Abstract.
Let E/k(T) be an elliptic curve defined over a rational function field of characteristic zero. Fix a Weierstrass equation for E. For points R ∈ E(k(T)), write xR=AR/DR2 with relatively prime polynomials A
R
(T),D
R
(T) ∈ k[T]. The sequence {D
nR
}
n
≥ 1 is called the elliptic divisibility sequence of R.
Let P,Q ∈ E(k(T)) be independent points. We conjecture that
deg (gcd(D
nP
, D
mQ
)) is bounded for m, n ≥ 1,
and that
gcd(D
nP
, D
nQ
) = gcdD
P
, D
Q
) for infinitely many n ≥ 1.
We prove these conjectures in the case that j(E) ∈ k. More generally, we prove analogous statements with k(T) replaced by the function field of any curve and with P and Q allowed to lie on different elliptic curves. If instead k is a finite field of characteristic p and again assuming that j(E) ∈ k, we show that deg (gcd(D
nP
, D
nQ
)) is as large as 
for infinitely many n≢0 (mod p).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ailon, N., Rudnick, Z.: Torsion points on curves and common divisors of ak-1 and bk-1. Acta Arithmetica, http://arxiv.org/abs/math.NT/0202102, to appear
Bézivin, J.P., Pethö, A., van der Poorten, A.J.: A full characterization of divisibility sequences. Amer. J. of Math. 112, 985–1001 (1990)
Bugeaud, Y., Corvaja, P., Zannier, U.: An upper bound for the G.C.D. of an -1 and bn -1. Math. Zeit. 243 (1), 79–84 (2003)
Chambert-Loir, A.: Points de petite hauteur sur les variétés semi-abéliennes. Ann. Sci. École Norm. Sup. 33, 789–821 (2000)
Corvaja, P., Zannier, U.: On the greatest prime factor of (ab+1)(ac+1). Proc. Amer. Math. Soc. 131, 1705–1709 (2002)
Corvaja, P., Zannier, U.: A lower bound for the height of a rational function at S-unit points. Monatshefte Math. http://arxiv.org/abs/math.NT/0311030, To appear
Corvaja, P., Rudnick, Z., Zannier, U.: A lower bound for periods of matrices. http://www.arxiv.org/abs/math.NT/0309215
David, S., Philippon, P.: Sous-variétés de torsion des variétés semi-abéliennes. C. R. Acad. Sci. Paris Sér. I Math. 331, 587–592 (2000)
Durst, L.K.: The apparition problem for equianharmonic divisibility. Proc. Nat. Acad. Sci. USA 38, 330–333 (1952)
Einsiedler, M., Everest, G., Ward, T.: Primes in elliptic divisibility sequences. LMS J. Comput. Math. 4, 1–13 (2001), (electronic)
Everest, G., van der Poorten, A., Shparlinski, I., Ward, T.: Recurrence sequences. Mathematical Surveys and Monographs 104, AMS, Providence, RI, 2003
Everest, G., Ward, T.: Primes in divisibility sequences. Cubo Mat. Educ. 3, 245–259 (2001)
Everest, G., Ward, T.: The canonical height of an algebraic point on an elliptic curve. New York J. Math. 6, 331–342 (2000), (electronic)
McKinnon, D.: Vojta’s main conjecture for blowup surfaces. Proc. Amer. Math. Soc. 131, 1–12 (2003), (electronic)
Raynaud, M.: Courbes sur une variété abélienne et points de torsion. Invent. Math. 71, 207–233 (1983)
Raynaud, M.: Sous-variétés d’une variété abélienne et points de torsion. Arithmetic and geometry, Vol. I, Progr. Math. 35, Birkhäuser, Boston, MA, 327–352 (1983)
Rosen, M.: Number theory in function fields. GTM 210, Springer-Verlag, New York, 2002
Shipsey, R.: Elliptic divisibility sequences. Ph.D. thesis, Goldsmith’s College, University of London, 2000
Silverman, J.H.: The arithmetic of elliptic curves. GTM 106, Springer-Verlag, New York, 1986
Silverman, J.H.: Advanced topics in the arithmetic of elliptic curves. GTM 151, Springer-Verlag, New York, 1994
Silverman, J.H.: Algebraic divisibility sequences and Vojta’s conjecture for exceptional divisors. In preparation
Silverman, J.H.: Common divisors of an-1 and bn-1 over function fields. New York Journal of Math. 10, 37–43 (2004) (electronic)
Vojta, P.: Diophantine approximations and value distribution theory. Lecture Notes in Mathematics 1239, Springer-Verlag, Berlin, 1987
Ward, M.: Memoir on elliptic divisibility sequences. Amer. J. Math. 70, 31–74 (1948)
Ward, M.: The law of repetition of primes in an elliptic divisibility sequence. Duke Math. J. 15, 941–946 (1948)
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classification (2000): Primary: 11D61; Secondary: 11G35
Acknowledgements. I would like to thank Gary Walsh for rekindling my interest in the arithmetic properties of divisibility sequences and for bringing to my attention the articles [1] and [3], and David McKinnon for showing me his article [14]. I also want to thank Zeev Rudnick for his helpful comments concerning the first draft of this paper, especially for Remark 5, for pointing out [7], and for letting me know that he described conjectures similar to those made in this paper at CNTA 7 in 2002.
Rights and permissions
About this article
Cite this article
Silverman, J. Common divisors of elliptic divisibility sequences over function fields. manuscripta math. 114, 431–446 (2004). https://doi.org/10.1007/s00229-004-0468-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-004-0468-7