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Explicit quadratic Chabauty over number fields

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Abstract

We generalize the explicit quadratic Chabauty techniques for integral points on odd degree hyperelliptic curves and for rational points on genus 2 bielliptic curves to arbitrary number fields using restriction of scalars. This is achieved by combining equations coming from Siksek’s extension of classical Chabauty with equations defined in terms of p-adic heights attached to independent continuous idele class characters. We give several examples to show the practicality of our methods.

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References

  1. J. S. Balakrishnan and A. Besser, Coleman-Gross height pairings and the p-adic sigma function, Journal für die Reine und Angewandte Mathematik 698 (2015), 89–104.

    MathSciNet  MATH  Google Scholar 

  2. J. S. Balakrishnan, A. Besser and J. S. Müller, Quadratic Chabauty: p-adic heights and integral points on hyperelliptic curves, Journal für die Reine und Angewandte Mathematik 720 (2016), 51–79.

    MathSciNet  MATH  Google Scholar 

  3. J. S. Balakrishnan, A. Besser and J. S. Müller, Computing integral points on hyperelliptic curves using quadratic Chabauty, Mathematics of Computation 86 (2017), 1403–1434.

    Article  MathSciNet  Google Scholar 

  4. J. S. Balakrishnan, I. Dan-Cohen, M. Kim and S. Wewers, A non-abelian conjecture of Tate-Shafarevich type for hyperbolic curves, Mathematische Annalen 372 (2018), 369–428.

    Article  MathSciNet  Google Scholar 

  5. J. S. Balakrishnan and N. Dogra, Quadratic Chabauty and rational points II: Generalised height functions on Selmer varieties, International Mathematics Research Notices, Article no. rnz362, to appear.

  6. J. S. Balakrishnan and N. Dogra, Quadratic Chabauty and rational points, I: p-adic heights, Duke Mathematical Journal 167 (2018), 1981–2038.

    Article  MathSciNet  Google Scholar 

  7. J. S. Balakrishnan, N. Dogra, J. S. Müller, J. Tuitman and J. Vonk, Explicit Chabauty-Kim for the split Cartan modular curve of level 13, Annals of Mathematics 189 (2019), 885–944.

    Article  MathSciNet  Google Scholar 

  8. J. S. Balakrishnan, J. S. Müller and W. A. Stein, A p-adic analogue of the conjecture of Birch and Swinnerton-Dyer for modular abelian varieties, Mathematics of Computation 85 (2016), 983–1016.

    Article  MathSciNet  Google Scholar 

  9. J. S. Balakrishnan, M. Çiperiani, J. Lang, B. Mirza and R. Newton, Shadow lines in the arithmetic of elliptic curves, in Directions in Number Theory, Association for Women in Mathematics Series, Vol. 3, Springer, Cham, 2016, pp. 33–55.

    Chapter  Google Scholar 

  10. A. Besser, p-adic Arakelov theory, Journal of Number Theory 111 (2005), 318–371.

    Article  MathSciNet  Google Scholar 

  11. F. Bianchi, Quadratic Chabauty for (bi)elliptic curves and Kim’s conjecture, Algebra & Number Theory 14 (2020), 2369–2416.

    Article  MathSciNet  Google Scholar 

  12. W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, Journal of Symbolic Computation 24 (1997), 235–265.

    Article  MathSciNet  Google Scholar 

  13. N. Bruin, Chabauty methods using elliptic curves, Journal für die Reine und Angewandte Mathematik 562 (2003), 27–49.

    MathSciNet  MATH  Google Scholar 

  14. N. Bruin and M. Stoll, The Mordell-Weil sieve: proving non-existence of rational points on curves, LMS Journal of Computation and Mathematics 13 (2010), 272–306.

    Article  MathSciNet  Google Scholar 

  15. A. Brumer, On the units of algebraic number fields, Mathematika 14 (1967), 121–124.

    Article  MathSciNet  Google Scholar 

  16. C. Chabauty, Sur les points rationnels des courbes algébriques de genre supérieur à l’unité, Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 212 (1941), 882–885.

    MATH  Google Scholar 

  17. R. Coleman, Torsion points on curves and p-adic abelian integrals, Annals of Mathematics 121 (1985), 111–168.

    Article  MathSciNet  Google Scholar 

  18. R. Coleman and B. Gross, p-adic heights on curves, in Algebraic Number Theory, Advanced Studies in Pure Mathematics, Vol. 17, Academic Press, Boston, MA, 1989, pp. 73–81.

