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Weight filtrations on Selmer schemes and the effective Chabauty–Kim method

Published online by Cambridge University Press:  20 June 2023

L. Alexander Betts*
Affiliation:
Department of Mathematics, Harvard University, Science Center Room 325, 1 Oxford Street, Cambridge, MA 02138, USA abetts@math.harvard.edu

Abstract

We develop an effective version of the Chabauty–Kim method which gives explicit upper bounds on the number of $S$-integral points on a hyperbolic curve in terms of dimensions of certain Bloch–Kato Selmer groups. Using this, we give a new ‘motivic’ proof that the number of solutions to the $S$-unit equation is bounded uniformly in terms of $\#S$.

Type
Research Article
Copyright
© 2023 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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References

Asada, M., Matsumoto, M. and Oda, T., Local monodromy on the fundamental groups of algebraic curves along a degenerate stable curve, J. Pure Appl. Algebra 103 (1995), 235283.CrossRefGoogle Scholar
Balakrishnan, J., Dan-Cohen, I., Kim, M. and Wewers, S., A non-abelian conjecture of Tate–Shafarevich type for hyperbolic curves, Math. Ann. 372 (2018), 369428.10.1007/s00208-018-1684-xCrossRefGoogle Scholar
Balakrishnan, J. and Dogra, N., Quadratic Chabauty and rational points I: $p$-adic heights, Duke Math. J. 167 (2018), 19812038, with an appendix by J.S. Müller.10.1215/00127094-2018-0013CrossRefGoogle Scholar
Balakrishnan, J. and Dogra, N., An effective Chabauty–Kim theorem, Compos. Math. 155 (2019), 10571075.CrossRefGoogle Scholar
Balakrishnan, J., Dogra, N., Müller, J. S., Tuitman, J. and Vonk, J., Explicit Chabauty–Kim for the split Cartan modular curve of level 13, Ann. Math. 189 (2019), 885944.10.4007/annals.2019.189.3.6CrossRefGoogle Scholar
Berkovich, V., Étale cohomology for non-Archimedean analytic spaces, Publ. Math. Inst. Hautes Études Sci. 78 (1993), 5161.10.1007/BF02712916CrossRefGoogle Scholar
Berthelot, P., Cohomologie rigide et cohomologie rigide à supports propres, Première partie. Preprint 96-03, Institut de Recherche Mathématique de Rennes (1996), https://perso.univ-rennes1.fr/pierre.berthelot/.Google Scholar
Besser, A., Coleman integration using the Tannakian formalism, Math. Ann. 322 (2002), 1948.10.1007/s002080100263CrossRefGoogle Scholar
Betts, L. A., Heights via anabelian geometry and local Bloch–Kato Selmer sets, PhD thesis, University of Oxford (2018).Google Scholar
Betts, L. A. and Dogra, N., The local theory of unipotent Kummer maps and refined Selmer schemes (2019), arXiv:1909.05734v1.Google Scholar
Betts, L. A. and Litt, D., Semisimplicity of the Frobenius action on $\pi_1$, in p-adic Hodge theory, singular varieties and non-abelian aspects, SISYPHT 2019, Simons Symposia (Springer, Cham, 2023), 17–64.Google Scholar
Bloch, S. and Kato, K., L-functions and Tamagawa numbers of motives, in The Grothendieck Festschrift, Vol. 1, eds. P. Cartier, L. Illusie, N. M. Katz, G. Laumon, Y. I. Manin and K. A. Ribet (Birkhäuser, 2007), 333–400.Google Scholar
Brown, F., Integral points on curves, the unit equation and motivic periods, Preprint (2017), arXiv:1704.00555v1.Google Scholar
Chiarellotto, B. and Le Stum, B., $F$-isocristaux unipotents, Compos. Math. 116 (1999), 81110.10.1023/A:1000602824628CrossRefGoogle Scholar
Coleman, R., Effective Chabauty, Duke Math. J. 52 (1985), 765770.CrossRefGoogle Scholar
Deligne, P., Equations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, vol. 163 (Springer, 1970).10.1007/BFb0061194CrossRefGoogle Scholar
Deligne, P., Milne, J., Ogus, A. and Shih, K.-Y., Tannakian categories, in Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, vol. 900 (Springer, 1982), 101228.10.1007/978-3-540-38955-2_4CrossRefGoogle Scholar
Dimitrov, V., Gao, Z. and Habegger, P., Uniformity in Mordell–Lang for curves, Ann. Math. 194 (2021), 237298.10.4007/annals.2021.194.1.4CrossRefGoogle Scholar
Evertse, J.-H., On equations in $S$-units and the Thue–Mahler equation, Invent. Math. 75 (1984), 561584.10.1007/BF01388644CrossRefGoogle Scholar
Faltings, G., Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), 349366.CrossRefGoogle Scholar
Faltings, G., Mathematics around Kim's new proof of Siegel's theorem, in Diophantine geometry: proceedings, CRM Series, vol. 4 (Edizioni della Normale, 2007), 173188.Google Scholar
Flynn, E. V., Poonen, B. and Schaefer, E. F., Cycles and quadratic polynomials and rational points on a genus-2 curve, Duke Math. J. 90 (1997), 435463.10.1215/S0012-7094-97-09011-6CrossRefGoogle Scholar
Fresnel, J. and van der Put, M., Rigid analytic geometry and its applications, Progress in Mathematics, vol. 218 (Springer, 2004).10.1007/978-1-4612-0041-3CrossRefGoogle Scholar
