Ordinary elliptic curves of high rank over with constant j-invariant II

✍ Scribed by Claus Diem; Jasper Scholten

• Publisher: Elsevier Science
• Year: 2007
• Tongue: English
• Weight: 154 KB
• Volume: 124
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2006.08.005

✦ Synopsis

We show that for all odd primes p, there exist ordinary elliptic curves over 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 , there exists a hyperelliptic curve over F p of genus ( -1)/2 whose Jacobian is isogenous to the power of one ordinary elliptic curve.