Organization of the paper

Section 2 discusses the basic definitions and notation for modular abelian varieties, modular forms, Hecke algebras, the Shimura construction and modular degrees. Section 3 is a brief introduction to visibility theory for Shafarevich-Tate groups. In Section 4 we state and prove an equivariant version of a theorem of Agashe-Stein (see [AS05, Thm 3.1]) which guarantees existence of visible elements. The theorem is more general because it makes use of the action of the Hecke algebra on the modular Jacobian.

In Section 5 we introduce the notion of strong visibility which is relevant for visualizing cohomology classes in Jacobians of modular curves whose level is a multiple of the level of the original abelian variety. Theorem 5.1.3 guarantees existence of strongly visible elements of the Shafarevich-Tate group under some hypotheses on the component groups, a congruence condition between modular forms, and irreducibility of the Galois representation. In Section 5.4 we prove a variant of the same theorem (Theorem 5.4.2) with more stringent hypotheses that are easier to verify in specific cases.

Section 6 discusses in detail two computational examples for which strongly visible elements of certain order exist which provides evidence for the Birch and Swinnerton-Dyer conjecture. We state a general conjecture (Conjecture 7.1.1) in Section 7 according to which every element of the Shafarevich-Tate group of a modular abelian variety becomes visible at higher level. We provide evidence for the the conjecture in Section 7.2 and tables of computational data in Section 7.4.

Acknowledgement: The authors would like to thank David Helm, Ben Howard, Barry Mazur, Bjorn Poonen and Ken Ribet for discussions and comments on the paper.