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

. For a given genus g ≥ 1 , we give lower 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 ε > 0 and all q large enough, the existence of a curve of genus g over F q with at least 1 + q + (2 g − ε ) √ 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 √ q valid for g ≥ 3 and odd q ≥ 11 . Finally, explicit constructions of towers of curves improve this result, with a bound of the form 1 + q + 4 √ q − 32 valid for all g ≥ 2 and for all q .