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

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

2012-01-11

arxiv.org/abs/1201.2310v2

Mathematics

Number Theory

18 pages

Scientific paper

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.

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