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On the \(\mathrm{PGL}_{2}\)-invariant quadruples of torsion points of elliptic curves

  • Fedor A. Bogomolov
  • Hang FuEmail author
Research Article
  • 8 Downloads

Abstract

Let E be an elliptic curve and \(\pi :E\rightarrow {\mathbb {P}}^{1}\) a standard double cover identifying \(\pm \, P\in E\). It is known that for some torsion points \(P_{i}\in E\), \(1\leqslant i\leqslant 4\), the cross ratio of \(\{\pi (P_{i})\}_{i=1}^{4}\) is independent of E. We will give a complete classification of such quadruples.

Keywords

Elliptic curves Torsion points q-series Congruence subgroups Modular curves 

Mathematics Subject Classification

14H52 11G05 11F03 20H05 40A20 

Notes

Acknowledgements

The second author would like to express his gratitude for a pleasant stay at Laboratory of Algebraic Geometry, HSE, where a substantial part of this article was accomplished.

References

  1. 1.
    Bogomolov, F.A., Fu, H.: Division polynomials and intersection of projective torsion points. Eur. J. Math. 2(3), 644–660 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bogomolov, F.A., Fu, H.: Elliptic curves with large intersection of projective torsion points. Eur. J. Math. 4(2), 555–560 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bogomolov, F., Fu, H., Tschinkel, Yu.: Torsion of elliptic curves and unlikely intersections. In: Andersen, J.E., et al. (eds.) Geometry and Physics, vol. I, pp. 19–38. Oxford University Press, Oxford (2018)zbMATHGoogle Scholar
  4. 4.
    Bogomolov, F., Tschinkel, Yu.: Algebraic varieties over small fields. In: Zannier, U. (ed.) Diophantine Geometry. Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series, vol. 4, pp. 73–91. Edizioni della Normale, Pisa (2007)Google Scholar
  5. 5.
    DeMarco, L., Krieger, H., Ye, H.: Uniform Manin–Mumford for a family of genus 2 curves (2019). arXiv:1901.09945
  6. 6.
    Silverman, J.H.: Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 151. Springer, New York (1994)CrossRefGoogle Scholar
  7. 7.
    Suzuki, M.: Group Theory. I. Grundlehren der Mathematischen Wissenschaften, vol. 247. Springer, Berlin (1982)Google Scholar
  8. 8.
    Wolfram Research, Inc.: Mathematica, Version 11.0. Champaign, IL (2016)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.AG LaboratoryNational Research University Higher School of EconomicsMoscowRussia
  3. 3.National Center for Theoretical SciencesNational Taiwan UniversityTaipeiTaiwan

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