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

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

2003-05-04

arxiv.org/abs/math/0305064v1

Mathematics

Number Theory

15 pages, 0 figures, LaTeX

Scientific paper

We show that under the assumption of Artin's Primitive Root Conjecture, for all primes p there exist ordinary elliptic curves over $\bar F_p(x)$ with arbitrary high rank and constant j-invariant. For odd primes p, this result follows from a theorem which states that whenever p is a generator of (Z/ell Z)^*/<-1> (ell an odd prime) there exists a hyperelliptic curve over $\bar F_p$ whose Jacobian is isogenous to a power of one ordinary elliptic curve.

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