Ordinary elliptic curves of high rank over $\bar F_p(x)$ with constant j-invariant

Bouw, Irene I.; Diem, Claus; Scholten, Jasper

Ordinary elliptic curves of high rank over $\bar F_p(x)$ with constant j-invariant

Authors

Bouw, Irene I.

Diem, Claus

Scholten, Jasper

Description

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.
15 pages, 0 figures, LaTeX