Journal of number theory vol:124 issue:1 pages:31-41
We show that for all odd primes p, there exist ordinary elliptic curves over 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 l, there exists a hyperelliptic curve over F-p of genus (l - 1)/2 whose Jacobian is isogenous to the power of one ordinary elliptic curve. (C) 2006 Elsevier Inc. All rights reserved.