Ordinary elliptic curves of high rank over ${\bar{F}}_p(x)$ 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 ${\bar{F}}_p(x)$ 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.