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

  • Fedor A. Bogomolov
  • Hang FuEmail author
Research Article


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.


Elliptic curves Torsion points q-series Congruence subgroups Modular curves 

Mathematics Subject Classification

14H52 11G05 11F03 20H05 40A20 



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.


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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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