Papers › Chabauty-Coleman computations on rank 1 Picard curves
Chabauty-Coleman computations on rank 1 Picard curves
Sachi Hashimoto, Travis Morrison
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We provably compute the full set of rational points on 1403 Picard curves defined over 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.
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