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Published in:

2021 | OriginalPaper | Chapter

# Chabauty–Coleman Computations on Rank 1 Picard Curves

Authors : Sachi Hashimoto, Travis Morrison

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## 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.
Appendix
Available only for authorised users
Footnotes
1
For practical purposes, we terminated RankBounds if it ran for 120 s without returning an answer. We discarded 520 curves from the total 5335 curves in the database provided by Sutherland either from terminating RankBounds or due to a “Runtime error”.

2
For some heuristics for choosing e given a fixed N see Appendix 5.

3
Due to a minor bug in the code of Balakrishnan and Tuitman for computing local coordinates at very infinite points, which in certain cases yields a Runtime Error instead of computing the coordinates, we sometimes choose the next largest possible prime.

4
For a given curve there are often multiple choices of g, and we pick the one with the smallest first split prime.

5
For the remaining 544 curves, it can be quite computationally expensive to compute $$X(\mathbb {Q})$$ depending on the parameters N, e, and p. Individual curves can require up to several hours. This computation was run on a single core of a 28-core 2.2 GHz Intel 2 Xeon Gold server with 256 GB RAM.

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