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