  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 F_q come either from Serre's refinement of the Weil bound
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 descent,
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, isolated improvements are
obtained for q=3,8,9.

Algebraic Geometry;Number Theory