Large automorphism groups compared to the p-rank of algebraic curves in characteristic p
Abstract
Let cX be a (projective, geometrically irreducible, non-singular) algebraic curve of genus ≥ 2 and positive p-rank γ(cX), defined over an algebraically closed field 𝕂 of positive characteristic p>0. Contrary to what occurs for the genus, no function h(γ) exists such that |aut(cX)|≤ h(γ) whenever γ=γ(cX). Thus, to have a bound on |aut(cX)| only depending on γ(cX), some restrictions on cX and aut(cX) are needed. In this context, the following theorem is proven. Let Γ be a subgroup of aut(cX). Assume that there is a point P∈ cX such that the quotient curve cX/SP is rational, where SP denotes the Sylow p-subgroup of the stabilizer ΓP of P in Γ. Then the following p-rank analog of the Riemann-Hurwitz bound |Γ|≤ 24 (p/(p−1))4 γ(cX)4 holds, unless a subgroup of index ≤ 2 of Γ fixes P. This bound is sharp up to a constant depending only on p.
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BibTeX
@misc{giulietti2026large,
title = {{Large automorphism groups compared to the $p$-rank of algebraic curves in characteristic $p$}},
author = {Massimo Giulietti and Gábor Korchmáros and Marco Timpanella},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.10065},
eprint = {2606.10065},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.10065},
}
RIS
TY - UNPB
TI - Large automorphism groups compared to the p-rank of algebraic curves in characteristic p
AU - Massimo Giulietti
AU - Gábor Korchmáros
AU - Marco Timpanella
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2606.10065
UR - https://arxiv.org/abs/2606.10065
AB - Let cX be a (projective, geometrically irreducible, non-singular) algebraic curve of genus ≥ 2 and positive p-rank γ(cX), defined over an algebraically closed field 𝕂 of positive characteristic p>0. Contrary to what occurs for the genus, no function h(γ) exists such that |aut(cX)|≤ h(γ) whenever γ=γ(cX). Thus, to have a bound on |aut(cX)| only depending on γ(cX), some restrictions on cX and aut(cX) are needed. In this context, the following theorem is proven. Let Γ be a subgroup of aut(cX). Assume that there is a point P∈ cX such that the quotient curve cX/S_P is rational, where S_P denotes the Sylow p-subgroup of the stabilizer Γ_P of P in Γ. Then the following p-rank analog of the Riemann-Hurwitz bound |Γ|≤ 24 (p/(p−1))^4 γ(cX)^4 holds, unless a subgroup of index ≤ 2 of Γ fixes P. This bound is sharp up to a constant depending only on p.
ER -