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An effective Chabauty–Kim theorem

  • Jennifer S. Balakrishnan (a1) and Netan Dogra (a2)

Abstract

The Chabauty–Kim method allows one to find rational points on curves under certain technical conditions, generalising Chabauty’s proof of the Mordell conjecture for curves with Mordell–Weil rank less than their genus. We show how the Chabauty–Kim method, when these technical conditions are satisfied in depth 2, may be applied to bound the number of rational points on a curve of higher rank. This provides a non-abelian generalisation of Coleman’s effective Chabauty theorem.

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Balakrishnan is supported in part by NSF grant DMS-1702196, the Clare Boothe Luce Professorship (Henry Luce Foundation), and Simons Foundation grant #550023.

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[BBM16] Balakrishnan, J. S., Besser, A. and Müller, J. S., Quadratic Chabauty: p-adic height pairings and integral points on hyperelliptic curves , J. Reine Angew. Math. 720 (2016), 5179.
[BD17] Balakrishnan, J. S. and Dogra, N., Quadratic Chabauty and rational points II: Generalised height functions on Selmer varieties, Preprint (2017), arXiv:arXiv:1705.00401.
[BK90] Bloch, S. and Kato, K., L-functions and Tamagawa numbers of motives , in The Grothendieck Festschrift, Vol. I (Birkhäuser, Boston, 1990).
[Cha41] Chabauty, C., Sur les points rationnels des courbes algébriques de genre supérieur à l’unité , C. R. Acad. Sci. Paris 212 (1941), 882885.
[CK10] Coates, J. and Kim, M., Selmer varieties for curves with CM Jacobians , Kyoto J. Math. 50 (2010), 827852.
[Col85] Coleman, R., Effective Chabauty , Duke Math. J. 52 (1985), 765770.
[Del89] Deligne, P., Le groupe fondamental de la droite projective moins trois points , in Galois groups over ℚ, Mathematical Sciences Research Institute Publications, vol. 16, eds Ihara, Y., Ribet, K. and Serre, J.-P. (Springer, New York, 1989).
[DDLR15] Darmon, H., Daub, M., Lichtenstein, S. and Rotger, V., Algorithms for Chow–Heegner points via iterated integrals , Math. Comput. 84 (2015), 25052547.
[EH17] Ellenberg, J. S. and Hast, D. R., Rational points on solvable curves over $\mathbb{Q}$ via non-abelian Chabauty, Preprint, 2017, arXiv:1706.00525.
[KRZ16] Katz, E., Rabinoff, J. and Zureick-Brown, D., Uniform bounds for the number of rational points on curves of small Mordell–Weil rank , Duke. Math. J. 165 (2016), 31893240.
[Kim05] Kim, M., The motivic fundamental group of ℙ1 -{ 0, 1, } and the theorem of Siegel , Invent. Math. 161 (2005), 629656.
[Kim09] Kim, M., The unipotent Albanese map and Selmer varieties for curves , Publ. Res. Inst. Math. Sci. 45 (2009), 89133.
[Kim10] Kim, M., Massey products for elliptic curves of rank 1 , J. Amer. Math. Soc. 23 (2010), 725747.
[KT08] Kim, M. and Tamagawa, A., The l-component of the unipotent Albanese map , Math. Ann. 340 (2008), 223235.
[Kob84] Koblitz, N., p-adic Numbers, p-adic analysis and zeta-functions, Graduate Texts in Mathematics, vol. 58 (Springer, New York, 1984).
[Oda95] Oda, T., A note on the ramification of the Galois representation on the fundamental group of an algebraic curve, II , J. Number Theory 53 (1995), 342355.
[Ser97] Serre, J.-P., Galois cohomology (Springer, Berlin, 1997).
[Sto19] Stoll, M., Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell–Weil rank , J. Eur. Math. Soc. (JEMS) 21 (2019), 923956.
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An effective Chabauty–Kim theorem

  • Jennifer S. Balakrishnan (a1) and Netan Dogra (a2)

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