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

TL;DR

This paper introduces a fast algorithm to compute the number of rational points on a curve over extension fields, given initial point counts over the base field, with detailed proofs and Python implementation.

Contribution

The paper presents a novel, efficient algorithm for calculating extension field rational points on curves, expanding computational tools in algebraic geometry.

Findings

Algorithm accurately computes $N_r$ for various $r$

Implementation verified with multiple examples

Provides detailed proof and Python code

Abstract

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.