Ordinary elliptic curves of high rank over Fp(x) with constant j-invariant
Irene I. Bouw, Claus Diem, Jasper Scholten
Abstract
We show that under the assumption of Artin's Primitive Root Conjecture, for all primes p there exist ordinary elliptic curves over Fp(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 Fp whose Jacobian is isogenous to a power of one ordinary elliptic curve.
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