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

Abstract

For a given genus $g\ge 1$, 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 $\epsilon >0$ and all $q$ large enough, the existence of a curve of genus $g$ over ${\mathbb{F}}_q$ 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 $g\ge 3$ and odd $q\ge 11$. 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 $q$.

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