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PreprintCC BY 4.0arXiv

Nonexistence of maximal curves of genus five over F64

  • Gilberto B. Almeida Filho
  • Saeed Tafazolian
  • Stéfani C. Vieira
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Abstract

We show that there is no maximal curve of genus five over F64. As a consequence, N64(5)=140, and the genus spectrum of maximal curves over F64 is determined. The proof uses the vanishing of the third iterate of the Cartier operator. We prove that a nonhyperelliptic curve of genus five in characteristic two satisfying this condition is nontrigonal. Its canonical theta characteristic defines a separable cover of degree four with one geometric branch value. The two possible ramification types give either a rational subcanonical point or a point bound obtained from the cubic resolvent. Both cases exclude maximality over F64.

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Cite

BibTeXRIS
BibTeX
@misc{filho2026nonexistence,
  title = {{Nonexistence of maximal curves of genus five over $\F_64$}},
  author = {Gilberto B. Almeida Filho and Saeed Tafazolian and Stéfani C. Vieira},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.36456},
  eprint = {2609.36456},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.36456},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Nonexistence of maximal curves of genus five over F_64
AU  - Gilberto B. Almeida Filho
AU  - Saeed Tafazolian
AU  - Stéfani C. Vieira
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.36456
UR  - https://arxiv.org/abs/2609.36456
AB  - We show that there is no maximal curve of genus five over F_64. As a consequence, N_64(5)=140, and the genus spectrum of maximal curves over F_64 is determined. The proof uses the vanishing of the third iterate of the Cartier operator. We prove that a nonhyperelliptic curve of genus five in characteristic two satisfying this condition is nontrigonal. Its canonical theta characteristic defines a separable cover of degree four with one geometric branch value. The two possible ramification types give either a rational subcanonical point or a point bound obtained from the cubic resolvent. Both cases exclude maximality over F_64.
ER  -