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

Abstract

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.