A bootstrap for the number of ${\mathbb{F}}_{q^r}$-rational points on a curve over ${\mathbb{F}}_q$

In this note we present a fast algorithm that finds for any $r$ the number $N_r$ of ${\mathbb{F}}_{q^r}$ rational points on a smooth absolutely irreducible curve $C$ defined over ${\mathbb{F}}_q$ assuming that we know $N_1,\cdots ,N_g$, where $g$ is the genus of $C$. The proof of its validity is given in detail and its working are illustrated with several examples. In an Appendix we list the Python function in which we have implemented the algorithm together with other routines used in the examples.