Bounds for the number of points on curves over finite fields

Let $\mathcal{X}$ be a projective irreducible nonsingular algebraic curve defined over a finite field ${\mathbb{F}}_q$. This paper presents a variation of the St"orh-Voloch theory and sets new bounds to the number of ${\mathbb{F}}_{q^r}$-rational points on $\mathcal{X}$. In certain cases, where comparison is possible, the results are shown to improve other bounds such as Weil’s, St"orh-Voloch’s and Ihara’s.