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Journal Articles Mathematical Proceedings of the Cambridge Philosophical Society Year : 2023

Lower bounds on the maximal number of rational points on curves over finite fields

Abstract

Abstract For a given genus $g \geq 1$ , we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus g over $\mathbb{F}_q$ . As a consequence of Katz–Sarnak theory, we first get for any given $g>0$ , any $\varepsilon>0$ and all q large enough, the existence of a curve of genus g over $\mathbb{F}_q$ with at least $1+q+ (2g-\varepsilon) \sqrt{q}$ rational points. Then using sums of powers of traces of Frobenius of hyperelliptic curves, we get a lower bound of the form $1+q+1.71 \sqrt{q}$ valid for $g \geq 3$ and odd $q \geq 11$ . Finally, explicit constructions of towers of curves improve this result: We show that the bound $1+q+4 \sqrt{q} -32$ is valid for all $g\ge 2$ and for all q .

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hal-04221916 , version 1 (28-09-2023)

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Jonas Bergström, Everett Howe, Elisa Lorenzo García, Christophe Ritzenthaler. Lower bounds on the maximal number of rational points on curves over finite fields. Mathematical Proceedings of the Cambridge Philosophical Society, 2023, pp.1-26. ⟨10.1017/S0305004123000476⟩. ⟨hal-04221916⟩
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