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

Abstract

We show that under the assumption of Artin’s Primitive Root Conjecture, for all primes p there exist ordinary elliptic curves over Open image in new window 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 Open image in new window whose Jacobian is isogenous to a power of one ordinary elliptic curve.