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On the Lang–Trotter conjecture for two elliptic curves

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Abstract

Following Lang and Trotter, we describe a probabilistic model that predicts the distribution of primes p with given Frobenius traces at p for two fixed elliptic curves over \(\mathbb {Q}\). In addition, we propose explicit Euler product representations for the constant in the predicted asymptotic formula and describe in detail the universal component of this constant. A new feature is that in some cases the \(\ell \)-adic limits determining the \(\ell \)-factors of the universal constant, unlike the Lang–Trotter conjecture for a single elliptic curve, do not stabilize. We also prove the conjecture on average over a family of elliptic curves, which extends the main results of Fouvry and Murty (Supersingular primes common to two elliptic curves, number theory (Paris, 1992), London Mathematical Society Lecture Note Series, vol 215, Cambridge University Press, Cambridge, 1995) and Akbary et al. (Acta Arith 111(3):239–268, 2004), following the work of David et al. (Math Ann 368(1–2):685–752, 2017).

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References

  1. Akbary, A., David, C., Juricevic, R.: Average distributions and products of special values of L-series. Acta Arith. 111(3), 239–268 (2004). https://doi.org/10.4064/aa111-3-3.

    Article  MathSciNet  MATH  Google Scholar 

  2. Baier, S., Jones, N.: A refined version of the Lang-Trotter conjecture. Int. Math. Res. Not. IMRN 3, 433–461 (2009). https://doi.org/10.1093/imrn/rnn136.

    MathSciNet  MATH  Google Scholar 

  3. Castryck, W., Hubrechts, H.: The distribution of the number of points modulo an integer on elliptic curves over finite fields. Ramanujan J. 30(2), 223–242 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chandee, V., David, C., Koukoulopoulos, D., Smith, E.: The frequency of elliptic curve groups over prime finite fields. Can. J. Math. 68(4), 721–761 (2016). https://doi.org/10.4153/CJM-2015-013-1.

    Article  MathSciNet  MATH  Google Scholar 

  5. Cojocaru, A., Davis, R., Silverberg, A., Stange, K.: Arithmetic properties of the Frobenius traces defined by a rational abelian variety, with two appendices by J-P Serre. Int. Math. Res. Not. IMRN 12, 3557–3602 (2017). https://doi.org/10.1093/imrn/rnw058

    MathSciNet  MATH  Google Scholar 

  6. David, C., Koukoulopoulos, D., Smith, E.: Sums of Euler products and statistics of elliptic curves. Math. Ann. 368(1–2), 685–752 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. David, C., Pappalardi, F.: Average Frobenius distributions of elliptic curves. Int. Math. Res. Not. 4, 165–183 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Elkies, N.: Distribution of supersingular primes. Journées Arithmétiques, 1989 (Luminy, 1989), Astérisque No. 198–200 (1991), 127–132 (1992)

  9. Faltings, G.: Endlichkeitssaẗze fur̈ abelsche Varietaẗen ub̈er Zahlkor̈pern. Invent. Math. 73(3), 349–366 (1983). https://doi.org/10.1007/BF01388432

    Article  MathSciNet  Google Scholar 

  10. Fouvry, É., Murty, M.R.: On the distribution of supersingular primes. Can. J. Math. 48(1), 81–104 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fouvry, E., Murty, M.R.: Supersingular primes common to two elliptic curves. In: Supersingular Primes Common to Two Elliptic Curves, Number Theory (Paris, 1992). London Mathematical Society Lecture Note Series, vol. 215. Cambridge University Press, Cambridge (1995), pp. 91–102. https://doi.org/10.1017/CBO9780511661990.007

  12. Gekeler, E.-U.: Frobenius distributions of elliptic curves over finite prime fields. Int. Math. Res. Not. 37, 1999–2018 (2003). https://doi.org/10.1155/S1073792803211272.

