Improved upper bounds for the number of rational points on algebraic curves over finite fields

Currently, the best known bounds on the number of rational points on an absolutely irreducible, smooth, projective algebraic curve of genus g, defined over a finite field %plane1D;53D;_q, generally come either from Serre's refinement of the Weil bound if the genus is small compared to q, or from the optimization of the explicit formulae if the genus is large. We give methods for improving these bounds in both cases. Examples of improvements on the bounds include lowering them for a wide range of small genus when q = 8, 32, 2¹³, 27, 243, 125, and when q = 2^s, s > 1. For large genera, isolated improvements are obtained for q = 3, 8, 9.