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The relative Bogomolov conjecture for fibered products of elliptic curves

  • Lars Kühne EMAIL logo
Veröffentlicht/Copyright: 27. Januar 2023

Abstract

We deduce an analogue of the Bogomolov conjecture for non-degenerate subvarieties in fibered products of families of elliptic curves from the author’s recent theorem on equidistribution in families of abelian varieties. This generalizes results of DeMarco and Mavraki and improves certain results of Manin–Mumford type proven by Masser and Zannier to results of Bogomolov type, yielding the first results of this type for subvarieties of relative dimension > 1 in families of abelian varieties with trivial trace.

Funding statement: The author was supported by an Ambizione Grant of the Swiss National Science Foundation during the early stages of this project. He also received funding from the European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 101027237.

Acknowledgements

The author thanks the anonymous referee for their attentive reading and their many suggestions that helped to improve the exposition substantially. He also thanks Laura DeMarco, Ziyang Gao, Philipp Habegger, Myrto Mavraki, and Fabien Pazuki for their advice, discussion and encouragement. Finally, he thanks Jakob Stix for pointing out some inaccuracies in the article.

References

[1] Y. André, Mumford–Tate groups of mixed Hodge structures and the theorem of the fixed part, Compos. Math. 82 (1992), no. 1, 1–24. Suche in Google Scholar

[2] Y. André, P. Corvaja and U. Zannier, The Betti map associated to a section of an abelian scheme, Invent. Math. 222 (2020), no. 1, 161–202. 10.1007/s00222-020-00963-wSuche in Google Scholar

[3] C. Birkenhake and H. Lange, Complex abelian varieties, 2nd ed., Grundlehren Math. Wiss. 302, Springer, Berlin 2004. 10.1007/978-3-662-06307-1Suche in Google Scholar

[4] F. Bogomolov, H. Fu and Y. Tschinkel, Torsion of elliptic curves and unlikely intersections, Geometry and physics. Vol. I, Oxford University, Oxford (2018), 19–37. 10.1093/oso/9780198802013.003.0002Suche in Google Scholar

[5] A. Borel, Linear algebraic groups, 2nd ed., Grad. Texts in Math. 126, Springer, New York 1991. 10.1007/978-1-4612-0941-6Suche in Google Scholar

[6] P. Deligne, Théorie de Hodge. III, Publ. Math. Inst. Hautes Études Sci. 44 (1974), 5–77. 10.1007/BF02685881Suche in Google Scholar

[7] L. DeMarco, H. Krieger and H. Ye, Uniform Manin–Mumford for a family of genus 2 curves, Ann. of Math. (2) 191 (2020), no. 3, 949–1001. 10.4007/annals.2020.191.3.5Suche in Google Scholar

[8] L. DeMarco and N. M. Mavraki, Elliptic surfaces and intersections of adelic -divisors, preprint (2020), https://arxiv.org/abs/2012.14529; to appear in J. Eur. Math. Soc. Suche in Google Scholar

[9] L. DeMarco and N. M. Mavraki, Variation of canonical height and equidistribution, Amer. J. Math. 142 (2020), no. 2, 443–473. 10.1353/ajm.2020.0012Suche in Google Scholar

[10] F. Diamond and J. Shurman, A first course in modular forms, Grad. Texts in Math. 228, Springer, New York 2005. Suche in Google Scholar

[11] V. Dimitrov, Z. Gao and P. Habegger, Uniform bound for the number of rational points on a pencil of curves, Int. Math. Res. Not. IMRN 2021 (2021), no. 2, 1138–1159. 10.1093/imrn/rnz248Suche in Google Scholar

[12] V. Dimitrov, Z. Gao and P. Habegger, Uniformity in Mordell–Lang for curves, Ann. of Math. (2) 194 (2021), no. 1, 237–298. 10.4007/annals.2021.194.1.4Suche in Google Scholar

[13] V. Dimitrov, Z. Gao and P. Habegger, A consequence of the relative Bogomolov conjecture, J. Number Theory 230 (2022), 146–160. 10.1016/j.jnt.2021.03.028Suche in Google Scholar

[14] Z. Gao, Generic rank of Betti map and unlikely intersections, Compos. Math. 156 (2020), no. 12, 2469–2509. 10.1112/S0010437X20007435Suche in Google Scholar

