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Constants in Titchmarsh divisor problems for elliptic curves

Abstract

Inspired by the analogy between the group of units \(\mathbb {F}_p^{\times }\) of the finite field with p elements and the group of points \(E(\mathbb {F}_p)\) of an elliptic curve \(E/\mathbb {F}_p\), E. Kowalski, A. Akbary & D. Ghioca, and T. Freiberg & P. Kurlberg investigated the asymptotic behaviour of elliptic curve sums analogous to the Titchmarsh divisor sum \(\sum \nolimits _{p \le x} \tau (p + a) \sim C x\). In this paper, we present a comprehensive study of the constants C(E) emerging in the asymptotic study of these elliptic curve divisor sums in place of the constant C above. Specifically, by analyzing the division fields of an elliptic curve \(E/\mathbb {Q}\), we prove bounds for the constants C(E) and, in the generic case of a Serre curve, we prove explicit closed formulae for C(E) amenable to concrete computations. Moreover, we compute the moments of the constants C(E) over two-parameter families of elliptic curves \(E/\mathbb {Q}\). Our methods and results complement recent studies of average constants occurring in other conjectures about reductions of elliptic curves by addressing not only the average behaviour, but also the individual behaviour of these constants, and by providing explicit tools towards the computational verifications of the expected asymptotics.

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References

  1. Akbary, A., Felix, A.T.: On invariants of elliptic curves on average. Acta Arith. 168(1), 31–70 (2015)

    MathSciNet  Article  Google Scholar 

  2. Akbary, A., Ghioca, D.: A geometric variant of Titchmarsh divisor problem. Int. J. Number Theory 8(1), 53–69 (2012)

    MathSciNet  Article  Google Scholar 

  3. Akhtari, S., David, C., Hahn, H., Thompson, L.: Distribution of squarefree values of sequences associated with elliptic curves. Contemp. Math. 606, 171–188 (2013)

    MathSciNet  Article  Google Scholar 

  4. Balog, A., Cojocaru, A.C., David, C.: Average twin prime conjecture for elliptic curves. Am. J. Math. 133(5), 1179–1229 (2011)

    MathSciNet  Article  Google Scholar 

  5. Banks, W.D., Shparlinski, I.E.: Sato-Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height. Isr. J. Math. 173, 253–277 (2009)

    MathSciNet  Article  Google Scholar 

  6. Bombieri, E., Friedlander, J., Iwaniec, H.: Primes in arithmetic progressions to large moduli. Acta Math. 156, 203–251 (1986)

    MathSciNet  Article  Google Scholar 

  7. Cojocaru, A.C.: Primes, elliptic curves and cyclic groups: a synopsis. Revue Roumain de Mathématiques Pures et Appliquées, Invited contributions to the Eighth Congress of Romanian Mathematicians (Iasi, 2015). Tome LXII No. 1 (2017)

  8. Cojocaru, A.C., Murty, M.R.: Cyclicity of elliptic curves modulo \(p\) and elliptic curve analogues of Linnik’s problem. Math. Ann. 330, 601–625 (2004)

    MathSciNet  Article  Google Scholar 

  9. Cojocaru, A.C., Fitzpatrick, M., Insley, T., Yilmaz, H.: Reductions modulo primes of Serre curves: computational data, appendix to primes, elliptic curves and cyclic groups by A.C. Cojocaru. Contemp. Math. (to appear)

  10. Cojocaru, A.C., Iwaniec, H., Jones, N.: The average asymptotic behaviour of the Frobenius fields of an elliptic curve (preprint)

  11. Cox, D.A.: Primes of the Form \(x^2 + n y^2\). Fermat, Class Field Theory, and Complex Multiplication. Pure and Applied Mathematics, 2nd edn. Wiley, Hoboken (2013)

    Book  Google Scholar 

  12. David, C., Koukoulopoulos, D., Smith, E.: Sums of Euler products and statistics on elliptic curves. Math. Ann. http://www.mathstat.concordia.ca/faculty/cdavid/PAPERS/random-euler-products.pdf (to appear)

  13. Felix, A.T.: Generalizing the Titchmash divisor problem. Int. J. Number Theory 8(3), 613–629 (2012)

    MathSciNet  Article  Google Scholar 

  14. Felix, A.T., Murty, M.R.: On the asymptotics for invariants of elliptic curves modulo \(p\). J. Ramanujan Math. Soc. 28(3), 271–298 (2013)

    MathSciNet  MATH  Google Scholar 

  15. Fouvry, É.: Sur le problem des diviseurs de Titchmarsh. J. Reine Angew. Math. 357, 51–76 (1984)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  17. Freiberg, T., Kulberg, P.: On the average exponent of elliptic curves modulo \(p\). Int. Math. Res. Not. 8, 2265–2293 (2014)

    MathSciNet  Article  Google Scholar 

  18. Freiberg, T., Pollack, P.: The average of the first invariant factor for reductions of CM elliptic curves mod \(p\). Int. Math. Res. Not. 21, 11333–11350 (2015)

    MathSciNet  Article  Google Scholar 

  19. Gekeler, E.-U.: Statistics about elliptic curves over finite prime fields. Manuscr. Math. 127(1), 55–67 (2008)

    MathSciNet  Article  Google Scholar 

  20. Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 6th edn. Oxford University Press, Oxford (2008). Revised by D. R. Heath-Brown and J. H. Silverman, with a foreword by A. Wiles

