A classification of zero p-rank curves with large automorphism groups in odd characteristic
Abstract
Let cX be a curve of genus g≥2 and p-rank zero over an algebraically closed field of odd characteristic p, and let G≤Aut(cX) satisfy |G|>24g(g−1). We prove that G fixes a point unless (cX,G) belongs to an explicit list. For g≥4, the curve is a cyclic quotient of a Hermitian curve, of a Giulietti--Korchmáros curve, or of Skabelund's cover of a Ree curve. We determine the equations and all the groups G that occur, including their central kernels. The exceptional A7 action in characteristic $5$ is on the Hermitian curve of degree $6$; genera $2$ and $3$ are treated separately. The equations are recovered from the group action by two rigidity lemmas for equivariant one-point p-group covers of the affine line. If G has no common fixed point and its Sylow p-subgroups are noncyclic, then, unless G induces 2G2(3) on the orbit of Sylow fixed points, the bound |G|>max{84(g−1),2g2} gives a rational quotient with exactly one wild and one tame short orbit, and a rational Sylow quotient.
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BibTeX
@misc{tafazolian2026classification,
title = {{A classification of zero $p$-rank curves with large automorphism groups in odd characteristic}},
author = {Saeed Tafazolian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.33185},
eprint = {2609.33185},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33185},
}
RIS
TY - UNPB
TI - A classification of zero p-rank curves with large automorphism groups in odd characteristic
AU - Saeed Tafazolian
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2609.33185
UR - https://arxiv.org/abs/2609.33185
AB - Let cX be a curve of genus g≥2 and p-rank zero over an algebraically closed field of odd characteristic p, and let G≤Aut(cX) satisfy |G|>24g(g−1). We prove that G fixes a point unless (cX,G) belongs to an explicit list. For g≥4, the curve is a cyclic quotient of a Hermitian curve, of a Giulietti--Korchmáros curve, or of Skabelund's cover of a Ree curve. We determine the equations and all the groups G that occur, including their central kernels. The exceptional A_7 action in characteristic $5$ is on the Hermitian curve of degree $6$; genera $2$ and $3$ are treated separately. The equations are recovered from the group action by two rigidity lemmas for equivariant one-point p-group covers of the affine line. If G has no common fixed point and its Sylow p-subgroups are noncyclic, then, unless G induces ^2G_2(3) on the orbit of Sylow fixed points, the bound |G|>max{84(g−1),2g^2} gives a rational quotient with exactly one wild and one tame short orbit, and a rational Sylow quotient.
ER -