Duke Mathematical Journal

Quadratic Chabauty and rational points, I: p-adic heights

Jennifer S. Balakrishnan and Netan Dogra

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We give the first explicit examples beyond the Chabauty–Coleman method where Kim’s nonabelian Chabauty program determines the set of rational points of a curve defined over Q or a quadratic number field. We accomplish this by studying the role of p-adic heights in explicit non-Abelian Chabauty.

Article information

Duke Math. J., Volume 167, Number 11 (2018), 1981-2038.

Received: 3 May 2016
Revised: 9 March 2018
First available in Project Euclid: 20 July 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14G05: Rational points
Secondary: 11G50: Heights [See also 14G40, 37P30] 14G40: Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30]

rational points on higher genus curves non-Abelian Chabauty p-adic heights


Balakrishnan, Jennifer S.; Dogra, Netan. Quadratic Chabauty and rational points, I: $p$ -adic heights. Duke Math. J. 167 (2018), no. 11, 1981--2038. doi:10.1215/00127094-2018-0013. https://projecteuclid.org/euclid.dmj/1532073621

Export citation


  • [1] J. S. Balakrishnan and A. Besser, Computing local $p$-adic height pairings on hyperelliptic curves, Int. Math. Res. Not. IMRN 2012, no. 11, 2405–2444.
  • [2] J. S. Balakrishnan and A. Besser, Coleman-Gross height pairings and the $p$-adic sigma function, J. Reine Angew. Math. 698 (2015), 89–104.
  • [3] J. S. Balakrishnan, A. Besser, and J. S. Müller, Quadratic Chabauty: $p$-adic height pairings and integral points on hyperelliptic curves, J. Reine Angew. Math. 720 (2016), 51–79.
  • [4] J. S. Balakrishnan, A. Besser, and J. S. Müller, Computing integral points on hyperelliptic curves using quadratic Chabauty, Math. Comp. 86 (2017), 1403–1434.
  • [5] J. S. Balakrishnan, I. Dan-Cohen, M. Kim, and S. Wewers, A non-abelian conjecture of Tate-Shafarevich type for hyperbolic curves, to appear in Math Ann.
  • [6] J. S. Balakrishnan and N. Dogra, Sage code, September 2017, https://github.com/jbalakrishnan/QCI.
  • [7] J. S. Balakrishnan and N. Dogra, Quadratic Chabauty and rational points, I: $p$-adic heights, preprint, arXiv:1601.00388v2 [math.NT].
  • [8] J. S. Balakrishnan and N. Dogra, Quadratic Chabauty and rational points, II: Generalised height functions on Selmer varieties, preprint, arXiv:1705.00401v1 [math.NT].
  • [9] J. S. Balakrishnan, N. Dogra, J. S. Muller, J. Tuitman, and J. Vonk, Explicit Chabauty-Kim for the split Cartan modular curve of level 13, preprint, arXiv:1711.05846v1 [math.NT].
  • [10] A. Besser, “The $p$-adic height pairings of Coleman-Gross and of Nekovář,” in Number Theory, CRM Proc. Lecture Notes 36, Amer. Math. Soc., Providence, 2004, 13–25.
  • [11] Y. Bilu and P. Parent, Serre’s uniformity problem in the split Cartan case, Ann. of Math. (2) 173 (2011), 569–584.
  • [12] Y. Bilu, P. Parent, and M. Rebolledo, Rational points on $x_{0}^{+}(p^{r})$, Ann. Inst. Fourier (Grenoble) 63 (2013), 957–984.
  • [13] S. Bloch and K. Kato, “ ${L}$-functions and Tamagawa numbers of motives,” in The Grothendieck Festschrift, Vol. I, Progr. Math. 86, Birkhäuser, Boston, 1990, 333–400.
  • [14] N. Bruin and M. Stoll, The Mordell-Weil sieve: proving non-existence of rational points on curves, LMS J. Comput. Math. 13 (2010), 272–306.
  • [15] C. Chabauty, Sur les points rationnels des courbes algébriques de genre supérieur à l’unité, C. R. Acad. Sci. Paris 212 (1941), 882–885.
  • [16] J. Coates and M. Kim, Selmer varieties for curves with CM Jacobians, Kyoto J. Math. 50 (2010), 827–852.
  • [17] R. F. Coleman, Effective Chabauty, Duke Math. J. 52 (1985), 765–770.
  • [18] R. F. Coleman and B. H. Gross, “$p$-adic heights on curves,” in Algebraic Number Theory, Adv. Stud. Pure Math. 17, Academic Press, Boston, 1989, 73–81.
  • [19] H. Daniels and Á. Lozano-Robledo, personal communication, March 2015.
