Motivation

Mazur suggested that the Shafarevich-Tate group ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(K, E)$ of an abelian variety $A$ over a number field $K$ could be studied via a collection of finite subgroups (the visible subgroups) corresponding to different embeddings of the variety into larger abelian varieties $C$ over $K$ (see [Maz99] and [CM00]). The advantage of this approach is that the isomorphism classes of principal homogeneous spaces, for which one has à priori little geometric information, can be given a much more explicit description as $K$ -rational points on the quotient abelian variety $C / A$ (the reason why they are called visible elements).

Agashe, Cremona, Klenke and the second author built upon the ideas of Mazur and developed a systematic theory of visibility of Shafarevich-Tate groups of abelian varieties over number fields (see [Aga99b,AS02,AS05,CM00,Kle01,Ste00]). More precisely, Agashe and Stein provided sufficient conditions for the existence of visible sugroups of certain order in the Shafarevich-Tate group and applied their general theory to the case of newform subvarieties ${A_f}_{/\mathbb{Q}}$ of the Jacobian $J_0(N)_{/\mathbb{Q}}$ of the modular curve $X_0(N)_{/\mathbb{Q}}$ (here, $f$ is a newform of level $N$ and weight 2 which is an eigenform for the Hecke operators acting on the space $S_2(\Gamma_0(N))$ of cuspforms of level $N$ and weight 2). Unfortunately, there is no guarantee that a non-trivial element of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, A_f)$ is visible for the embedding $A_f \hookrightarrow J_0(N)$ .

In this paper we consider the case of modular abelian varieties over $\mathbb{Q}$ and make use of the algebraic and arithmetic properties of the corresponding newforms to provide sufficient conditions for the existence of visible elements of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, A_f)$ in modular Jacobians of level a multiple of the base level $N$ . More precisely, we consider morphism of the form $A_f \hookrightarrow J_0(N) \xrightarrow{\phi} J_0(MN)$ , where $\phi$ is a suitable linear combination of degeneracy maps which makes the kernel of the composition morphism almost trivial (i.e., trivial away from the 2-part). For specific examples, the sufficient conditions can be verified explicitely. We also provide a table of examples where certain elements of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, A_f)$ which are invisible in $J_0(N)$ become visible at a suitably chosen higher level. At the end, we state some general conjectures inspired by our results.