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Pré-Publication, Document De Travail Année : 2022

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

Résumé

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, with a bound of the form $1+q+4 \sqrt{q} -32$ valid for all $g\ge 2$ and for all $q$.

Dates et versions

hal-03685062 , version 1 (01-06-2022)

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Jonas Bergström, Everett W. Howe, Elisa Lorenzo García, Christophe Ritzenthaler. Lower bounds on the maximal number of rational points on curves over finite fields. 2022. ⟨hal-03685062⟩
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