Ordinary elliptic curves of high rank over with constant j-invariant II

We show that for all odd primes p, there exist ordinary elliptic curves over 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 ℓ, there exists a hyperelliptic curve over of genus (ℓ−1)/2 whose Jacobian is isogenous to the power of one ordinary elliptic curve.