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

Claus Diem·Jasper Scholten
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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.

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Claus Diem, Jasper Scholten. 2006-07-18. Ordinary elliptic curves of high rank over $\bar F_p(x)$ with constant j-invariant II. https://arxiv.org/abs/math/0509600

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