Visibility of Shafarevich-Tate group of abelian varieties over number field extensions

TL;DR

This paper investigates the behavior of visible elements in the Shafarevich-Tate group of an abelian subvariety over number field extensions, focusing on their existence and properties under Galois cohomology restrictions.

Contribution

It introduces new conditions for the production of visible elements of order p in quadratic and degree p extensions, expanding understanding of the Shafarevich-Tate group's visibility.

Findings

Conditions for visible elements of order p over quadratic extensions

Conditions for visible elements of order p over degree p extensions

Analysis of the restriction map in Galois cohomology

Abstract

Given an abelian variety J and an abelian subvariety A of J over a number field K, we study the visible elements of the Shafarevich-Tate group of A with respect to J over certain number field extension M of K. The notion of visible elements in Shafarevich-Tate group of an abelian variety was introduced by Mazur. In this article, we study the image of Visible elements of A with respect to J under the natural restriction map of the Galois cohomology of A over K to the Galois cohomology of A over M. In particular, for a fixed odd prime p, we investigate the conditions under which visible elements of order p can be produced over a quadratic extension or a degree p extension M of K.