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 Störh-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, Störh-Voloch's and Ihara's.

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