Some Examples

This section contains some examples of visible and invisible elements of Shafarevich-Tate groups. Section 4.1 uses Theorem 3.1 to produce nontrivial visible elements of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A)$, where $A$ is a $20$-dimensional modular abelian variety, thus giving evidence for the BSD conjecture. In Section 4.2 we show that an invisible Shafarevich-Tate group from [CM00] becomes visible at a higher level.

In [AS02], we describe the notation used below (which is standard) and the algorithms that we used to carry out the computations described below. We also report on a large number of similar computations, which were performed using the second author's modular symbols package, which is part of MAGMA (see [BCP97]).