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

2003/05/04 by Irene I. Bouw, Bouw, Irene I., Claus Diem +3
Computer Science · Mathematics · Social Sciences · #11G05 (Primary) #11G20 #14H40 #14H52 (Secondary) #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #FOS: Mathematics #Historical and Political Studies #Number Theory (math.NT) #math.AG #math.NT #msc:11G05 #msc:11G20 #msc:14H40 #msc:14H52

paper · pdf · doi:10.48550/arxiv.math/0305064

15 pages, 0 figures, LaTeX

arxiv created 2003/05/04 · openalex publication_date 2003/05/04 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that under the assumption of Artin's Primitive Root Conjecture, for all primes p there exist ordinary elliptic curves over 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)^*/ (ell an odd prime) there exists a hyperelliptic curve over F_p whose Jacobian is isogenous to a power of one ordinary elliptic curve.

Related

• Ordinary elliptic curves of high rank over F_p(x) with constant j-invariant II
• Infinitely many elliptic curves over ℚ(i) with rank 2 and j-invariant 1728
• On Ranks of Jacobian Varieties in Prime Degree Extensions
• Isogenies of non-CM elliptic curves with rational j-invariants over number fields
• Elliptic curves of high rank over function fields