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The a-number of the Serre-type maximal curves of Abdón-Bezerra-Quoos
Abstract
For every prime power q, we determine the a-number of the Serre-type maximal curves X(q,n): yq2−y=x(qn+1)/(q+1) over 𝔽q2n, n≥ 3 odd, introduced by Abdón, Bezerra and Quoos as a generalization of Serre's curve y4+y=x3 over 𝔽64. The Weierstrass semigroup at the unique point at infinity gives an explicit basis of holomorphic differentials, on which the Cartier operator acts by monomials; this reduces the a-number to a lattice-point count, which we evaluate in closed form.
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BibTeX
@misc{shafighi2026anumber,
title = {{The a-number of the Serre-type maximal curves of Abdón-Bezerra-Quoos}},
author = {Mahyar Shafighi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15266},
eprint = {2610.15266},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15266},
}
RIS
TY - UNPB
TI - The a-number of the Serre-type maximal curves of Abdón-Bezerra-Quoos
AU - Mahyar Shafighi
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15266
UR - https://arxiv.org/abs/2610.15266
AB - For every prime power q, we determine the a-number of the Serre-type maximal curves X(q,n): y^q^2−y=x^(q^n+1)/(q+1) over 𝔽_q^2n, n≥ 3 odd, introduced by Abdón, Bezerra and Quoos as a generalization of Serre's curve y^4+y=x^3 over 𝔽_64. The Weierstrass semigroup at the unique point at infinity gives an explicit basis of holomorphic differentials, on which the Cartier operator acts by monomials; this reduces the a-number to a lattice-point count, which we evaluate in closed form.
ER -