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

Jonas Bergström, Everett W. Howe, Elisa Lorenzo García, Christophe Ritzenthaler

Published 2022-04-18Version 1

For a given genus $g \geq 1$, we give minimal bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus $g$ over $\F_q$. As a consequence of Katz-Sarnak theory, we first get for any given $g>0$, any $\epsilon>0$ and all $q$ large enough, the existence of a curve of genus $g$ over $\F_q$ with at least $1+q+ (2g-\epsilon) \sqrt{q}$ 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~$q$.