On the Finiteness of $\Sha$ for Motives Associated to Modular Forms

Let $f$ be a modular form of even weight on $\Gamma_0(N)$ with associated motive $\mathcal{M}_f$. Let $K$ be a quadratic imaginary field satisfying certain standard conditions. We improve a result of \Nekovar{} and prove that if a rational prime $p$ is outside a finite set of primes depending only on the form $f$, and if the image of the Heegner cycle associated with $K$ in the $p$-adic intermediate Jacobian of $\mathcal{M}_f$ is not divisible by $p$, then the $p$-part of the Tate-\shafarevic{} group of $\mathcal{M}_f$ over $K$ is trivial. An important ingredient of this work is an analysis of the behavior of ``Kolyvagin test classes'' at primes dividing the level $N$. In addition, certain complications, due to the possibility of $f$ having a Galois conjugate self-twist, have to be dealt with.

1991 Mathematics Subject Classification: 11G18, 11F66, 11R34, 14C15.

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