Common Divisors of Elliptic Divisibility Sequences over Function Fields

Abstract

Let E/k(T) be an elliptic curve defined over a rational function field and fix a Weierstrass equation for E. For a point P € E(k(T)), we can write .....with relatively prime polynomials Aр, Bр € k[T]. The sequence (Βnρ )n ≥1 is called the elliptic divisibility sequence of P € E. For two such elliptic divisibility sequences (Βnρ ) n ≥1 and (ΒnQ) n ≥1, we consider the degree of the greatest common divisor of terms in the elliptic divisibility sequences, deg gcd(Βnρ , ΒnQ) ≥

We conjecture a complete theory for how this degree is bounded as n increases, and we support this conjecture with proofs and experiments. In characteristic 0, Silverman already conjectured that this degree is always bounded by a constant, and he gave a proof for curves with constant j- invariant. In characteristic p, Silverman conjectured that there is always a constant c such that there are in?nitely many n with

deg gcd(BnP, BnQ) ≥ cn2

We conjecture that there are curves that do as well as curves that don't satisfy the stronger bound that

deg gcd (ВnP, BnQ) ≥ cn2

for infinitely many n, and that this is still true when we do not allow the field characteristic р to divide n.