Kolyvagin Derivatives of Modular Points on Elliptic Curves

TL;DR

This paper explores the construction of modular points on elliptic curves and uses Kolyvagin's derivatives to identify elements in Shafarevich-Tate groups, revealing new insights into their structure and properties.

Contribution

It introduces a novel application of Kolyvagin derivatives to modular points on elliptic curves, establishing their infinite order and non-divisibility by primes.

Findings

Modular points derived from $A$ are of infinite order.

Identifies elements in Shafarevich-Tate groups of order $p^n$.

Provides conditions under which these points are non-divisible by $p$.

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 $p^{n}$.