Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-26T00:04:33.311Z Has data issue: false hasContentIssue false
  • English
  • Français

Torsion points on isogenous abelian varieties

Part of: Arithmetic problems. Diophantine geometry Abelian varieties and schemes Arithmetic algebraic geometry

Published online by Cambridge University Press:  20 July 2022

Gabriel A. Dill
Gabriel A. Dill*
Affiliation:
Leibniz Universität Hannover, Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Welfengarten 1, 30167 Hannover, Germany [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Investigating a conjecture of Zannier, we study irreducible subvarieties of abelian schemes that dominate the base and contain a Zariski dense set of torsion points that lie on pairwise isogenous fibers. If everything is defined over the algebraic numbers and the abelian scheme has maximal variation, we prove that the geometric generic fiber of such a subvariety is a union of torsion cosets. We go on to prove fully or partially explicit versions of this result in fibered powers of the Legendre family of elliptic curves. Finally, we apply a recent result of Galateau and Martínez to obtain uniform bounds on the number of maximal torsion cosets in the Manin–Mumford problem across a given isogeny class. For the proofs, we adapt the strategy, due to Lang, Serre, Tate, and Hindry, of using Galois automorphisms that act on the torsion as homotheties to the family setting.

Keywords

isogenyabelian schemeManin–Mumfordeffectivityuniformityhomothety

MSC classification

Primary: 11G05: Elliptic curves over global fields 14K02: Isogeny 14K12: Subvarieties
Secondary: 11G50: Heights 14G40: Arithmetic varieties and schemes; Arakelov theory; heights
Type
Research Article
Information
Compositio Mathematica , Volume 158 , Issue 5 , May 2022 , pp. 1020 - 1051
Check for updates
Copyright
© 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

