Mathematische Zeitschrift

, Volume 285, Issue 3–4, pp 795–820 | Cite as

Anatomy of torsion in the CM case

Article
  • 93 Downloads

Abstract

Let \(T_{\mathrm {CM}}(d)\) denote the maximum size of a torsion subgroup of a CM elliptic curve over a degree d number field. We initiate a systematic study of the asymptotic behavior of \(T_{\mathrm {CM}}(d)\) as an arithmetic function. Whereas a recent result of the last two authors computes the upper order of \(T_{\mathrm {CM}}(d)\), here we determine the lower order, the typical order and the average order of \(T_{\mathrm {CM}}(d)\) as well as study the number of isomorphism classes of groups G of order \(T_{\mathrm {CM}}(d)\) which arise as the torsion subgroup of a CM elliptic curve over a degree d number field. To establish these analytic results we need to extend some prior algebraic results. Especially, if \(E_{/F}\) is a CM elliptic curve over a degree d number field, we show that d is divisible by a certain function of \(\# E(F)[{\text {tors}}]\), and we give a complete characterization of all degrees d such that every torsion subgroup of a CM elliptic curve defined over a degree d number field already occurs over \(\mathbb {Q}\).

Mathematics Subject Classification

Primary 11G15 Secondary 11G05 11N25 11N37 

Notes

Acknowledgments

We thank Filip Najman for explaining how to obtain stronger lower bounds on T(6) and T(10). We thank Robert S. Rumely for suggesting we investigate prime power Olson degrees. The exposition in Sect. 7 benefitted from talks by Carl Pomerance on the material in [8]. The first author was supported in part by NSF Grant DMS-1344994 (RTG in Algebra, Algebraic Geometry, and Number Theory, at the University of Georgia). The third author is supported by NSF award DMS-1402268.

