2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/102295We 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 versionNumber TheoryAlgebraic Geometry11G05; 11G20; 14H40; 14H52text