i will begin by introducing the birch and swinnerton-dyer conjecture in the context of abelian varieties attached to modular forms, and discuss some of the main results about it. i will then introduce mazur's notion of visibility of shafarevich-tate groups and explain some of the basic facts and theorems. cremona, mazur, agashe, and myself carried out large computations about visibility for modular abelian varieties of level $n$ in $j_0(n)$. these computations addressed the following question: if $a$ is a modular abelian variety of level $n$, how much of the shafarevich-tate group of $a$ is modular of level $n$, i.e., visible in $j_0(n)$. the results of these computations suggest that often much of the shafarevich-tate group is not modular of level $n$. this suggests asking if every element is modular of level $n*m$, for some auxiliary integer $m$, and if so, what can one say about the set of such $m$? i will finish the talk with some new data and thoughts about this last question, which is still very much open.