Brice Miayoka Moussolo

arXiv2025

Rational and isolated quadratic points on hyperelliptic curves of rank 0 and small genus

In this article, we present a method for computing rational points on hyperelliptic curves of genus~3 and isolated quadratic points on hyperelliptic curves of genus~2 and~3 whose Jacobians have rank~0. Our approach begins by computing the image of the Mordell--Weil group on the associated Kummer variety and then determining which of these points correspond to rational or isolated quadratic points on the curve. We have developed and implemented this algorithm using the computer algebra system \texttt{\texttt{Magma}}. The method takes advantage of structural properties specific to hyperelliptic curves and their Jacobians. We applied our algorithm to a large dataset, analyzing 7,396 genus~3 hyperelliptic curves and 12,075 genus~2 hyperelliptic curves.

math.NT↗

arXiv2023

Rational Points of some genus $3$ curves from the rank $0$ quotient strategy

In 1922, Mordell conjectured that the set of rational points on a smooth curve $C$ over $\mathbb{Q}$ with genus $g \ge 2$ is finite. This has been proved by Faltings in 1983. However, Coleman determined in 1985 an upper bound of $#C(\mathbb{Q})$ by following Chabauty's approach which considers the special case when the Jacobian variety of $C$ has Mordell-Weil rank $< g$. In 2006, Stoll improved the Coleman's bound. Balakrishnan with her co-authors in [1] implemented the Chabauty-Coleman method to compute the rational points of genus $3$ hyperelliptic curves. Then, Hashimoto and Morrison [8] did the same work for Picard curves. But it happens that this work has not yet been done for all genus 3 curves. In this paper, we describe an algorithm to compute the complete set of rational points $C(\mathbb{Q})$ for any genus $3$ curve $C/\mathbb{Q}$ that is a degree-$2$ cover of a genus $1$ curve whose Jacobian has rank $0$. We implemented this algorithm in Magma, and we ran it on approximately $40, 000$ curves selected from databases of plane quartics and genus $3$ hyperellitic curves. We discuss some interesting examples, and we exhibit curves for which the number of rational points meets the Stoll's bound

math.NT↗