References

  1. 1.
    Bhowmik, G.: Average orders of certain functions connected with arithmetic of matrices. J. Indian Math. Soc. (N.S.) 59, 97–105 (1993)MathSciNetMATHGoogle Scholar
  2. 2.
    Bourdon, A., Clark, P.L., Stankewicz, J.: Torsion points on CM elliptic curves over real number fields. Trans. Am. Math. Soc. (to appear)Google Scholar
  3. 3.
    Breuer, F.: Torsion bounds for elliptic curves and Drinfeld modules. J. Number Theory 130, 1241–1250 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Childress, N.: Class Field Theory, Universitext. Springer, New York (2009)CrossRefMATHGoogle Scholar
  5. 5.
    Clark, P.L., Cook, B., Stankewicz, J.: Torsion points on elliptic curves with complex multiplication (with an appendix by Alex Rice). Int. J. Number Theory 9, 447–479 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Clark, P.L., Corn, P., Rice, A., Stankewicz, J.: Computation on elliptic curves with complex multiplication. LMS J. Comput. Math. 17, 509–535 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Clark, P.L., Pollack, P.: The truth about torsion in the CM case. C. R. Math. Acad. Sci. Paris 353, 683–688 (2015)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Erdős, P.: On the normal number of prime factors of \(p-1\) and some related problems concerning Euler’s \(\varphi \)-function. Q. J. Math. 6, 205–213 (1935)CrossRefMATHGoogle Scholar
  9. 9.
    Erdős, P., Pomerance, C.: On the normal number of prime factors of \(\phi (n)\). Rocky Mt. J. Math. 15, 343–352 (1985)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Erdős, P., Szekeres, G.: Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem. Acta Sci. Math. (Szeged) 7, 95–102 (1935)MATHGoogle Scholar
  11. 11.
    Erdős, P., Wagstaff Jr., S.S.: The fractional parts of the Bernoulli numbers. Illinois J. Math. 24, 104–112 (1980)MathSciNetMATHGoogle Scholar
  12. 12.
    Ford, K., Luca, F., Pomerance, C.: The image of Carmichael’s \(\lambda \)-function. Algebra Number Theory 8, 2009–2025 (2014)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Hall, R.R.: Sets of Multiples, Cambridge Tracts in Mathematics, vol. 118. Cambridge University Press, Cambridge (1996)CrossRefGoogle Scholar
  14. 14.
    Hall, R.R., Tenenbaum, G.: Divisors, Cambridge Tracts in Mathematics, vol. 90. Cambridge University Press, Cambridge (1988)Google Scholar
  15. 15.
    Hardy, G.H., Ramanujan, S.: The Normal Number of Prime Factors of a Number \(n\) [Q. J. Math. 48, 76–92 (1917)]. Collected Papers of Srinivasa Ramanujan. AMS Chelsea Publ., Providence (2000)Google Scholar
  16. 16.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 6th edn. Oxford University Press, Oxford (2008)MATHGoogle Scholar
  17. 17.
    Hindry, M., Silverman, J.: Sur le nombre de points de torsion rationnels sur une courbe elliptique. C. R. Acad. Sci. Paris Sér. I Math. 329, 97–100 (1999)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    van Hoeij, M.: Low degree places on the modular curve \(X_1(N)\). arXiv:1202.4355 [math.NT]
  19. 19.
    Iwaniec, H., Kowalski, E.: Analytic Number Theory, American Mathematical Society Colloquium Publications, vol. 53. American Mathematical Society, Providence (2004)CrossRefGoogle Scholar
  20. 20.
    Jacobson Jr., M.J., Ramachandran, S., Williams, H.C.: Numerical results on class groups of imaginary quadratic fields. In: Hess, F., Pauli, S., Pohst, M (eds.) Algorithmic Number Theory, Lecture Notes in Computer Science, vol. 4076, pp. 87–101. Springer, Berlin (2006)Google Scholar
  21. 21.
    Jeon, D., Kim, C.H., Park, E.: On the torsion of elliptic curves over quartic number fields. J. Lond. Math. Soc. (2) 74, 1–12 (2006)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Jeon, D., Kim, C.H., Schweizer, A.: On the torsion of elliptic curves over cubic number fields. Acta Arith. 113(3), 291–301 (2004)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Lenstra Jr., H.W., Pomerance, C.: A rigorous time bound for factoring integers. J. Am. Math. Soc. 5, 483–516 (1992)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Luca, F., Pizarro-Madariaga, A., Pomerance, C.: On the counting function of irregular primes. Indag. Math. (N.S.) 26, 147–161 (2015)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Luca, F., Pomerance, C.: Irreducible radical extensions and Euler-function chains. Integers 7, (2007). article #A25. http://www.integers-ejcnt.org/vol7-2.html
  26. 26.
    Luca, F., Pomerance, C.: On the range of Carmichael’s universal-exponent function. Acta Arith. 162, 289–308 (2014)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Mazur, B.: Modular curves and the Eisenstein ideal. Publ. Math. I.H.E.S. 47, 33–186 (1977)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    McNew, N., Pollack, P., Pomerance, C.: Numbers divisible by a large shifted prime and large torsion subgroups of CM elliptic curves. Int. Math. Res. Notices (2016). doi:10.1093/imrn/rnw173
  29. 29.
    Merel, L.: Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math. 124, 437–449 (1996)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Olson, L.D.: Points of finite order on elliptic curves with complex multiplication. Manuscr. Math. 14, 195–205 (1974)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Parent, P.: Bornes effectives pour la torsion des courbes elliptiques sur les corps de nombres. J. Reine Angew. Math. 506, 85–116 (1999)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Pomerance, C., Sárközy, A.: On homogeneous multiplicative hybrid problems in number theory. Acta Arith. 49, 291–302 (1988)MathSciNetMATHGoogle Scholar
  33. 33.
    Prasad, D., Yogananda, C.S.: Bounding the torsion in CM elliptic curves. C. R. Math. Acad. Sci. Soc. R. Can. 23, 1–5 (2001)MathSciNetMATHGoogle Scholar
  34. 34.
    Schwarz, W., Spilker, J.: Arithmetical Functions, London Mathematical Society Lecture Note Series, vol. 184. Cambridge University Press, Cambridge (1994)CrossRefGoogle Scholar
  35. 35.
    Silverberg, A.: Torsion points on abelian varieties of CM-type. Compos. Math. 68, 241–249 (1988)MathSciNetMATHGoogle Scholar
  36. 36.
    Silverberg, A.: Points of finite order on abelian varieties. In: Adolphson, A.C., Sperber, S., Tretkoff, M.D. (eds.) \(p\)-Adic Methods in Number Theory and Algebraic Geometry, Contemporary Mathematics, vol. 133, pp. 175–193. Am. Math. Soc., Providence (1992)Google Scholar
  37. 37.
    Sutherland, A.V.: Torsion subgroups of elliptic curves over number fields (in preprint)Google Scholar
  38. 38.
    Timofeev, N.M.: Hardy-Ramanujan and Halasz inequalities for shifted prime numbers. Mat. Zametki 57, 747–764 (1995). (Russian)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Mathematics Department, Boyd Graduate Studies Research CenterUniversity of GeorgiaAthensUSA

Personalised recommendations