André, Y., Shimura varieties, subvarieties, and CM points, Six lectures at the University of Hsinchu (Taiwan), August–September 2001 (with an appendix by C.-L. Chai), http://math.cts.nthu.edu.tw/Mathematics/english/lecnotes/andre2001all.ps, 2001.Google Scholar
Barroero, F. and Dill, G. A., On the Zilber-Pink conjecture for complex abelian varieties, Ann. Sci. Éc. Norm. Supér. (4) 55 (2022), 261–282.10.24033/asens.2496CrossRefGoogle Scholar
Binyamini, G., Density of algebraic points on Noetherian varieties, Geom. Funct. Anal. 29 (2019), 72–118.CrossRefGoogle Scholar
Binyamini, G., Point counting for foliations over number fields, Forum Math. Pi 10 (2022), e6.CrossRefGoogle Scholar
Bogomolov, F. A., Points of finite order on abelian varieties, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), 782–804.Google Scholar
Bombieri, E. and Gubler, W., Heights in Diophantine geometry, New Mathematical Monographs, vol. 4 (Cambridge University Press, Cambridge, 2006).Google Scholar
Bosser, V. and Gaudron, E., Logarithmes des points rationnels des variétés abéliennes, Canad. J. Math. 71 (2019), 247–298.CrossRefGoogle Scholar
Bourdon, A. and Clark, P. L., Torsion points and Galois representations on CM elliptic curves, Pacific J. Math. 305 (2020), 43–88.10.2140/pjm.2020.305.43CrossRefGoogle Scholar
Cassels, J. W. S., An introduction to the geometry of numbers, Classics in Mathematics (Springer, Berlin, 1997). Corrected reprint of the 1971 edition.Google Scholar
D'Andrea, C., Krick, T. and Sombra, M., Heights of varieties in multiprojective spaces and arithmetic Nullstellensätze, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), 549–627.10.24033/asens.2196CrossRefGoogle Scholar
David, S. and Philippon, P., Minorations des hauteurs normalisées des sous-variétés des puissances des courbes elliptiques, Int. Math. Res. Pap. IMRP 3 (2007), Art. ID rpm006.Google Scholar
DeMarco, L., Krieger, H. and Ye, H., Uniform Manin–Mumford for a family of genus 2 curves, Ann. of Math. (2) 191 (2020), 949–1001.10.4007/annals.2020.191.3.5CrossRefGoogle Scholar
Dill, G. A., Unlikely intersections with isogeny orbits in a product of elliptic schemes, Math. Ann. 377 (2020), 1509–1545.CrossRefGoogle Scholar
Dill, G. A., Unlikely intersections between isogeny orbits and curves, J. Eur. Math. Soc. (JEMS) 23 (2021), 2405–2438.CrossRefGoogle Scholar
Dimitrov, V., Gao, Z. and Habegger, P., Uniformity in Mordell–Lang for curves, Ann. of Math. (2) 194 (2021), 237–298.10.4007/annals.2021.194.1.4CrossRefGoogle Scholar
Eckstein, C., Homothéties, à chercher dans l'action de Galois sur des points de torsion, vol. 2005/7 of Prépublication de l'Institut de Recherche Mathématique Avancée. Thèse, Université Louis Pasteur, Institut de Recherche Mathématique Avancée (IRMA), Strasbourg (2005).Google Scholar
Grothendieck, A., Éléments de géométrie algébrique. I. Le langage des schémas, Publ. Math. Inst. Hautes Études Sci. 4 (1960), 5–228.10.1007/BF02684778CrossRefGoogle Scholar
Grothendieck, A., Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Publ. Math. Inst. Hautes Études Sci. 11 (1961), 5–167.Google Scholar
Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Publ. Math. Inst. Hautes Études Sci. 24 (1965), 5–231.Google Scholar
Faltings, G., Diophantine approximation on abelian varieties, Ann. of Math. (2) 133 (1991), 549–576.CrossRefGoogle Scholar
Faltings, G. and Chai, C.-L., Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 22 (Springer, Berlin, 1990). With an appendix by D. Mumford.10.1007/978-3-662-02632-8CrossRefGoogle Scholar
Fulton, W., Intersection theory, second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 2 (Springer, Berlin, 1998).10.1007/978-1-4612-1700-8CrossRefGoogle Scholar
Galateau, A. and Martínez, C., A bound for the torsion on subvarieties of abelian varieties, Preprint (2017), arXiv:1710.04577.Google Scholar
Galateau, A. and Martínez, C., Homothéties explicites des représentations $\ell$-adiques, Preprint (2020), arXiv:2006.07401.Google Scholar
Gao, Z., A special point problem of André–Pink–Zannier in the universal family of Abelian varieties, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (2017), 231–266.Google Scholar