Gouvêa, F. Q., p-adic numbers, an introduction, second edition, Universitext (Springer, 1997).10.1007/978-3-642-59058-0CrossRefGoogle Scholar
Hadian, M., Motivic fundamental groups and integral points, Duke Math. J. 160 (2011), 503565.10.1215/00127094-1444296CrossRefGoogle Scholar
Hain, R., The geometry of the mixed Hodge structure on the fundamental group, in Algebraic geometry–Bowdoin 1985, part 2, Proceedings of Symposia in Mathematics, vol. 46 (American Mathematical Society, 1987), 247282.CrossRefGoogle Scholar
Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52 (Springer, 1977).CrossRefGoogle Scholar
de Jong, A. J., Étale fundamental groups of non-archimedean analytic spaces, Compos. Math. 97 (1995), 89118.Google Scholar
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.CrossRefGoogle Scholar
Kim, M., The motivic fundamental group of the projective line minus three points and the theorem of Siegel, Invent. Math. 161 (2005), 629656.CrossRefGoogle Scholar
Kim, M., The unipotent Albanese map and Selmer varieties for curves, Publ. Res. Inst. Math. Sci. 45 (2009), 89133.CrossRefGoogle Scholar
Kim, M., Tangential localization for Selmer varieties, Duke Math. J. 161 (2012), 173199.10.1215/00127094-1507332CrossRefGoogle Scholar
Kim, M. and Coates, J., Selmer varieties for curves with CM Jacobians, Kyoto J. Math. 50 (2010), 827852.Google Scholar
Kim, M. and Tamagawa, A., The $l$-component of the unipotent Albanese map, Math. Ann. 340 (2008), 223235.10.1007/s00208-007-0151-xCrossRefGoogle Scholar
Labute, J. P., On the descending central series of groups with a single defining relation, J. Algebra 14 (1970), 1623.CrossRefGoogle Scholar
Laksov, D., Weierstrass points on curves, Astérisque 87–88 (1981), 221247.Google Scholar
Liu, Q., Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6 (Oxford University Press, 2002); translated by R. Erné.CrossRefGoogle Scholar
Liu, Q. and Tong, J., Néron models of algebraic curves, Trans. Amer. Math. Soc. 368 (2016), 70197043.CrossRefGoogle Scholar
Loeffler, D. and Zerbes, S., Euler systems (Arizona Winter School 2018 notes) (2018), https://www.math.arizona.edu/~swc/aws/2018/2018LoefflerZerbesNotes.pdf (downloaded 08/01/21).Google Scholar
May, J. and Ponto, K., More concise algebraic topology: localization, completion, and model categories, Chicago Lectures in Mathematics Series (University of Chicago Press, 2011).10.7208/chicago/9780226511795.001.0001CrossRefGoogle Scholar
Metropolis, N. and Rota, G.-C., Witt vectors and the algebra of necklaces, Adv. Math. (NY) 50 (1983), 95125.10.1016/0001-8708(83)90035-XCrossRefGoogle Scholar
Milne, J., Algebraic groups, Cambridge Studies in Advanced Mathematics, vol. 170 (Cambridge University Press, 2017).10.1017/9781316711736CrossRefGoogle Scholar
Neukirch, J., Schmidt, A. and Wingberg, K., Cohomology of number fields, second edition, Grundlehren der Mathematischen Wissenschaften, vol. 323 (Springer, 2013); corrected second printing.Google Scholar
Olsson, M., Towards non-Abelian $p$-adic Hodge theory in the good reduction case, Mem. Amer. Math. Soc. 210 (2011), 8164.Google Scholar
Olsson, M., The bar construction and affine stacks, Comm. Algebra 44 (2016), 30883121.10.1080/00927872.2015.1082578CrossRefGoogle Scholar
Poonen, B., The classification of rational preperiodic points of quadratic polynomials over $\mathbb {Q}$: a refined conjecture, Math. Z. 228 (1998), 1129.CrossRefGoogle Scholar
Poonen, B. and McCallum, W., The method of Chabauty and Coleman, in Méthodes explicites en théorie des nombres, points rationnels et Équations diophantiennes, Panoramas et Synthèses, vol. 36 (Société Mathématique de France, 2012), 99117.Google Scholar
Reutenauer, C., Free Lie algebras, Handbook of Algebra, vol. 3 (North-Holland, 2003), 887903.Google Scholar
Serre, J.-P., Semisimplicity and tensor products of group representations: converse theorems, J. Algebra 194 (1997), 496520.10.1006/jabr.1996.6929CrossRefGoogle Scholar
Serre, J.-P., Galois cohomology, Springer Monographs in Mathematics (Springer, 2002); translated from the French by P. Ion.Google Scholar
Raynaud, M., Grothendieck, A. and Rim, D. S., Séminaire de Géométrie Algébrique du Bois Marie – 1967–69 – Groupes de Monodromie en Géométrie Algébrique, Vol. I (SGA 7 I), Lecture Notes in Mathematics, vol. 288 (Springer, 1972).Google Scholar
Shiho, A., Crystalline fundamental groups I — isocrystals on log crystalline site and log convergent site, J. Math. Sci. Univ. Tokyo 7 (2000), 509656.Google Scholar
Soulé, C., $K$-théorie des anneaux d'entiers de crops de nombres et cohomologie étale, Invent. Math. 55 (1979), 251295.CrossRefGoogle Scholar
Wojtkowiak, Z., Cosimplicial objects in algebraic geometry, in Algebraic K-theory and algebraic topology, eds. P. G. Goerss and J. F. Jardine, NATO ASI Series, vol. 47 (Springer, Dordrecht, 1993), 287327.10.1007/978-94-017-0695-7_15CrossRefGoogle Scholar