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arXiv · 2609.36456

Nonexistence of maximal curves of genus five over $\F_{64}$

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 $\F_{64}$. 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 $\F_{64}$.

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Gilberto B. Almeida Filho, Saeed Tafazolian, Stéfani C. Vieira. 2026-09-29. Nonexistence of maximal curves of genus five over $\F_{64}$. https://arxiv.org/abs/2609.36456

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