  We show that for all odd primes $p$, there exist ordinary elliptic curves
over $\bar{\mathbb{F}}_p(x)$ with arbitrarily high rank and constant
$j$-invariant. This shows in particular that there are elliptic curves with
arbitrarily high rank over these fields for which the corresponding elliptic
surface is not supersingular. The result follows from a theorem which states
that for all odd prime numbers $p$ and $\ell$, there exists a hyperelliptic
curve over $\bar{\mathbb{F}}_p$ of genus $(\ell-1)/2$ whose Jacobian is
isogenous to the power of one ordinary elliptic curve.
