Author:
BERGSTRÖM JONAS,HOWE EVERETT W.,LORENZO GARCÍA ELISA,RITZENTHALER CHRISTOPHE
Abstract
AbstractFor 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.
Publisher
Cambridge University Press (CUP)