Currently available as an official version and as a preprint.

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 every g > 0, every ε > 0, and all q large enough, the existence of a curve of genus g over F_q with at least 1 + q + (2g − 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.