Geometric methods for improving the upper bounds on the number of rationalpoints on algebraic curves over finite fields

Currently, the best upper bounds on the number of rational points on an abs olutely irreducible, smooth, projective algebraic curve of genus g defined over a finite field F-q come either from Serre's refinement of the Well bou nd if the genus is small compared to q, or from Oesterle'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 des cent, and Honda-Tate theory. Examples of improvements on the bounds include lowering them for a wide range of small genus when q = 2(3), 2(5), 2(13), 3(3), 3(5), 5(3),5(7), and when q = 2(2s), s > 1. For large genera, isolate d improvements are obtained for q = 3, 8, 9.