Let X be an irreducible algebraic curve defined over the finite field F_q. The number M_q of rational points on X satisfied the Hasse-Weil upper bound,

where g is the genus of X.

This works well when q is large compared to g but not otherwise. In the latter case, the Stöhr–Voloch bound works better. A special case of this is the following, for a plane curve of degree n with not all points as inflexions and q odd:

Various aspects of these results will be discussed.