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

Abstract

In 1922, Mordell conjectured that the set of rational points on a smooth curve C over Q with genus g ≥ 2 is finite. This has been proved by Faltings in 1983. However, Coleman determined in 1985 an upper bound of #C(Q) by following Chabauty'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(Q) for any genus 3 curve C 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.