Visibility of ideal classes

TL;DR

This paper explores the concept of visibility of ideal classes in number fields, developing new methods to analyze capitulation and providing data that suggests a higher likelihood of visibility of nontrivial Shafarevich-Tate elements in elliptic curves with positive rank.

Contribution

It introduces a novel approach to study capitulation of ideal classes and applies it to gather data, offering insights into the visibility of Shafarevich-Tate elements for elliptic curves.

Findings

Visibility of nontrivial Shafarevich-Tate elements may be more common in elliptic curves of positive rank.

Developed a new method to study capitulation of ideal classes.

Numerical data supports the connection between ideal class capitulation and elliptic curve properties.

Abstract

Cremona, Mazur, and others have studied what they call visibility of elements of Shafarevich-Tate groups of elliptic curves. The analogue for an abelian number field $K$ is capitulation of ideal classes of $K$ in the minimal cyclotomic field containing $K$. We develop a new method to study capitulation and use it and classical methods to compute data with the hope of gaining insight into the elliptic curve case. For example, the numerical data for number fields suggests that visibility of nontrivial Shafarevich-Tate elements might be much more common for elliptic curves of positive rank than for curves of rank 0.