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Transitive automorphism groups of maximal curves
Abstract
Let q be an odd prime power and let X be a maximal curve of genus at least two over Fq2. We prove that the automorphism group of X acts transitively on its rational points if and only if X is the Hermitian curve, or q=5 and X is the unique F25-maximal curve of genus $3$, namely the maximal Klein quartic. Together with the result of Giulietti and Korchmáros in characteristic two, this gives the classification for every prime power q. In particular, the Hermitian curve is the only maximal curve of genus at least two whose automorphism group is doubly transitive on its rational points.
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BibTeX
@misc{tafazolian2026transitive,
title = {{Transitive automorphism groups of maximal curves}},
author = {Saeed Tafazolian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.17611},
eprint = {2609.17611},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17611},
}
RIS
TY - UNPB
TI - Transitive automorphism groups of maximal curves
AU - Saeed Tafazolian
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2609.17611
UR - https://arxiv.org/abs/2609.17611
AB - Let q be an odd prime power and let X be a maximal curve of genus at least two over F_q^2. We prove that the automorphism group of X acts transitively on its rational points if and only if X is the Hermitian curve, or q=5 and X is the unique F_25-maximal curve of genus $3$, namely the maximal Klein quartic. Together with the result of Giulietti and Korchmáros in characteristic two, this gives the classification for every prime power q. In particular, the Hermitian curve is the only maximal curve of genus at least two whose automorphism group is doubly transitive on its rational points.
ER -