Chabauty–Coleman Computations on Rank 1 Picard Curves

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.