  • [20] H. Darmon, V. Rotger, and I. Sols, “Iterated integrals, diagonal cycles and rational points on elliptic curves” in Publications mathématiques de Besançon: Algèbre et théorie des nombres, 2012/2, Publ. Math. Besançon Algèbre Théorie Nr. 2012, Presses Univ. Franche-Comté, Besançon, 2012, 19–46.
  • [21] P. Deligne, “Le groupe fondamental de la droite projective moins trois points,” in Galois Groups over $\mathbb{Q}$ (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ. 16, Springer, New York, 1989, 79–297.
  • [22] P. Deligne and A. B. Goncharov, Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. Éc. Norm. Supér. (4) 38 (2005), 1–56.
  • [23] N. Dogra, Topics in the theory of Selmer varieties, Ph.D. dissertation, Oxford University, Oxford, 2015.
  • [24] J. S. Ellenberg and D. R. Hast, Rational points on solvable curves over $\mathbb{Q}$ via non-abelian Chabauty, preprint, arXiv:1706.00525v2 [math.NT].
  • [25] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), 349–366.
  • [26] E. V. Flynn and J. L. Wetherell, Finding rational points on bielliptic genus 2 curves, Manuscripta Math. 100 (1999), 519–533.
  • [27] J.-M. Fontaine and B. Perrin-Riou, “Autour des conjectures de Bloch et Kato: cohomologie Galoisienne et valeurs de fonctions $L$,” in Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math 55, Amer. Math. Soc., Providence, 1994, 599–706.
  • [28] W. Fulton, Intersection Theory, 2nd ed., Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin, 2013.
  • [29] A. Grothendieck, P. Deligne, and N. Katz, Groupes de monodromie en géométrie algébrique, I, Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I), Lecture Notes in Math. 288, Springer, New York, 1972; II, SGA 7 II, 340, 1973.
  • [30] R. H. Kaenders, The mixed Hodge structure on the fundamental group of a punctured Riemann surface, Proc. Amer. Math. Soc. 129 (2001), 1271–1281.
  • [31] M. Kim, The motivic fundamental group of $\mathbf{P}^{1}\setminus\{0,1,\infty\}$ and the theorem of Siegel, Invent. Math. 161 (2005), 629–656.
  • [32] M. Kim, The unipotent Albanese map and Selmer varieties for curves, Publ. Res. Inst. Math. Sci. 45 (2009), 89–133.
  • [33] M. Kim, Tangential localization for Selmer varieties, Duke Math. J. 161 (2012), 173–199.
  • [34] M. Kim and A. Tamagawa, The $l$-component of the unipotent Albanese map, Math. Ann. 340 (2008), 223–235.
  • [35] B. Mazur, W. Stein, and J. Tate, Computation of $p$-adic heights and log convergence, Doc. Math. Extra Vol. (2006), 577–614.
  • [36] J. Nekovář, “On $p$-adic height pairings,” in Séminaire de Théorie des Nombres, Paris, 1990–91, Progr. Math. 108, Birkhäuser, Boston, 1993, 127–202.
  • [37] M. C. Olsson, Towards non-abelian $p$-adic Hodge theory in the good reduction case, Mem. Amer. Math. Soc. 210 (2011), no. 990.
  • [38] B. Poonen, E. F. Schaefer, and M. Stoll, Twists of $X(7)$ and primitive solutions to $x^{2}+y^{3}=z^{7}$, Duke Math. J. 137 (2007), 103–158.
  • [39] M. Raynaud, “$1$-motifs et monodromie géométrique,” in Périodes $p$-adiques (Bures-sur-Yvette, 1988), Astérisque 223, Soc. Math. France, Montrouge, 1994, 295–319.
  • [40] V. Scharaschkin, Local-global problems and the Brauer-Manin obstruction, Ph.D. dissertation, University of Michigan, Ann Arbor, 1999.
  • [41] A. J. Scholl, “Height pairings and special values of $L$-functions,” in Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math. 55, Amer. Math. Soc., Providence, 1994, 571–598.
  • [42] J.-P. Serre, Galois Cohomology, Springer, Berlin, 1997.
  • [43] S. Siksek, Explicit Chabauty over number fields, Algebra Number Theory 7 (2013), 765–793.
  • [44] J. H. Silverman, Computing heights on elliptic curves, Math. Comp. 51 (1988), 339–358.
  • [45] The Sage Developers, Sagemath, the Sage Mathematics Software System (Version 8.0), 2017, http://www.sagemath.org.
  • [46] M. Waldschmidt, On the $p$-adic closure of a subgroup of rational points on an Abelian variety, Afr. Mat. 22 (2011), 79–89.