Gao, Z., Towards the Andre–Oort conjecture for mixed Shimura varieties: the Ax–Lindemann theorem and lower bounds for Galois orbits of special points, J. Reine Angew. Math. 732 (2017), 85–146.10.1515/crelle-2014-0127CrossRefGoogle Scholar
Gao, Z., Ge, T. and Kühne, L., The uniform Mordell–Lang conjecture, Preprint (2021), arXiv:2105.15085.Google Scholar
Gaudron, E. and Rémond, G., Théorème des périodes et degrés minimaux d'isogénies, Comment. Math. Helv. 89 (2014), 343–403.CrossRefGoogle Scholar
Gaudron, E. and Rémond, G., Nouveaux théorèmes d'isogénie, Mém. Soc. Math. France, to appear. Preprint (2020), http://math.univ-bpclermont.fr/~gaudron/art20.pdf.Google Scholar
Görtz, U. and Wedhorn, T., Algebraic geometry I: schemes. With examples and exercises, Advanced Lectures in Mathematics (Vieweg + Teubner, Wiesbaden, 2010).CrossRefGoogle Scholar
Grothendieck, A., Technique de descente et théorèmes d'existence en géométrie algébrique. V. Les schémas de Picard: théorèmes d'existence, in Séminaire Bourbaki, Vol. 7 (Société Mathématique de France, Paris, 1995), 143–161.Google Scholar
Habegger, P., Special points on fibered powers of elliptic surfaces, J. Reine Angew. Math. 685 (2013), 143–179.Google Scholar
Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52 (Springer, New York–Heidelberg, 1977).10.1007/978-1-4757-3849-0CrossRefGoogle Scholar
Hindry, M., Autour d'une conjecture de Serge Lang, Invent. Math. 94 (1988), 575–603.10.1007/BF01394276CrossRefGoogle Scholar
Hrushovski, E., The Manin–Mumford conjecture and the model theory of difference fields, Ann. Pure Appl. Logic 112 (2001), 43–115.10.1016/S0168-0072(01)00096-3CrossRefGoogle Scholar
Kühne, L., Equidistribution in families of abelian varieties and uniformity, Preprint (2021), arXiv:2101.10272.Google Scholar
Lang, S., Division points on curves, Ann. Mat. Pura Appl. (4) 70 (1965), 229–234.CrossRefGoogle Scholar
Lang, S., Fundamentals of Diophantine geometry (Springer, New York, 1983).CrossRefGoogle Scholar
Lang, S., Elliptic functions, second edition, Graduate Texts in Mathematics, vol. 112 (Springer, New York, 1987). With an appendix by J. Tate.CrossRefGoogle Scholar
Laurent, M., Équations diophantiennes exponentielles, Invent. Math. 78 (1984), 299–327.10.1007/BF01388597CrossRefGoogle Scholar
Lin, Q. and Wang, M.-X., Isogeny orbits in a family of abelian varieties, Acta Arith. 170 (2015), 161–173.10.4064/aa170-2-4CrossRefGoogle Scholar
Lombardo, D., Bounds for Serre's open image theorem for elliptic curves over number fields, Algebra Number Theory 9 (2015), 2347–2395.10.2140/ant.2015.9.2347CrossRefGoogle Scholar
Lombardo, D., Galois representations attached to abelian varieties of CM type, Bull. Soc. Math. France 145 (2017), 469–501.CrossRefGoogle Scholar
Lombardo, D. and Tronto, S., Effective Kummer theory for elliptic curves, Int. Math. Res. Not. IMRN (2021), rnab216, https://doi.org/10.1093/imrn/rnab216.CrossRefGoogle Scholar
Lombardo, D. and Tronto, S., Some uniform bounds for elliptic curves over $\mathbb {Q}$, Preprint (2021), arXiv:2106.09950.Google Scholar
Masser, D. W. and Wüstholz, G., Estimating isogenies on elliptic curves, Invent. Math. 100 (1990), 1–24.10.1007/BF01231178CrossRefGoogle Scholar
Masser, D. W. and Wüstholz, G., Isogeny estimates for abelian varieties, and finiteness theorems, Ann. of Math. (2) 137 (1993), 459–472.10.2307/2946529CrossRefGoogle Scholar
Masser, D. W. and Wüstholz, G., Periods and minimal abelian subvarieties, Ann. of Math. (2) 137 (1993), 407–458.10.2307/2946542CrossRefGoogle Scholar
Masser, D. W. and Zannier, U., Bicyclotomic polynomials and impossible intersections, J. Théor. Nombres Bordeaux 25 (2013), 635–659.CrossRefGoogle Scholar
Masser, D. W. and Zannier, U., Torsion points, Pell's equation, and integration in elementary terms, Acta Math. 225 (2020), 227–313.10.4310/ACTA.2020.v225.n2.a2CrossRefGoogle Scholar
Milne, J. S., Abelian varieties, in Arithmetic geometry (Storrs, Conn., 1984) (Springer, New York, 1986), 103–150.CrossRefGoogle Scholar
