Common divisors of elliptic divisibility sequences over function fields

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 RE(k(T)), write x_R=A_R/D_R² with relatively prime polynomials A_R(T),D_R(T)kT]. The sequence {D_nR}_{n 1} is called the elliptic divisibility sequence of R. Let P,QE(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 n0 (mod p).Mathematics Subject Classification (2000): Primary: 11D61; Secondary: 11G35Acknowledgements. 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.