Ordinary elliptic curves of high rank over (F)over-barp(x) with constant j-invariant

We show that under the assumption of Artin's Primitive Root Conjecture, for all primes p there exist ordinary elliptic curves over (F) over bar (p)(x) with arbitrarily high rank and constant j-invariant. For odd primes p, this result follows from a theorem we prove which states that whenever p is a generator of (Z/lZ)*/<-1> (l an odd prime) there exists a hyperelliptic curve over (D) over bar (p) whose Jacobian is isogenous to a power of one ordinary elliptic curve.