Abstract
The main result of this paper states the subpolynomial growth of the number of integral points with bounded height of a variety over a number field whose fundamental group is large. This generalizes a recent paper of Ellenberg, Lawrence and Venkatesh and replies to two questions asked therein.
Funding source: Agence Nationale de la Recherche
Award Identifier / Grant number: ANR-18-CE40-0017
Funding statement: Marco Maculan was supported by the Agence Nationale de la Recherche, ANR-18-CE40-0017.
Acknowledgements
We warmly thank Pascal Autissier, Ariyan Javanpeykar and the anonymous referee for their useful comments.
References
[1] B. Bakker, Y. Brunebarbe, B. Klingler and J. Tsimerman, Definability of mixed period maps, preprint (2020), https://arxiv.org/abs/2006.12403. Search in Google Scholar
[2] B. Bakker, Y. Brunebarbe and J. Tsimerman, o-minimal GAGA and a conjecture of Griffiths, Invent. Math. 232 (2023), no. 1, 163–228. 10.1007/s00222-022-01166-1Search in Google Scholar
[3] V. V. Batyrev and Y. Tschinkel, Rational points on some Fano cubic bundles, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), no. 1, 41–46. Search in Google Scholar
[4] A. Beauville, Le problème de Torelli, Séminaire Bourbaki. Volume 1985/86, 38ème année. Exposés Nos. 651–668, Astérisque 145/146, Société Mathématique de France, Paris (1987), 7–20. Search in Google Scholar
[5] P. Berthelot, A. Grothendieck, and L. Illusie Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6). Théorie des intersections et théorème de Riemann–Roch, Lecture Notes in Math. 225, Springer, Berlin 1971. 10.1007/BFb0066283Search in Google Scholar
[6] E. Bombieri and W. Gubler, Heights in Diophantine geometry, New Math. Monogr. 4, Cambridge University, Cambridge 2006. Search in Google Scholar
[7] E. Bombieri and J. Pila, The number of integral points on arcs and ovals, Duke Math. J. 59 (1989), no. 2, 337–357. 10.1215/S0012-7094-89-05915-2Search in Google Scholar
[8] N. Broberg, A note on a paper by R. Heath-Brown: “The density of rational points on curves and surfaces”, J. reine angew. Math. 571 (2004), 159–178. 10.1515/crll.2004.039Search in Google Scholar
[9] Y. Brunebarbe, Increasing hyperbolicity of varieties supporting a variation of hodge structures with level structures, preprint (2020), https://arxiv.org/abs/2007.12965. Search in Google Scholar
[10] Y. Brunebarbe, Hyperbolicity in presence of a large local system, preprint (2022), https://arxiv.org/abs/2207.03283. Search in Google Scholar
[11] F. Campana, Fundamental group and positivity of cotangent bundles of compact Kähler manifolds, J. Algebraic Geom. 4 (1995), no. 3, 487–502. Search in Google Scholar
[12] F. Campana, B. Claudon and P. Eyssidieux, Représentations linéaires des groupes kählériens: factorisations et conjecture de Shafarevich linéaire, Compos. Math. 151 (2015), no. 2, 351–376. 10.1112/S0010437X14007751Search in Google Scholar
[13] H. Chen, Explicit uniform estimation of rational points I. Estimation of heights, J. reine angew. Math. 668 (2012), 59–88. 10.1515/CRELLE.2011.138Search in Google Scholar
[14] H. Chen, Explicit uniform estimation of rational points II. Hypersurface coverings, J. reine angew. Math. 668 (2012), 89–108. 10.1515/CRELLE.2011.139Search in Google Scholar
[15] B. Claudon, P. Corvaja, J.-P. Demailly, S. Diverio, J. Duval, C. Gasbarri, S. Kebekus, M. Pau, E. Rousseau, N. Sibony, B. Taji and C. Voisin, Propriétés d’hyperbolicité des variétés algébriques, Panor. Synthèses 56, Société Mathématique de France, Paris 2021. Search in Google Scholar
[16] P. Deligne, Théorie de Hodge. II, Publ. Math. Inst. Hautes Études Sci. 40 (1971), 5–57. 10.1007/BF02684692Search in Google Scholar
[17] J. S. Ellenberg, B. Lawrence and A. Venkatesh, Sparsity of integral points on moduli spaces of varieties, Int. Math. Res. Not. IMRN 2023 (2023), no. 17, 15073–15101. 10.1093/imrn/rnac243Search in Google Scholar
[18] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366. 10.1007/BF01388432Search in Google Scholar
[19] G. Faltings, Erratum: “Finiteness theorems for abelian varieties over number fields”, Invent. Math. 75 (1984), no. 2, 381. 10.1007/BF01388572Search in Google Scholar
[20] G. Faltings, Diophantine approximation on abelian varieties, Ann. of Math. (2) 133 (1991), no. 3, 549–576. 10.2307/2944319Search in Google Scholar
[21] B. Fantechi, L. Göttsche, L. Illusie, S. L. Kleiman, N. Nitsure and A. Vistoli, Fundamental algebraic geometry, Math. Surveys Monogr. 123, American Mathematical Society, Providence 2005. 10.1090/surv/123Search in Google Scholar
