Maximal curves with trivial automorphism group
Abstract
We construct algebraic curves with trivial geometric automorphism group by taking cyclic scalar quotients of curves of the form yr−y=xR(x), r=pe. We give a condition on the genus under which the quotient has no nontrivial automorphism. The proof uses the ramification of a degree-r morphism to the projective line, Castelnuovo--Severi, and explicit automorphisms of the covering curve. The quotient also has gonality r, with a unique morphism of degree r to the projective line up to an automorphism of the latter. Applying the construction to maximal additive curves gives explicit maximal curves with trivial geometric automorphism group in every positive characteristic, including families defined over the prime field and of unbounded genus. To the best of our knowledge, these are the first examples of maximal curves with trivial geometric automorphism group. Every maximal curve in these families is covered by the Hermitian curve. For every odd prime p and every n≥2, we prove that the curve Xp,n, maximal over Fp4n+2, is not a Galois subcover of the corresponding Hermitian curve, even over the algebraic closure. Thus every odd characteristic yields an infinite family of curves covered but not Galois covered by the Hermitian curve. To the best of our knowledge, these are the first such infinite families.
Related
- Perfect 1-factorisations of K18 with a non-trivial automorphism group
- Fermat quotients with CM Jacobians and trivial automorphism group
- Transitive automorphism groups of maximal curves
- A non-trivially non-trivial automorphism at an inaccessible cardinal
- Uniform Unicellular Dessins d'Enfants with Trivial Automorphism Groups
BibTeX
@misc{bartoli2026maximal,
title = {{Maximal curves with trivial automorphism group}},
author = {Daniele Bartoli and Herivelto Borges and Marco Timpanella and Giovanni Zini},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35788},
eprint = {2610.35788},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35788},
}
RIS
TY - UNPB
TI - Maximal curves with trivial automorphism group
AU - Daniele Bartoli
AU - Herivelto Borges
AU - Marco Timpanella
AU - Giovanni Zini
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35788
UR - https://arxiv.org/abs/2610.35788
AB - We construct algebraic curves with trivial geometric automorphism group by taking cyclic scalar quotients of curves of the form y^r−y=xR(x), r=p^e. We give a condition on the genus under which the quotient has no nontrivial automorphism. The proof uses the ramification of a degree-r morphism to the projective line, Castelnuovo--Severi, and explicit automorphisms of the covering curve. The quotient also has gonality r, with a unique morphism of degree r to the projective line up to an automorphism of the latter. Applying the construction to maximal additive curves gives explicit maximal curves with trivial geometric automorphism group in every positive characteristic, including families defined over the prime field and of unbounded genus. To the best of our knowledge, these are the first examples of maximal curves with trivial geometric automorphism group. Every maximal curve in these families is covered by the Hermitian curve. For every odd prime p and every n≥2, we prove that the curve X_p,n, maximal over F_p^4n+2, is not a Galois subcover of the corresponding Hermitian curve, even over the algebraic closure. Thus every odd characteristic yields an infinite family of curves covered but not Galois covered by the Hermitian curve. To the best of our knowledge, these are the first such infinite families.
ER -