A Quadratic Bound for Automorphism Groups of Curves of Positive p-Rank in Odd Characteristic
Abstract
Let X be a nonsingular projective curve of genus g≥2 over an algebraically closed field of odd characteristic p. Giulietti and Korchmáros established a quadratic genus bound with constant $900$ forcing zero p-rank for curves of even genus. We extend this bound to odd genus: for every g≥2, positive p-rank implies |Aut(X)|<900g2. Starting from Montanucci's two-short-orbit reduction, we combine ramification estimates and a minimum-genus argument to reduce a counterexample to an almost-simple group acting on a cover of the projective line with two branch points. The main ingredients are an intrinsic description of the second ramification group via Hasse--Arf and genus bounds for intermediate quotients associated with parabolic subgroups. These, together with order and local-structure estimates, exclude every possible simple socle. The proof uses the classification of finite simple groups.
Related
- Large automorphism groups compared to the p-rank of algebraic curves in characteristic p
- A classification of zero p-rank curves with large automorphism groups in odd characteristic
- K3 surfaces with finite automorphism group in positive characteristic
- Finite Symplectic Automorphism Groups of K3 Surfaces in Characteristic Seven
- Nagata's conjecture on a polynomial automorphism in positive characteristic
BibTeX
@misc{tafazolian2026quadratic,
title = {{A Quadratic Bound for Automorphism Groups of Curves of Positive $p$-Rank in Odd Characteristic}},
author = {Saeed Tafazolian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05562},
eprint = {2610.05562},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05562},
}
RIS
TY - UNPB
TI - A Quadratic Bound for Automorphism Groups of Curves of Positive p-Rank in Odd Characteristic
AU - Saeed Tafazolian
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05562
UR - https://arxiv.org/abs/2610.05562
AB - Let X be a nonsingular projective curve of genus g≥2 over an algebraically closed field of odd characteristic p. Giulietti and Korchmáros established a quadratic genus bound with constant $900$ forcing zero p-rank for curves of even genus. We extend this bound to odd genus: for every g≥2, positive p-rank implies |Aut(X)|<900g^2. Starting from Montanucci's two-short-orbit reduction, we combine ramification estimates and a minimum-genus argument to reduce a counterexample to an almost-simple group acting on a cover of the projective line with two branch points. The main ingredients are an intrinsic description of the second ramification group via Hasse--Arf and genus bounds for intermediate quotients associated with parabolic subgroups. These, together with order and local-structure estimates, exclude every possible simple socle. The proof uses the classification of finite simple groups.
ER -