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p-adic properties of division polynomials and elliptic divisibility sequences

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Abstract.

For a fixed rational point P ∈ E (K) on an elliptic curve, we consider the sequence of values (F n (P)) n ≥1 of the division polynomials of E at P. For a finite field we prove that the sequence is periodic. For a local field we prove (under certain hypotheses) that there is a power q=pe so that for all m≥1, the limit of exists in K and is algebraic over We apply this result to prove an analogous p-adic limit and algebraicity result for elliptic divisibility sequences.

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References

  1. Bézivin, J.P., Pethö, A., van der Poorten, A.J.: A full characterization of divisibility sequences. Am. J. Math. 112, 985–1001 (1990)

    Google Scholar 

  2. Cheon, J., Hahn, S.: Explicit valuations of division polynomials of an elliptic curve. Manuscripta Math. 97, 319–328 (1998)

    Article  Google Scholar 

  3. Chudnovsky, D.V., Chudnovsky, G.V.: Sequences of numbers generated by addition in formal groups and new primality and factorization tests. Advances in Applied Mathematics 7, 385–434 (1986)

    Article  Google Scholar 

  4. Durst, L.K.: The apparition problem for equianharmonic divisibility. Proc. Nat. Acad. Sci. U. S. A. 38, 330–333 (1952)

    Google Scholar 

  5. Einsiedler, M., Everest, G., Ward, T.: Primes in elliptic divisibility sequences. LMS J. Comput. Math. 4, 1–13 (electronic), 2001

    Google Scholar 

  6. Elkies, N. : The existence of infinitely many supersingular primes for every elliptic curve over Invent. Math. 89, 561–567 (1987)

    Google Scholar 

  7. Elkies, N.: Distribution of supersingular primes. Journées Arithmétiques, 1989 (Luminy, 1989), Astérisque 198-200 (1991), pp. 127–132, 1992

    Google Scholar 

  8. Everest, G., van der Poorten, A., Shparlinski, I., Ward, T.: Recurrence sequences. Mathematical Surveys and Monographs 104, AMS, Providence, RI, 2003

  9. Everest, G., Ward, T.: Primes in divisibility sequences. Cubo Mat. Educ. 3, 245–259 (2001)

    Google Scholar 

  10. Everest, G., Ward, T.: The canonical height of an algebraic point on an elliptic curve. New York J. Math. 6, 331–342 (electronic), 2000

    Google Scholar 

  11. Gale, D. : The Strange and Surprising Saga of the Somos Sequences. Mathematical Intelligencer 13, 40–42, 49–50 (1991)

    Google Scholar 

  12. Mazur, B., Tate, J.: The p-adic sigma function. Duke Math. J. 62, 663–688 (1991)

    Article  Google Scholar 

  13. Propp, J.: The Somos Sequence Site. http://www.math.wisc.edu/∼propp/somos.html.

  14. Robinson, R.: Periodicity of Somos sequences. Proc. Am. Math. Soc. 116, 613–619 (1992)

    Google Scholar 

  15. Schoof, R.: Elliptic curves over finite fields and the computation of square roots mod p. Math. Comp. 44, 483–494 (1985)

    Google Scholar 

  16. Schoof, R.: Counting points on elliptic curves over finite fields. Les Dix-huitièmes Journées Arithmétiques (Bordeaux, 1993), J. Théor. Nombres Bordeaux 7, 219–254 (1995)

    Google Scholar 

  17. Serre, J.-P.: Abelian l-adic representations and elliptic curves. W. A. Benjamin, Inc., New York-Amsterdam 1968

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

    Google Scholar 

  19. Shipsey, R.: Elliptic divisibility sequences. Ph.D. thesis, Goldsmith’s College (University of London), 2000

  20. Silverman, J.H.: The arithmetic of elliptic curves. GTM 106, Springer-Verlag, New York, 1986

  21. Silverman, J.H.: Advanced topics in the arithmetic of elliptic curves. GTM 151, Springer-Verlag, New York, 1994

  22. Silverman, J.H., Stephens, N.: The sign of an elliptic divisibility sequence. Preprint 2004, (arXiv:mathNT/0402415)

  23. Swart, C.S.: Elliptic divisibility sequences. Ph.D. thesis, Royal Holloway (University of London), 2003

  24. Ward, M.: Memoir on elliptic divisibility sequences. Am. J. Math. 70, 31–74 (1948)

    Google Scholar 

  25. Ward, M.: The law of repetition of primes in an elliptic divisibility sequence. Duke Math. J. 15, 941–946 (1948)

    Article  Google Scholar 

  26. Whittaker, E.T., Watson, G.N.: A course in modern analysis. Cambridge Univ. Press, Cambridge, 4th ed., 1927

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Authors and Affiliations

  1. Mathematics Department, Brown University, Box 1917, Providence, RI, 02912, USA

    Joseph H. Silverman

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  1. Joseph H. Silverman
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Correspondence to Joseph H. Silverman.

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Mathematics Subject Classification (1991): 11G07, 11D61, 14G20, 14H52

The author’s research supported by NSA grant H98230-04-1-0064.

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Silverman, J. p-adic properties of division polynomials and elliptic divisibility sequences. Math. Ann. 332, 443–471 (2005). https://doi.org/10.1007/s00208-004-0608-0

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  • DOI: https://doi.org/10.1007/s00208-004-0608-0

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