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Cartan images and \(\ell \)-torsion points of elliptic curves with rational j-invariant

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Abstract

Let \(\ell \) be an odd prime and d a positive integer. We determine when there exists a degree-d number field K and an elliptic curve E / K with \(j(E)\in \mathbb {Q}\setminus \{0,1728\}\) for which \(E(K)_\mathrm {tors}\) contains a point of order \(\ell \), conditionally on a conjecture of Sutherland. We likewise determine when there exists such a pair (K, E) for which the image of the associated mod-\(\ell \) Galois representation is contained in a Cartan subgroup or its normalizer. We do the same under the stronger assumption that E is defined over \(\mathbb {Q}\).

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Notes

  1. For \(\ell =2\), all conjugacy classes of subgroups of \({\text {GL}}_2(\mathbb {Z}/\ell \mathbb {Z})\) occur as the image of \(\rho _{E,\ell }\) for some elliptic curve \(E/\mathbb {Q}\), so we ignore this case henceforth.

References

  1. Anderson, D.D., Camillo, V.: Subgroups of direct products of groups, ideals and subrings of direct products of rings, and Goursat’s lemma. In: Rings, modules and representations, of Contemp. Math., vol. 480, pp. 1–12. American Mathematical Society, Providence (2009)

  2. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: the user language. J. Symb. Comput. 24, 235–265 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bourdon, A., Clark, P.L.: Torsion points and Galois representations on CM elliptic curves. arXiv:1612.03229 (2016)

  4. Bourdon, A., Clark, P.L., Stankewicz, J.: Torsion points on CM elliptic curves over real number fields. Trans. Amer. Math. Soc. 369(12), 8457–8496 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bourdon, A., Pollack, P.: Torsion subgroups of CM elliptic curves over odd degree number fields. Int. Math. Res. Not. 2017(16), 4923–4961 (2016)

    MathSciNet  Google Scholar 

  6. Bruin, P., Najman, F.: A criterion to rule out torsion groups for elliptic curves over number fields. Res. Number Theory 2, 23 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chou, M.: Torsion of rational elliptic curves over quartic Galois number fields. J. Number Theory 160, 603–628 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Clark, P.L., Corn, P., Rice, A., Stankewicz, J.: Computation on elliptic curves with complex multiplication. LMS J. Comput. Math. 17(1), 509–535 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cremona, J.E.: Elliptic curves over \({\mathbb{Q}}\). (2016). http://www.lmfdb.org/EllipticCurve/Q/. Accessed 14 Feb 2017

  10. Derickx, M., Sutherland, A.: Torsion subgroups of elliptic curves over quintic and sextic number fields. Proc. Amer. Math. Soc. 145(10), 4233–4245 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dickson, L.E.: Linear Groups with an Exposition of Galois Field Theory. B. G. Teubner, Leipzig (1901)

    MATH  Google Scholar 

  12. González-Jiménez, E.: Complete classification of the torsion structures of rational elliptic curves over quintic number fields. J. Algebra 478, 484–505 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. González-Jiménez, E., Lozano-Robledo, Á.: On the minimal degree of definition of \(p\)-primary torsion subgroups of elliptic curves. Math. Res. Lett. 24(4), 1067–1096 (2017)

  14. González-Jiménez, E., Lozano-Robledo, Á.: On the torsion of rational elliptic curves over quartic fields. Math. Comp. (2017). https://doi.org/10.1090/mcom/3235

  15. González-Jiménez, E., Lozano-Robledo, Á.: Elliptic curves with abelian division fields. Math. Z. 283(3–4), 835–859 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. González-Jiménez, E., Najman, F.: Growth of torsion groups of elliptic curves upon base change. arXiv:1609.02515 (2016)

  17. Jeon, D.: Families of elliptic curves over cyclic cubic number fields with prescribed torsion. Math. Comput. 85(299), 1485–1502 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jeon, D., Kim, C.H., Lee, Y.: Families of elliptic curves over cubic number fields with prescribed torsion subgroups. Math. Comput. 80(273), 579–591 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jeon, D., Kim, C.H., Lee, Y.: Infinite families of elliptic curves over dihedral quartic number fields. J. Number Theory 133(1), 115–122 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jeon, D., Kim, C.H., Park, E.: On the torsion of elliptic curves over quartic number fields. J. London Math. Soc. 74(1), 1–12 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jeon, D., Kim, C.H., Schweizer, A.: On the torsion of elliptic curves over cubic number fields. Acta Arith. 113(3), 291–301 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kamienny, S.: Torsion points on elliptic curves and \(q\)-coefficients of modular forms. Invent. Math. 109(2), 221–229 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kenku, M.A., Momose, F.: Torsion points on elliptic curves defined over quadratic fields. Nagoya Math. J. 109, 125–149 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lozano-Robledo, Á.: On the field of definition of \(p\)-torsion points on elliptic curves over the rationals. Math. Ann. 357(1), 279–305 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mazur, B.: Modular curves and the Eisenstein ideal. Inst. Hautes Études Sci. Publ. Math. 47(1), 33–186 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mazur, B.: Rational isogenies of prime degree (with an appendix by D. Goldfeld). Invent. Math. 44(2), 129–162 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  27. Najman, F.: Isogenies of non-CM elliptic curves with rational \(j\)-invariants over number fields. Math. Proc. Cambridge Philos. Soc. 164(1), 179–184 (2018)

  28. Najman, F.: Exceptional elliptic curves over quartic fields. Int. J. Number Theory 8(5), 1231–1246 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Najman, Filip: Torsion of elliptic curves over cubic fields. J. Number Theory 132(1), 26–36 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Najman, F.: Torsion of rational elliptic curves over cubic fields and sporadic points on \(X_1(n)\). Math. Res. Lett. 23, 245–272 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Propp, O.Y.: GitHub repository related to Cartan images and \(\ell \)-torsion points of elliptic curves with rational \(j\)-invariant. (2017). https://github.com/oronpropp/ecdegsets. Accessed 21 July 2017

  32. Rebolledo, M., Wuthrich, C.: A moduli interpretation for the non-split Cartan modular curve. Glasgow Math. J. (2017). https://doi.org/10.1017/S0017089517000180

  33. Serre, J.P.: Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15(4), 259–331 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  34. Sutherland, A.V.: Computing images of Galois representations attached to elliptic curves. In: Forum of Mathematics, Sigma, vol. 4, pp. 79. Cambridge University Press, Cambridge (2016)

  35. Wang, J.: On the cyclic torsion of elliptic curves over cubic number fields. arXiv:1502.06873 (2017)

  36. Zywina, D.J.: On the possible images of the mod-\(\ell \) representations associated to elliptic curves over \({\mathbb{Q}}\). arXiv:1508.07660 (2015)

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Acknowledgements

The author wishes to thank Andrew Sutherland for supervising this research and providing many helpful suggestions and insights; Filip Najman and Álvaro Lozano-Robledo for their comments on an early draft of this paper; and the referees for their comments and careful review. This research was generously supported by MIT’s UROP program and the Paul E. Gray (1954) Endowed Fund for UROP.

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  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Oron Y. Propp

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Propp, O.Y. Cartan images and \(\ell \)-torsion points of elliptic curves with rational j-invariant. Res. number theory 4, 12 (2018). https://doi.org/10.1007/s40993-018-0097-y

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