Geometric Methods for Improving the Upper Bounds on the Number of Rational Points on Algebraic Curves over Finite Fields

Currently, the best upper bounds on the number of rational points on an absolutely irreducible, smooth, projective algebraic curve of genus g defined over a finite field Fq come either from Serre’s refinement of the Weil bound if the genus is small compared to q, or from Oesterlé’s optimization of the explicit formulae method if the genus is large. This paper presents three methods for improving these bounds. The arguments used are the indecomposability of the theta divisor of a curve, Galois descent, and Honda-Tate theory. Examples of improvements on the bounds include lowering them for a wide range of small genus when q = 2, 2, 2, 3, 3, 5, 5, and when q = 2, s > 1. For large genera, isolated improvements are obtained for q = 3, 8, 9.