## Common Divisors of Elliptic Divisibility Sequences over Function Fields

##### Summary

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.