[15] Z. Gao, Mixed Ax–Schanuel for the universal abelian varieties and some applications, Compos. Math. 156 (2020), no. 11, 2263–2297. 10.1112/S0010437X20007447Suche in Google Scholar

[16] T. Gauthier, Good height functions on quasi-projective varieties: Equidistribution and applications in dynamics, preprint (2021), https://arxiv.org/abs/2105.02479. Suche in Google Scholar

[17] P. Habegger and J. Pila, Some unlikely intersections beyond André–Oort, Compos. Math. 148 (2012), no. 1, 1–27. 10.1112/S0010437X11005604Suche in Google Scholar

[18] M. Kashiwara, A study of variation of mixed Hodge structure, Publ. Res. Inst. Math. Sci. 22 (1986), no. 5, 991–1024. 10.2977/prims/1195177264Suche in Google Scholar

[19] N. M. Katz and B. Mazur, Arithmetic moduli of elliptic curves, Ann. of Math. Stud. 108, Princeton University, Princeton 1985. 10.1515/9781400881710Suche in Google Scholar

[20] L. Kühne, Equidistribution in families of abelian varieties and uniformity, preprint (2021), https://arxiv.org/abs/2101.10272. Suche in Google Scholar

[21] J. D. Lewis, A survey of the Hodge conjecture, 2nd ed., CRM Monogr. Ser. 10, American Mathematical Society, Providence 1999. Suche in Google Scholar

[22] D. Masser and U. Zannier, Torsion points on families of squares of elliptic curves, Math. Ann. 352 (2012), no. 2, 453–484. 10.1007/s00208-011-0645-4Suche in Google Scholar

[23] D. Masser and U. Zannier, Torsion points on families of products of elliptic curves, Adv. Math. 259 (2014), 116–133. 10.1016/j.aim.2014.03.016Suche in Google Scholar

[24] D. Masser and U. Zannier, Torsion points, Pell’s equation, and integration in elementary terms, Acta Math. 225 (2020), no. 2, 227–313. 10.4310/ACTA.2020.v225.n2.a2Suche in Google Scholar

[25] J. S. Milne, Shimura varieties and moduli, Handbook of moduli. Vol. II, Adv. Lect. Math. (ALM) 25, International Press, Somerville (2013), 467–548. Suche in Google Scholar

[26] M. Olsson, Algebraic spaces and stacks, Amer. Math. Soc. Colloq. Publ. 62, American Mathematical Society, 2016. 10.1090/coll/062Suche in Google Scholar

[27] C. A. M. Peters and J. H. M. Steenbrink, Mixed Hodge structures, Ergeb. Math. Grenzgeb. (3) 52, Springer, Berlin 2008. Suche in Google Scholar

[28] R. Pink, Arithmetical compactification of mixed Shimura varieties, Bonner Math. Schr. 209, Universität Bonn, Bonn 1990. Suche in Google Scholar

[29] R. Pink, A common generalization of the conjectures of André–Oort, Manin–Mumford, and Mordell–Lang, http://www.math.ethz.ch/~pink, 2005. Suche in Google Scholar

[30] M. Raynaud, Faisceaux amples sur les schémas en groupes et les espaces homogènes, Lecture Notes in Math. 119, Springer, Berlin 1970. 10.1007/BFb0059504Suche in Google Scholar

[31] E. Ullmo, Positivité et discrétion des points algébriques des courbes, Ann. of Math. (2) 147 (1998), no. 1, 167–179. 10.2307/120987Suche in Google Scholar

[32] J. Wildeshaus, Realizations of polylogarithms, Lecture Notes in Math. 1650, Springer, Berlin 1997. 10.1007/BFb0093051Suche in Google Scholar

[33] X. Yuan, Arithmetic bigness and a uniform Bogomolov-type result, preprint (2021), https://arxiv.org/abs/2108.05625. Suche in Google Scholar

[34] X. Yuan and S.-W. Zhang, Adelic line bundles over quasi-projective varieties, preprint (2021), https://arxiv.org/abs/2105.13587. Suche in Google Scholar

[35] S.-W. Zhang, Equidistribution of small points on abelian varieties, Ann. of Math. (2) 147 (1998), no. 1, 159–165. 10.2307/120986Suche in Google Scholar

[36] S.-W. Zhang, Small points and Arakelov theory, Doc. Math. 2 (1998), 217–225. Suche in Google Scholar

Received: 2021-03-10
Revised: 2022-10-18
Published Online: 2023-01-27
Published in Print: 2023-02-01

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