Visibility of the Shafarevich--Tate Group at Higher Level

We study visibility of Shafarevich--Tate groups of modular abelian varieties in Jacobians of modular curves of higher level. We prove a theorem about the existence of visible elements at a specific higher level under certain hypothesis which can be verified explicitly. We also provide a table of examples of visible subgroups at higher level and state a conjecture inspired by our data.

2000 Mathematics Subject Classification: 11G05, 11G10, 11G18, 11Y40

Keywords and Phrases: Elliptic Curves, Abelian Varieties, Mordell-Weil Groups, Shafarevich-Tate Groups, Visibility

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