  21. Halberstam, H.: Footnote to the Titchmarsh-Linnik divisor problem. Proc. Am. Math. Soc. 18, 187–188 (1967)

    MathSciNet  Article  Google Scholar 

  22. Howe, E.W.: On the group orders of elliptic curves over finite fields. Compos. Math. 85, 229–247 (1993)

    MathSciNet  MATH  Google Scholar 

  23. Jones, N.: Averages of elliptic curve constants. Math. Ann. 345, 685–710 (2009)

    MathSciNet  Article  Google Scholar 

  24. Jones, N.: Almost all elliptic curves are Serre curves. Trans. Am. Math. Soc. 362(3), 1547–1570 (2010)

    MathSciNet  Article  Google Scholar 

  25. Jones, N.: A bound for the conductor of an open subgroup of \(\text{GL}_2\) associated to an elliptic curve. arXiv:1904.10431 (preprint)

  26. Kaplan, N., Petrow, I.: Elliptic curves over a finite field and the trace formula. arXiv:1510.03980 (preprint)

  27. Kawamura, T.: The effective surjectivity of mod \(\ell \) Galois representations of 1- and 2-dimensional abelian varieties with trivial endomorphism ring. Comment. Math. Helv. 78, 486–493 (2003)

    MathSciNet  Article  Google Scholar 

  28. Kim, S.: Average behaviors of invariant factors in Mordell-Weil groups of CM elliptic curves modulo \(p\). Finite Fields Appl. 30, 178–190 (2014)

    MathSciNet  Article  Google Scholar 

  29. Kowalski, E.: Analytic problems for elliptic curves. J. Ramanujan Math. Soc. 21(1), 19–114 (2006)

    MathSciNet  MATH  Google Scholar 

  30. Linnik, J.V.: The Dispersion Method in Binary Additive Problems. Translations of Mathematical Monographs, vol. 4. American Mathematical Society, Providence (1963)

    Google Scholar 

  31. Masser, D., Wüstholz, G.: Galois properties of division fields of elliptic curves. Bull. Lond. Math. Soc. 25, 247–254 (1993)

    MathSciNet  Article  Google Scholar 

  32. Pollack, P.: A Titchmarsh divisor problem for elliptic curves. Math. Proc. Camb. Philos. Soc. 160(1), 167–189 (2016)

    MathSciNet  Article  Google Scholar 

  33. Rodriquez, G.: Sul problema dei divisori di Titchmarsh. Boll. Unione Math. Ital. Serie 3 20, 358–366 (1965)

    MathSciNet  MATH  Google Scholar 

  34. Barkley Rosser, J., Schoenfeld, L.: Approximate formulas for some functions of prime numbers. Ill. J. Math. 6, 64–94 (1962)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  36. Silverman, J.H.: The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 106. Springer, New York (2000)

    Google Scholar 

  37. Titchmarsh, E.C.: A divisor problem. Rend. Circ. Mat. Palermo 54, 414–429 (1930)

    Article  Google Scholar 

  38. Vladut, S.G.: Cyclicity statistics for elliptic curves over finite fields. Finite Fields Appl. 5, 13–25 (1999)

    MathSciNet  Article  Google Scholar 

  39. Weil, A.: On a certain type of characters of the idèle-class group of an algebraic number-field. In: Proceedings of the International Symposium on Algebraic Number Theory, Tokyo-Nikko, pp. 1–7 (1955)

  40. Weil, A.: On the theory of complex multiplication. In: Proceedings of the International Symposium on Algebraic Number Theory, Tokyo-Nikko, pp. 9–22 (1955)

  41. Wu, J.: The average exponent of elliptic curves modulo \(p\). J. Number Theory 135, 28–35 (2014)

    MathSciNet  Article  Google Scholar 

  42. Zywina, D.: Bounds for Serre’s open image theorem. http://www.math.cornell.edu/~zywina/papers/Serre-Bound.pdf (preprint)

  43. Zywina, D.: Possible indices for the Galois image of elliptic curves over \(\mathbb{Q}\). http://pi.math.cornell.edu/~zywina/papers/PossibleIndices/PossibleIndices.pdf (preprint)

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Author's contributions

Acknowlegements

This research started during the Arizona Winter School 2016: Analytic Methods in Arithmetic Geometry, organized at the University of Arizona, Tucson, USA, during March 12-16, 2016. We thank the conference organizers Alina Bucur, David Zureick-Brown, Bryden Cais, Mirela Ciperiani, and Romyar Sharifi for all their time and support, and we thank the National Science Foundation for sponsoring our participation in this conference. Moreover, we thank the referees for carefully reading the original manuscript and for all their comments and suggestions, which enabled us to improve the results of the paper.

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Correspondence to Alina Carmen Cojocaru.

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ACC’s work on this material was partially supported by the Simons Collaboration Grant under Award No. 318454. IV’s work was partially supported by the NSF Graduate Research Fellowship Program and Grant DMS-1601946

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Bell, R., Blakestad, C., Cojocaru, A.C. et al. Constants in Titchmarsh divisor problems for elliptic curves. Res. number theory 6, 1 (2020). https://doi.org/10.1007/s40993-019-0175-9

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  • DOI: https://doi.org/10.1007/s40993-019-0175-9

Keywords

  • Titchmarsh divisor
  • Divisor sum
  • Serre curve
  • Elliptic curve
  • Galois representation

Mathematics Subject Classification

  • 11A25: arithmetic functions, related numbers, inversion formulas
  • 11G05: elliptic curves over global fields
  • 11G20: curves over finite and local fields
  • 11N37: asymptotic results on arithmetic functions
  • 11Y60: evaluation of constants