    Article  MathSciNet  MATH  Google Scholar 

  13. Harris, M., Shepherd-Barron, N., Taylor, R.: A family of Calabi-Yau varieties and potential automorphy. Ann. Math. 171(2), 779–813 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory. Graduate Texts in Mathematics, vol. 84, 2nd edn. Springer, New York (1990)

    Book  MATH  Google Scholar 

  15. Johansson, C.: On the Sato-Tate conjecture for non-generic abelian surfaces. Trans. Am. Math. Soc. 369(9), 6303–6325 (2017). https://doi.org/10.1090/tran/6847. With an appendix by Francesc Fité

    Article  MathSciNet  MATH  Google Scholar 

  16. Jones, N.: Pairs of elliptic curves with maximal Galois representations. J. Number Theory 133(10), 3381–3393 (2013). https://doi.org/10.1016/j.jnt.2013.03.002.

    Article  MathSciNet  MATH  Google Scholar 

  17. Katz, N.M.: Lang-Trotter revisited. Bull. Am. Math. Soc. (N.S.) 46(3), 413–457 (2009). https://doi.org/10.1090/S0273-0979-09-01257-9

    Article  MathSciNet  MATH  Google Scholar 

  18. Kiyosi, I.: Introduction to Probability Theory. Cambridge University Press, Cambridge (1984)

    MATH  Google Scholar 

  19. Lang, S., Trotter, H.: Distribution of Frobenius automorphisms in GL\(\_2\)-extensions of the rational numbers. In: Frobenius Distributions in GL2-Extensions. Lecture Notes in Mathematics, vol. 504. Springer, Berlin (1976)

  20. Murty, M.R., Pujahari, S.: Distinguishing Hecke eigenforms. Proc. Am. Math. Soc. 145, 1899–1904 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Murty, V.K.: Modular Forms and the Chebotarev Density Theorem. II. Analytic Number Theory (Kyoto, 1996), pp. 287–308. Cambridge University Press, Cambridge (1997)

  22. Oesterlé, J.: Réduction modulo \(p^{n}\) des sous-ensembles analytiques fermés de \({\bf Z}^{N}_{p}\). Invent. Math. 66(2), 325–341 (1982). https://doi.org/10.1007/BF01389398. (French)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Serre, J.-P.: Quelques applications du théorème de densité de Chebotarev, French. Inst. Hautes Études Sci. Publ. Math. 54(323–401), 0073–8301 (1981)

    Google Scholar 

  25. Silverman, J.H.: The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 106. Springer, New York (1986)

    Book  MATH  Google Scholar 

  26. Thorner, J., Zaman, A.: A Chebotarev variant of the Brun-Titchmarsh theorem and bounds for the Lang-Trotter conjectures. Int. Math. Res. Not. (2017). https://doi.org/10.1093/imrn/rnx031

  27. Zywina, D.: Bounds for the Lang-Trotter conjecture. Contemp. Math. Centre Rech. Math. 655, 235–256 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the referee for their valuable comments and suggestions. They also thank Julia Gordon for correspondence on an earlier version of this paper and for her clarifying comments regarding the stability of \(\mathcal {S}_k\). We also would like to thank Jeff Achter and Jesse Thorner for correspondence and comments. The first author thanks Forrest Francis for help with computations in Lemma 4.5; he also thanks Vijay Patankar for several helpful comments regarding the paper.

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  1. James Parks

    Present address: Department of Mathematics, KTH Royal Institute of Technology, Lindstedtsvägen 25, 100 44, Stockholm, Sweden

Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive, Lethbridge, AB, T1K 3M4, Canada

    Amir Akbary & James Parks

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  1. Amir Akbary
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  2. James Parks
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Correspondence to Amir Akbary.

Additional information

Research of the first author is partially supported by NSERC. Research of the second author is partially supported by a PIMS postdoctoral fellowship.

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Akbary, A., Parks, J. On the Lang–Trotter conjecture for two elliptic curves. Ramanujan J 49, 585–623 (2019). https://doi.org/10.1007/s11139-018-0050-7

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