We study Shafarevich-Tate groups of motives attached to modular forms on Γ 0 (N ) of weight > 2. We deduce a criterion for the existence of nontrivial elements of these Shafarevich-Tate groups, andgive 16 examples in which a strong form of the Beilinson-Bloch conjecture would imply 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 ≤ 12. Our methods build upon the idea of visibility due to Cremona and Mazur, but in the context of motives rather than abelian varieties.

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We prove an explicit inner product formula for vector-valued Yoshida lifts by an explicit calculation of local zeta integrals in the Rallis inner product formula for O(4) and Sp(4). As a consequence , we obtain the non-vanishing of Yoshida lifts.

Abstract. We present a general conjecture on congruences between Hecke eigenvalues of parabolically induced and cuspidal automorphic representations of split reductive groups, modulo divisors of critical values of certain L-functions. We examine the consequences in several special cases, and use the Bloch-Kato conjecture to further motivate a belief in the congruences.

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