Kolyvagin Derivatives of Modular Points on Elliptic Curves

Abstract

Let $E/\mathbb{Q}$ and $A/\mathbb{Q}$ be elliptic curves. We can construct modular points derived from $A$ via the modular parametrisation of $E$. With certain assumptions we can show that these points are of infinite order and are not divisible by a prime $p$. In particular, using Kolyvagin's construction of derivative classes, we can find elements in certain Shafarevich-Tate groups of order $p^n$.

Final Version. To appear in Journal of Number Theory. 23 pages

References in corpus (2)

• Similarity classes of 3x3 matrices over a local principal ideal ring
• Asymptotics of conductors of elliptic curves over Q