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 consequenceof 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) root 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 root qvalid for g >= 3 and odd q >= 11. Finally, explicit constructions of towers of curves improve this result: We show that the bound 1 + q + 4 root q - 32 is valid for allg >= 2 and for all q.