Quadratic Chabauty and rational points. I: \(p\)-adic heights. (English) Zbl 1401.14123

Let \(X\) be a smooth projective curve of genus \(g\geq 2\) defined over a number field \(K\). The \(p\)-adic method of Chabauty and Coleman gives an effective way of computing the finite set \(X(K)\), under some conditions, the main one being that the Mordell-Weil rank of the Jacobian be less than \(g\). Kim’s non abelian approach replaces the Jacobian by the Selmer variety [M. Kim, Invent. Math. 161, No. 3, 629–656 (2005; Zbl 1090.14006)]. In the paper under review, the authors introduce new techniques for studying Selmer varieties which enable them to produce new methods for determining the rational points of a variety over \({\mathbb{Q}}\) or over a quadratic field when the Mordell-Weil rank is equal to \(g\). The methods generalize those which were used for the study of integral points on hyperelliptic curves using \(p\)-adic heights in [J. S. Balakrishnan et al., J. Reine Angew. Math. 720, 51–79 (2016; Zbl 1350.11067)]. They also use the results of the PhD Thesis of the second author [Topics in the theory of Selmer varieties. Oxford: Oxford University (2015)]. A crucial role in the proof is played by Nekovář’s approach to the \(p\)-adic height pairing [J. Nekovář, Prog. Math. 108, 127–202 (1993; Zbl 0859.11038)]. As an example, the authors show how to compute explicitly a finite set containing the rational points over \({\mathbb{Q}}\) or over a quadratic number field for a genus \(2\) bielliptic curve of Mordell-Weil rank \(2\). The example of \(X_0(37)\) over \({\mathbb{Q}}(i)\) is given; in an appendix to this paper, Stephen Müller carries out the Mordell-Weil sieve computations.


14G05 Rational points
11G50 Heights
14G40 Arithmetic varieties and schemes; Arakelov theory; heights


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[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. · Zbl 1284.11097
[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. · Zbl 1348.11091
[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. · Zbl 1350.11067
[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. · Zbl 1376.11053
[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. · Zbl 1153.11316
[11] Y. Bilu and P. Parent, Serre’s uniformity problem in the split Cartan case, Ann. of Math. (2) 173 (2011), 569–584. · Zbl 1278.11065
[12] Y. Bilu, P. Parent, and M. Rebolledo, Rational points on \(x_{0}^{+}(p^{r})\), Ann. Inst. Fourier (Grenoble) 63 (2013), 957–984. · Zbl 1307.11075
[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. · Zbl 0768.14001
[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. · Zbl 1278.11069
[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. · Zbl 0025.24902
[16] J. Coates and M. Kim, Selmer varieties for curves with CM Jacobians, Kyoto J. Math. 50 (2010), 827–852. · Zbl 1283.11092
[17] R. F. Coleman, Effective Chabauty, Duke Math. J. 52 (1985), 765–770. · Zbl 0588.14015
[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. · Zbl 0758.14009
[19] 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. · Zbl 1332.11054
[20] 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.
[21] P. Deligne and A. B. Goncharov, Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. Éc. Norm. Supér. (4) 38 (2005), 1–56. · Zbl 1084.14024
[22] N. Dogra, Topics in the theory of Selmer varieties, Ph.D. dissertation, Oxford University, Oxford, 2015.
[23] 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].
[24] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), 349–366.
[25] E. V. Flynn and J. L. Wetherell, Finding rational points on bielliptic genus 2 curves, Manuscripta Math. 100 (1999), 519–533. · Zbl 1029.11024
[26] 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.
[27] W. Fulton, Intersection Theory, 2nd ed., Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin, 2013. · Zbl 0541.14005
[28] 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.
[29] R. H. Kaenders, The mixed Hodge structure on the fundamental group of a punctured Riemann surface, Proc. Amer. Math. Soc. 129 (2001), 1271–1281. · Zbl 0971.14027
[30] M. Kim, The motivic fundamental group of \(\mathbf{P}^{1}⧵\{0,1,∞\}\) and the theorem of Siegel, Invent. Math. 161 (2005), 629–656. · Zbl 1090.14006
[31] M. Kim, The unipotent Albanese map and Selmer varieties for curves, Publ. Res. Inst. Math. Sci. 45 (2009), 89–133. · Zbl 1165.14020
[32] M. Kim, Tangential localization for Selmer varieties, Duke Math. J. 161 (2012), 173–199. · Zbl 1268.11079
[33] M. Kim and A. Tamagawa, The \(l\)-component of the unipotent Albanese map, Math. Ann. 340 (2008), 223–235. · Zbl 1126.14035
[34] B. Mazur, W. Stein, and J. Tate, Computation of \(p\)-adic heights and log convergence, Doc. Math. Extra Vol. (2006), 577–614. · Zbl 1135.11034
[35] J. Nekovář, “On \(p\)-adic height pairings,” in Séminaire de Théorie des Nombres, Paris, 1990–91, Progr. Math. 108, Birkhäuser, Boston, 1993, 127–202.
[36] M. C. Olsson, Towards non-abelian \(p\)-adic Hodge theory in the good reduction case, Mem. Amer. Math. Soc. 210 (2011), no. 990. · Zbl 1213.14002
[37] 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. · Zbl 1124.11019
[38] M. Raynaud, “\(1\)-motifs et monodromie géométrique,” in Périodes \(p\)-adiques (Bures-sur-Yvette, 1988), Astérisque 223, Soc. Math. France, Montrouge, 1994, 295–319.
[39] V. Scharaschkin, Local-global problems and the Brauer-Manin obstruction, Ph.D. dissertation, University of Michigan, Ann Arbor, 1999. · Zbl 0938.11053
[40] 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. · Zbl 0817.14008
[41] J.-P. Serre, Galois Cohomology, Springer, Berlin, 1997.
[42] S. Siksek, Explicit Chabauty over number fields, Algebra Number Theory 7 (2013), 765–793. · Zbl 1330.11043
[43] J. H. Silverman, Computing heights on elliptic curves, Math. Comp. 51 (1988), 339–358. · Zbl 0656.14016
[44] The Sage Developers, Sagemath, the Sage Mathematics Software System (Version 8.0), 2017, http://www.sagemath.org.
[45] M. Waldschmidt, On the \(p\)-adic closure of a subgroup of rational points on an Abelian variety, Afr. Mat. 22 (2011), 79–89. · Zbl 1291.11104
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