Ordinary elliptic curves of high rank over ${\bar{F}}_p(x)$ with constant j-invariant II

Abstract

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.

14 pages, new version