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 R ∈ E(k(T )), write xR = AR/D2 R with relatively prime polynomials AR(T ), DR(T ) ∈ k[T ]. The sequence {DnR}n≥1 is called the elliptic divisibility sequence of R. Let P, Q ∈ E(k(T )) be independent points. We conjecture that deg ( gcd(DnP , DmQ) ) is bounded for m, n ≥ 1, and that gcd(DnP , DnQ) = gcd(DP , DQ) 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 (