Maximal singular curves over finite fields from elliptic and hyperelliptic curves
Abstract
We construct explicit families of maximal singular curves over finite fields. A maximal singular curve is a curve that attains the Aubry-Perret bound, which is the natural extension of the Hasse-Weil-Serre bound on the maximum number of rational points on a smooth curve over a finite field. Starting from a smooth elliptic or hyperelliptic curve C̃ over 𝔽q (q odd), we generate singular curves C with non-split nodes or cusps. In our approach we use linear projections of geometric embeddings, and we apply Stöhr's embedding of hyperelliptic Gorenstein curves and the Rosa-Stöhr theory of trigonal Gorenstein curves. The construction is applied to the Tafazolian and Tafazolian-Torres smooth maximal curves. Finally, we also determine the gonality of all obtained curves.
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BibTeX
@misc{hilario2026maximal,
title = {{Maximal singular curves over finite fields from elliptic and hyperelliptic curves}},
author = {Cesar Hilario and Rodrigo Salomão and Renato Vidal Martins},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22557},
eprint = {2609.22557},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22557},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Maximal singular curves over finite fields from elliptic and hyperelliptic curves
AU - Cesar Hilario
AU - Rodrigo Salomão
AU - Renato Vidal Martins
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.22557
UR - https://arxiv.org/abs/2609.22557
AB - We construct explicit families of maximal singular curves over finite fields. A maximal singular curve is a curve that attains the Aubry-Perret bound, which is the natural extension of the Hasse-Weil-Serre bound on the maximum number of rational points on a smooth curve over a finite field. Starting from a smooth elliptic or hyperelliptic curve C̃ over 𝔽_q (q odd), we generate singular curves C with non-split nodes or cusps. In our approach we use linear projections of geometric embeddings, and we apply Stöhr's embedding of hyperelliptic Gorenstein curves and the Rosa-Stöhr theory of trigonal Gorenstein curves. The construction is applied to the Tafazolian and Tafazolian-Torres smooth maximal curves. Finally, we also determine the gonality of all obtained curves.
ER -