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Maximal Curves with Few Automorphisms

  • Herivelto Borges
  • Lucas Reis
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Abstract

We consider the problem of constructing maximal curves with few automorphisms over finite fields. Our main ingredient is the determination of the PGL2 (𝔽̅p)-stabilizer of the roots of the Chebyshev polynomials, apart from two exceptional cases already treated in the literature. We prove that, in all other cases, the only nontrivial Möbius symmetry is the sign involution. This settles the general case for the automorphism groups of the corresponding Chebyshev curves. We further construct an infinite family of maximal curves of unbounded genus whose only nontrivial automorphism is the hyperelliptic involution. This reveals a contrasting phenomenon to the common heuristic that curves with many rational points tend to have large automorphism groups.

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Cite

BibTeXRIS
BibTeX
@misc{borges2026maximal,
  title = {{Maximal Curves with Few Automorphisms}},
  author = {Herivelto Borges and Lucas Reis},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.37777},
  eprint = {2610.37777},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.37777},
}
RIS
TY  - UNPB
TI  - Maximal Curves with Few Automorphisms
AU  - Herivelto Borges
AU  - Lucas Reis
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.37777
UR  - https://arxiv.org/abs/2610.37777
AB  - We consider the problem of constructing maximal curves with few automorphisms over finite fields. Our main ingredient is the determination of the PGL_2 (𝔽̅_p)-stabilizer of the roots of the Chebyshev polynomials, apart from two exceptional cases already treated in the literature. We prove that, in all other cases, the only nontrivial Möbius symmetry is the sign involution. This settles the general case for the automorphism groups of the corresponding Chebyshev curves. We further construct an infinite family of maximal curves of unbounded genus whose only nontrivial automorphism is the hyperelliptic involution. This reveals a contrasting phenomenon to the common heuristic that curves with many rational points tend to have large automorphism groups.
ER  -