Chabauty-Coleman computations on rank 1 Picard curves

TL;DR

This paper demonstrates the complete determination of rational points on 1403 Picard curves over $ ext{Q}$ with rank 1 Jacobians using an extended Chabauty-Coleman method, including new computational techniques.

Contribution

It extends Magma code for Coleman integration to compute $p$-adic integrals on curves over number fields, enabling effective Chabauty for a broader class of curves.

Findings

Successfully computed rational points on 1403 Picard curves

Extended Magma code for Coleman integrals over number fields

Identified points over number fields in Chabauty sets

Abstract

We provably compute the full set of rational points on 1403 Picard curves defined over $\mathbb{Q}$ with Jacobians of Mordell-Weil rank $1$ using the Chabauty-Coleman method. To carry out this computation, we extend Magma code of Balakrishnan and Tuitman for Coleman integration. The new code computes $p$-adic (Coleman) integrals on curves to points defined over number fields where the prime $p$ splits completely and implements effective Chabauty for curves whose Jacobians have infinite order points that are not the image of a rational point under the Abel-Jacobi map. We discuss several interesting examples of curves where the Chabauty-Coleman set contains points defined over number fields.