Constructing non-trivial elements of the Shafarevich-Tate group of an Abelian Variety over a Number Field

TL;DR

This paper extends methods to construct non-trivial elements of the Shafarevich-Tate group for abelian varieties over number fields, offering new evidence for the Birch and Swinnerton-Dyer conjecture.

Contribution

It generalizes Cremona and Mazur's construction techniques from elliptic curves to abelian varieties, providing precise conditions and theoretical support for BSD.

Findings

Conditions for constructing non-trivial Shafarevich-Tate elements

Extension of BSD conjecture evidence

Theoretical framework for abelian varieties

Abstract

The second part of the Birch and Swinnerton-Dyer (BSD) conjecture gives a conjectural formula for the order of the Shafarevich-Tate group of an elliptic curve in terms of other computable invariants of the curve. Cremona and Mazur initiated a theory that can often be used to verify the BSD conjecture by constructing non-trivial elements of the Shafarevich-Tate group of an elliptic curve by means of the Mordell-Weil group of an ambient curve. In this paper, we generalize Cremona and Mazur's work and give precise conditions under which such a construction can be made for the Shafarevich-Tate group of an abelian variety over a number field. We then give an extension of our general result that provides new theoretical evidence for the BSD conjecture.