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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 /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 and that deg ( gcd(DnP,DmQ) ) is bounded for m, n ≥ 1, 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 ( gcd(DnP,DnQ)) is as large as n + O ( √ n) for infinitely many n ̸ ≡ 0 (mod p).

Abstract. An elliptic divisibility sequence (EDS) is a sequence of integers (Wn)n≥0 generated by the nonlinear recursion satisfied by the division polyomials of an elliptic curve. We give a formula for the sign of Wn for unbounded nonsingular EDS, a typical case being Sign(Wn) = (−1) ⌊nβ ⌋ for an irrational number β ∈ R. As an application, we show that the associated sequence of absolute values (|Wn|) cannot be realized as the fixed point counting sequence of any abstract dynamical system.