Weierstrass Points and Curves Over Finite Fields

Abstract

For any projective embedding of a non‐singular irreducible complete algebraic curve defined over a finite field, we obtain an upper bound for the number of its rational points. The constants in the bound are related to the Weierstrass order‐sequence associated with the projective embedding. The bounds obtained lead to a proof of the Riemann hypothesis for curves over finite fields and yield several improvements on it.