Visualizing elements of order four in the Shafarevich-Tate group of an elliptic curve

TL;DR

This paper demonstrates that elements of order four in the Shafarevich-Tate group of an elliptic curve over a number field are visible in infinitely many abelian surfaces, revealing new insights into their structure.

Contribution

It proves the infinitude of abelian surfaces in which a given order four element of the Shafarevich-Tate group is visible, advancing understanding of the group's structure.

Findings

Elements of order four are visible in infinitely many abelian surfaces.

Visibility of these elements is linked to embeddings of elliptic curves.

The result applies to elliptic curves over arbitrary number fields.

Abstract

Let E be an elliptic curve defined over a number field K. Let h be an element of order 4 in the Shafarevich-Tate group of E. We prove that h is visible in infinitely many abelian surfaces up to isomorphism. This is to say that there are infinitely many abelian surfaces J such that E\hookrightarrow J and h lies in the kernel of the natural map H^1(K,E)\rightarrow H^1(K,J).