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A geometric linear Chabauty comparison theorem

Volume 202 / 2022

Sachi Hashimoto, Pim Spelier Acta Arithmetica 202 (2022), 67-88 MSC: Primary 11G30; Secondary 11D45. DOI: 10.4064/aa210616-6-10 Published online: 26 November 2021


The Chabauty–Coleman method is a $p$-adic method for finding all rational points on curves of genus $g$ whose Jacobians have Mordell–Weil rank $r \lt g$. Recently, Edixhoven and Lido developed a geometric quadratic Chabauty method that was adapted by Spelier to cover the case of geometric linear Chabauty. We compare the geometric linear Chabauty method and the Chabauty–Coleman method and show that geometric linear Chabauty can outperform Chabauty–Coleman in certain cases. However, as Chabauty–Coleman remains more practical for general computations, we discuss how to strengthen Chabauty–Coleman to make it theoretically equivalent to geometric linear Chabauty. We apply these methods to genus $2$ and genus $3$ curves.


  • Sachi HashimotoDepartment of Mathematics and Statistics
    Boston University
    111 Cummington Mall
    Boston, MA 02215, USA
  • Pim SpelierMathematical Institute
    Leiden University
    P.O. Box 9512
    2300 RA Leiden, The Netherlands

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