[22] P. A. Griffiths, Periods of integrals on algebraic manifolds. III. Some global differential-geometric properties of the period mapping, Publ. Math. Inst. Hautes Études Sci. 38 (1970), 125–180. 10.1007/BF02684654Search in Google Scholar
[23] B. H. Gross, A remark on tube domains, Math. Res. Lett. 1 (1994), no. 1, 1–9. 10.4310/MRL.1994.v1.n1.a1Search in Google Scholar
[24] A. Grothendieck, Séminaire de géométrie algébrique du Bois Marie 1960-61. Revêtements étales et groupe fondamental (SGA 1), Doc. Math. 3, Société Mathématique de France, Paris 2003. Search in Google Scholar
[25] R. Hartshorne, Algebraic geometry, Grad. Texts in Math. 52, Springer, New York 1977. 10.1007/978-1-4757-3849-0Search in Google Scholar
[26] D. R. Heath-Brown, The density of rational points on curves and surfaces, Ann. of Math. (2) 155 (2002), no. 2, 553–595. 10.2307/3062125Search in Google Scholar
[27] M. Hindry and J. H. Silverman, Diophantine geometry. An introduction, Grad. Texts in Math. 201, Springer, New York 2000. 10.1007/978-1-4612-1210-2Search in Google Scholar
[28] J. Kollár, Shafarevich maps and plurigenera of algebraic varieties, Invent. Math. 113 (1993), no. 1, 177–215. 10.1007/BF01244307Search in Google Scholar
[29] J. Kollár, Shafarevich maps and automorphic forms, M. B. Porter Lectures, Princeton University, Princeton 1995. 10.1515/9781400864195Search in Google Scholar
[30] A. Malcev, On isomorphic matrix representations of infinite groups, Rec. Math. [Mat. Sbornik] N. S. 8(50) (1940), 405–422. Search in Google Scholar
[31] M. Olsson, Algebraic spaces and stacks, Amer. Math. Soc. Colloq. Publ. 62, American Mathematical Society, Providence 2016. Search in Google Scholar
[32] E. Peyre, Points de hauteur bornée, topologie adélique et mesures de Tamagawa, J. Théor. Nombres Bordeaux 15 (2003), 319–349. 10.5802/jtnb.405Search in Google Scholar
[33] B. Poonen, Rational points on varieties, Grad. Stud. Math. 186, American Mathematical Society, Providence 2017. 10.1090/gsm/186Search in Google Scholar
[34] P. Salberger, On the density of rational and integral points on algebraic varieties, J. reine angew. Math. 606 (2007), 123–147. 10.1515/CRELLE.2007.037Search in Google Scholar
[35] S. H. Schanuel, Heights in number fields, Bull. Soc. Math. France 107 (1979), no. 4, 433–449. 10.24033/bsmf.1905Search in Google Scholar
[36] W. Schmid, Variation of Hodge structure: The singularities of the period mapping, Invent. Math. 22 (1973), 211–319. 10.1007/BF01389674Search in Google Scholar
[37] J.-P. Serre, Lectures on the Mordell–Weil theorem, Aspects of Math. E15, Friedrich Vieweg & Sohn, Braunschweig 1989. 10.1007/978-3-663-14060-3Search in Google Scholar
[38] C. L. Siegel, Über einige Anwendungen diophantischer Approximationen, On some applications of Diophantine approximations, Quad./Monogr. 2, Edizioni della Normale, Pisa (2014), 81–138. 10.1007/978-88-7642-520-2_2Search in Google Scholar
[39] J. Steenbrink and S. Zucker, Variation of mixed Hodge structure. I, Invent. Math. 80 (1985), no. 3, 489–542. 10.1007/BF01388729Search in Google Scholar
[40] D. Toledo, Projective varieties with non-residually finite fundamental group, Publ. Math. Inst. Hautes Études Sci. 77 (1993), 103–119. 10.1007/BF02699189Search in Google Scholar
[41] T. Stacks project authors, The stacks project, https://stacks.math.columbia.edu, 2022. Search in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Singular set and curvature blow-up rate of the level set flow
- Counting integral points of bounded height on varieties with large fundamental group
- 𝔸1-connected components and characterisation of 𝔸2
- New phenomena in deviation of Birkhoff integrals for locally Hamiltonian flows
- Torus counting and self-joinings of Kleinian groups
- Estimating the Morse index of free boundary minimal hypersurfaces through covering arguments
- On the integral Hodge conjecture for real abelian threefolds
- Weakly Kähler hyperbolic manifolds and the Green–Griffiths–Lang conjecture
Articles in the same Issue
- Frontmatter
- Singular set and curvature blow-up rate of the level set flow
- Counting integral points of bounded height on varieties with large fundamental group
- 𝔸1-connected components and characterisation of 𝔸2
- New phenomena in deviation of Birkhoff integrals for locally Hamiltonian flows
- Torus counting and self-joinings of Kleinian groups
- Estimating the Morse index of free boundary minimal hypersurfaces through covering arguments
- On the integral Hodge conjecture for real abelian threefolds
- Weakly Kähler hyperbolic manifolds and the Green–Griffiths–Lang conjecture