    Google Scholar 

  19. K. Conrad, A multivariable Hensel’s lemma, https://kconrad.math.uconn.edu/blurbs/gradnumthy/multivarhensel.pdf.

  20. N. Dogra, Unlikely intersections and the Chabauty-Kim method over number fields, preprint, https://arxiv.org/abs/1903.05032.

  21. B. H. Gross, Local heights on curves, in Arithmetic Geometry (Storrs, CT, 1984), Springer, New York, 1986, pp. 327–339.

    Chapter  Google Scholar 

  22. D. R. Hast, Functional transcendence for the unipotent Albanese map, Algebra & Number Theory, to appear, https://arxiv.org/abs/1911.00587.

  23. M. Kim, The motivic fundamental group of ℙ1 − {0, 1, ∞} and the theorem of Siegel, Inventiones Mathematicae 161 (2005), 629–656.

    Article  MathSciNet  Google Scholar 

  24. M. Kim, The unipotent Albanese map and Selmer varieties for curves, Kyoto University. Research Institute for Mathematical Sciences. Publications 45 (2009), 89–133.

    Article  MathSciNet  Google Scholar 

  25. F.-V. Kuhlmann, Maps on ultrametric spaces, Hensel’s lemma, and differential equations over valued fields, Communications in Algebra 39 (2011), 1730–1776.

    Article  MathSciNet  Google Scholar 

  26. H.-W. Leopoldt, Zur Arithmetik in abelschen Zahlkörpern, Journal für die Reine und Angewandte Mathematik 209 (1962), 54–71.

    MathSciNet  MATH  Google Scholar 

  27. The LMFDB Collaboration, The L-functions and Modular Forms Database, http://www.lmfdb.org.

  28. B. Mazur, Modular curves and arithmetic, in Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Warsaw, 1983), PWN, Warsaw, 1984, pp. 185–211.

    Google Scholar 

  29. B. Mazur and J. Tate, Canonical height pairings via biextensions, in Arithmetic and Geometry, Vol. I, Progress in Mathematics, Vol. 35, Birkhäuser, Boston, MA, 1983, pp. 195–237.

    Chapter  Google Scholar 

  30. W. McCallum and B. Poonen, The method of Chabauty and Coleman, in Explicit Methods in Number Theory, Panoramas et Synthèses, Vol. 36, Société Mathématique de France, Paris, 2012, pp. 99–117.

    Google Scholar 

  31. J. Nekovář, On p-adic height pairings, in Séminaire de Théorie des Nombres, Paris, 1990–91, Progress in Mathematics, Vol. 108, Birkhäuser, Boston, MA, 1993, pp. 127–202.

    Chapter  Google Scholar 

  32. B. Poonen, The p-adic closure of a subgroup of rational points on a commutative algebraic group, http://www-math.mit.edu/~poonen/papers/leopoldt.pdf.

  33. P. Schneider, p-adic height pairings. I, Inventiones Mathematicae 69 (1982), 401–409.

    Article  MathSciNet  Google Scholar 

  34. S. Siksek, Explicit Chabauty over number fields, Algebra & Number Theory 7 (2013), 765–793.

    Article  MathSciNet  Google Scholar 

  35. W. A. Stein et al., Sage Mathematics Software (Version 5.6), http://www.sagemath.org.

  36. N. Triantafillou, Restriction of scalars, the Chabauty-Coleman method, and ℙ1 {0, 1, ∞}, Ph.D. Thesis, MIT, 2019.

  37. M. Waldschmidt, On the p-adic closure of a subgroup of rational points on an Abelian variety, Afrika Matematika 22 (2011), 79–89.

    Article  MathSciNet  Google Scholar 

  38. J. L. Wetherell, Bounding the number of rational points on certain curves of high rank, Ph.D. Thesis, University of California, Berkeley, ProQuest, Ann Arbor, MI, 1997.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, Boston, MA, 02215, USA

    Jennifer S. Balakrishnan

  2. Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Be’er-Sheva, 84105, Israel

    Amnon Besser

  3. Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, P.O.B. 407, 9700 AK, Groningen, The Netherlands

    Francesca Bianchi & J. Steffen Müller

Authors
  1. Jennifer S. Balakrishnan
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  2. Amnon Besser
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  3. Francesca Bianchi
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  4. J. Steffen Müller
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Correspondence to Amnon Besser.

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Balakrishnan, J.S., Besser, A., Bianchi, F. et al. Explicit quadratic Chabauty over number fields. Isr. J. Math. 243, 185–232 (2021). https://doi.org/10.1007/s11856-021-2158-5

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  • DOI: https://doi.org/10.1007/s11856-021-2158-5

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