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

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

2005-09-26

arxiv.org/abs/math/0509600v2

Mathematics

Number Theory

14 pages, new version

Scientific paper

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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