Chabauty–Coleman Experiments for Genus 3 Hyperelliptic Curves

  • Jennifer S. Balakrishnan
  • Francesca Bianchi
  • Victoria Cantoral-Farfán
  • Mirela ÇiperianiEmail author
  • Anastassia Etropolski
Conference paper
Part of the Association for Women in Mathematics Series book series (AWMS, volume 19)


We describe a computation of rational points on genus 3 hyperelliptic curves C defined over \(\mathbb {Q}\) whose Jacobians have Mordell–Weil rank 1. Using the method of Chabauty and Coleman, we present and implement an algorithm in SageMath to compute the zero locus of two Coleman integrals and analyze the finite set of points cut out by the vanishing of these integrals. We run the algorithm on approximately 17,000 curves from a forthcoming database of genus 3 hyperelliptic curves and discuss some interesting examples where the zero set includes global points not found in \(C(\mathbb {Q})\).


Chabauty-Coleman method Rational points Hyperelliptic curves 


  1. 1.
    J. S. Balakrishnan, Explicitp-adic methods for elliptic and hyperelliptic curves, Advances on superelliptic curves and their applications, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., vol. 41, IOS, Amsterdam, 2015, pp. 260–285. MR 3525580Google Scholar
  2. 2.
    J. S. Balakrishnan, F. Bianchi, V. Cantoral-Farfán, M. Çiperiani, and A. Etropolski, Sage code,
  3. 3.
    J. S. Balakrishnan, R. W. Bradshaw, and K. S. Kedlaya, Explicit Coleman integration for hyperelliptic curves, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 6197, Springer, Berlin, 2010, pp. 16–31. MR 2721410Google Scholar
  4. 4.
    W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3–4, 235–265, Computational algebra and number theory (London, 1993). MR MR1484478Google Scholar
  5. 5.
    A. J. Best, Explicit Coleman integration in larger characteristic, to appear, Proceedings of ANTS XIII (2018).Google Scholar
  6. 6.
    A. Booker, D. Platt, J. Sijsling, and A. Sutherland, Genus 3 hyperelliptic curves,
  7. 7.
    N. Bruin and M. Stoll, Deciding existence of rational points on curves: An experiment, Experiment. Math. 17 (2008), no. 2, 181–189.MathSciNetCrossRefGoogle Scholar
  8. 8.
    _________ , The Mordell-Weil sieve: proving non-existence of rational points on curves, LMS J. Comput. Math. 13 (2010), 272–306. MR 2685127MathSciNetCrossRefGoogle Scholar
  9. 9.
    A. R. Booker, J. Sijsling, A. V. Sutherland, J. Voight, and D. Yasaki, A database of genus-2 curves over the rational numbers, LMS J. Comput. Math. 19 (2016), no. suppl. A, 235–254.Google Scholar
  10. 10.
    J. S. Balakrishnan and J. Tuitman, Magma code,
  11. 11.
    _________ , Explicit Coleman integration for curves, Arxiv preprint (2017).Google Scholar
  12. 12.
    C. Chabauty, Sur les points rationnels des courbes algébriques de genre supérieur à l’unité, C.R. Acad. Sci. Paris 212 (1941), 882–885.Google Scholar
  13. 13.
    E. Costa, N. Mascot, J. Sijsling, and J. Voight, Rigorous computation of the endomorphism ring of a Jacobian, Mathematics of Computation (2018).Google Scholar
  14. 14.
    R. F. Coleman, Effective Chabauty, Duke Math. J. 52 (1985), no. 3, 765–770. MR 808103MathSciNetCrossRefGoogle Scholar
  15. 15.
    _________ , Torsion points on curves andp-adic abelian integrals, Ann. of Math. (2) 121 (1985), no. 1, 111–168. MR 782557Google Scholar
  16. 16.
    G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366.MathSciNetCrossRefGoogle Scholar
  17. 17.
    D. Grant, On an analogue of the Lutz-Nagell theorem for hyperelliptic curves, J. Number Theory 133 (2013), no. 3, 963–969. MR 2997779MathSciNetCrossRefGoogle Scholar
  18. 18.
    D. Harvey and A. V. Sutherland, Computing Hasse-Witt matrices of hyperelliptic curves in average polynomial time, II, Frobenius distributions: Lang-Trotter and Sato-Tate conjectures, Contemp. Math., vol. 663, Amer. Math. Soc., Providence, RI, 2016, pp. 127–147.Google Scholar
  19. 19.
    N. Koblitz, p-adic numbers,p-adic analysis, and zeta-functions, second ed., Graduate Texts in Mathematics, vol. 58, Springer-Verlag, New York, 1984. MR 754003Google Scholar
  20. 20.
    B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. (1977), no. 47, 33–186 (1978).Google Scholar
  21. 21.
    J. S. Milne, Jacobian varieties, Arithmetic geometry (Storrs, Conn., 1984), Springer, New York, 1986, pp. 167–212. MR 861976CrossRefGoogle Scholar
  22. 22.
    W. McCallum and B. Poonen, The method of Chabauty and Coleman, Explicit methods in number theory, Panor. Synthèses, vol. 36, Soc. Math. France, Paris, 2012, pp. 99–117.Google Scholar
  23. 23.
    J. Paulhus, Decomposing Jacobians of curves with extra automorphisms, Acta Arithmetica 132 (2008), no. 3, 231–244 (eng).MathSciNetCrossRefGoogle Scholar
  24. 24.
    W. A. Stein et al., Sage Mathematics Software (Version 8.1), The Sage Development Team, 2017,
  25. 25.
    M. Stoll, Finite descent obstructions and rational points on curves, Algebra Number Theory 1 (2007), no. 4, 349–391, version of Scholar
  26. 26.
    M. Stoll, Arithmetic of Hyperelliptic Curves, 2014.Google Scholar
  27. 27.
    J. L. Wetherell, Bounding the number of rational points on certain curves of high rank, ProQuest LLC, Ann Arbor, MI, 1997, Thesis (Ph.D.)–University of California, Berkeley. MR 2696280Google Scholar

Copyright information

© The Author(s) and The Association for Women in Mathematics 2019

Authors and Affiliations

  • Jennifer S. Balakrishnan
    • 1
  • Francesca Bianchi
    • 2
  • Victoria Cantoral-Farfán
    • 3
  • Mirela Çiperiani
    • 4
    Email author
  • Anastassia Etropolski
    • 5
  1. 1.Department of Mathematics and StatisticsBoston UniversityBostonUSA
  2. 2.Mathematical InstituteUniversity of OxfordOxfordUK
  3. 3.The Abdus Salam International Center for Theoretical Physics, Mathematics SectionTriesteItaly
  4. 4.Department of MathematicsThe University of Texas at AustinAustinUSA
  5. 5.Department of MathematicsRice University MS 136HoustonUSA

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