Constructing Elements in Shafarevich-Tate Groups of Modular Motives

Neil Dummigan, William A. Stein, Mark Watkins

April 14, 2003


We study Shafarevich-Tate groups of motives attached to modular forms on Gamma0(N) of weight bigger than 2. We deduce a criterion for the existence of nontrivial elements of these Shafarevich-Tate groups, and give 16 examples in which the Beilinson-Bloch conjecture implies the existence of such elements. We also use modular symbols and observations about Tamagawa numbers to compute nontrivial conjectural lower bounds on the orders of the Shafarevich-Tate groups of modular motives of low level and weight at most 12. Our methods build upon Mazur's idea of visibility, but in the context of motives instead of abelian varieties.
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NOTE: Jan Nekovar's review of our article, he makes three extremely relevant remarks about this paper.