Milne, J. S., Algebraic groups, Cambridge Studies in Advanced Mathematics, vol. 170 (Cambridge University Press, Cambridge, 2017). The theory of group schemes of finite type over a field.10.1017/9781316711736CrossRefGoogle Scholar
Mumford, D., Fogarty, J. and Kirwan, F., Geometric invariant theory, third edition, Ergebnisse der Mathematik und ihrer Grenzgebiete (2), vol. 34 (Springer, Berlin, 1994).10.1007/978-3-642-57916-5CrossRefGoogle Scholar
Orr, M., Families of abelian varieties with many isogenous fibres, J. Reine Angew. Math. 705 (2015), 211–231.Google Scholar
Pazuki, F., Modular invariants and isogenies, Int. J. Number Theory 15 (2019), 569–584.10.1142/S1793042119500295CrossRefGoogle Scholar
Philippon, P., Sur des hauteurs alternatives. III, J. Math. Pures Appl. (9) 74 (1995), 345–365.Google Scholar
Pila, J., Special point problems with elliptic modular surfaces, Mathematika 60 (2014), 1–31.10.1112/S0025579313000168CrossRefGoogle Scholar
Pila, J. and Zannier, U., Rational points in periodic analytic sets and the Manin–Mumford conjecture, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19 (2008), 149–162.10.4171/RLM/514CrossRefGoogle Scholar
Pink, R., A combination of the conjectures of Mordell–Lang and André–Oort, in Geometric methods in algebra and number theory, Progress in Mathematics, vol. 235 (Birkhäuser, Boston, MA, 2005), 251–282.10.1007/0-8176-4417-2_11CrossRefGoogle Scholar
Pink, R., A common generalization of the conjectures of André–Oort, Manin–Mumford, and Mordell–Lang, Preprint (2005), https://people.math.ethz.ch/~pink/ftp/AOMMML.pdf.Google Scholar
Poonen, B., Rational points on varieties, Graduate Studies in Mathematics, vol. 186 (American Mathematical Society, Providence, RI, 2017).CrossRefGoogle Scholar
Raynaud, M., Courbes sur une variété abélienne et points de torsion, Invent. Math. 71 (1983), 207–233.CrossRefGoogle Scholar
Raynaud, M., Sous-variétés d'une variété abélienne et points de torsion, in Arithmetic and geometry, Vol. I, Progress in Mathematics, vol. 35 (Birkhäuser, Boston, MA, 1983), 327–352.10.1007/978-1-4757-9284-3_14CrossRefGoogle Scholar
Rémond, G., Géométrie diophantienne multiprojective, in Introduction to algebraic independence theory, Lecture Notes in Mathematics, vol. 1752 (Springer, Berlin, 2001), 95–131.Google Scholar
Rémond, G., Conjectures uniformes sur les variétés abéliennes, Q. J. Math. 69 (2018), 459–486.10.1093/qmath/hax042CrossRefGoogle Scholar
Rémond, G., Degré de définition des endomorphismes d'une variété abélienne, J. Eur. Math. Soc. (JEMS) 22 (2020), 3059–3099.10.4171/JEMS/981CrossRefGoogle Scholar
Robin, G., Estimation de la fonction de Tchebychef $\theta$ sur le $k$-ième nombre premier et grandes valeurs de la fonction $\omega (n)$ nombre de diviseurs premiers de $n$, Acta Arith. 42 (1983), 367–389.10.4064/aa-42-4-367-389CrossRefGoogle Scholar
Rosser, J. B. and Schoenfeld, L., Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94.10.1215/ijm/1255631807CrossRefGoogle Scholar
Scanlon, T., Automatic uniformity, Int. Math. Res. Not. IMRN 2004 (2004), 3317–3326.10.1155/S1073792804140816CrossRefGoogle Scholar
Serre, J.-P., Œuvres – Collected papers IV: 1985–1998 (Springer, Berlin, 2000).10.1007/978-3-642-41978-2CrossRefGoogle Scholar
Silverberg, A., Fields of definition for homomorphisms of abelian varieties, J. Pure Appl. Algebra 77 (1992), 253–262.10.1016/0022-4049(92)90141-2CrossRefGoogle Scholar
The Stacks Project Authors, Stacks Project (2020), https://stacks.math.columbia.edu.Google Scholar
Stevenhagen, P., Hilbert's 12th problem, complex multiplication and Shimura reciprocity, in Class field theory—its centenary and prospect (Tokyo, 1998), Advanced Studies in Pure Mathematics, vol. 30 (Mathematical Society of Japan, Tokyo, 2001), 161–176.10.2969/aspm/03010161CrossRefGoogle Scholar
Wintenberger, J.-P., Démonstration d'une conjecture de Lang dans des cas particuliers, J. Reine Angew. Math. 553 (2002), 1–16.Google Scholar
Zannier, U., Some problems of unlikely intersections in arithmetic and geometry, Annals of Mathematics Studies, vol. 181 (Princeton University Press, Princeton, NJ, 2012), with appendixes by D. W. Masser.Google Scholar