the classical hyperelliptic, Hermitian, or Ree curves. We determine the sharp boundary and the first gap below it. At equality, besides the classical curves, exactly three types occur: the ordinary curves yp−y=x+c/x, the zero-p-rank curves y3=xp−x for p≡2pmod3, and, in characteristic $3$, the ordinary genus-six family y3−y=c(x+1/(x3−x)). As a key ingredient, we classify zero-p-rank curves whose full automorphism group fixes no point and which admit a p-subgroup of order larger than the genus. We then prove that, when |S|>p, the boundary classification already follows from the weaker inequality |S|>p(g−p+1)/(p−1). For |S|≥ p3, the forbidden interval above m=(p−1)|S|/p has width at least p(p−1)/2, and below m+(p2−1)/2 at most one further genus can occur, with a rigid ramification structure. Finally, we classify the first admissible layer: it occurs only for |S|=p2, and the resulting ordinary curves admit a complete description in terms of two-dimensional Artin--Schreier spaces.
Related
- Finite subgroups of outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields
- Large cyclic automorphism groups and cyclic subgroups of index two in characteristic two
- Transitive automorphism groups of maximal curves
- A Quadratic Bound for Automorphism Groups of Curves of Positive p-Rank in Odd Characteristic
- Large automorphism groups compared to the p-rank of algebraic curves in characteristic p
BibTeX
@misc{tafazolian2026sharp,
title = {{Sharp bounds and gap phenomena for large $p$-subgroups of automorphism groups of curves}},
author = {Saeed Tafazolian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05539},
eprint = {2610.05539},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05539},
}
RIS
TY - UNPB
TI - Sharp bounds and gap phenomena for large p-subgroups of automorphism groups of curves
AU - Saeed Tafazolian
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05539
UR - https://arxiv.org/abs/2610.05539
AB - Let X be a curve of genus g≥2 over an algebraically closed field of odd characteristic p, and let G=Aut(X). A theorem of Giulietti and Korchmaros shows that if G fixes no point and a p-subgroup S≤ G_P satisfies |S|>p/(p−1)g, then X is one of the classical hyperelliptic, Hermitian, or Ree curves. We determine the sharp boundary and the first gap below it. At equality, besides the classical curves, exactly three types occur: the ordinary curves y^p−y=x+c/x, the zero-p-rank curves y^3=x^p−x for p≡2pmod3, and, in characteristic $3$, the ordinary genus-six family y^3−y=c(x+1/(x^3−x)). As a key ingredient, we classify zero-p-rank curves whose full automorphism group fixes no point and which admit a p-subgroup of order larger than the genus. We then prove that, when |S|>p, the boundary classification already follows from the weaker inequality |S|>p(g−p+1)/(p−1). For |S|≥ p^3, the forbidden interval above m=(p−1)|S|/p has width at least p(p−1)/2, and below m+(p^2−1)/2 at most one further genus can occur, with a rigid ramification structure. Finally, we classify the first admissible layer: it occurs only for |S|=p^2, and the resulting ordinary curves admit a complete description in terms of two-dimensional Artin--Schreier spaces.
ER -