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

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

We show that for all odd primes p, there exist ordinary elliptic curves over Fp(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 l, there ...

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

We show that under the assumption of Artin’s Primitive Root Conjecture, for all primes p there exist ordinary elliptic curves over Fp(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/lZ)∗/〈−1〉 (l an odd prime) there exists a hyperelliptic curve over Fp whose Jacobian is isogenous to a pow...

Ranks of Elliptic Curves over Q(t) with Constant J-invariant

Elliptic curves of high rank over function fields

By the Mordell-Weil theorem the group of Q(z)-rational points of an elliptic curve is finitely generated. It is not known whether the rank of this group can get arbitrary large as the curve varies. Mestre and Nagao have constructed examples of elliptic curves E with rank at least 13. In this paper a method is explained for finding a 14th independent point on E, which is defined over k(z), with ...

On the rank of certain parametrized elliptic curves

In this paper the family of elliptic curves over Q given by the equation Ep :Y2 = (X - p)3 + X3 + (X + p)3 where p is a prime number, is studied. Itis shown that the maximal rank of the elliptic curves is at most 3 and someconditions under which we have rank(Ep(Q)) = 0 or rank(Ep(Q)) = 1 orrank(Ep(Q))≥2 are given.