On the finiteness of Ш for motives associated to modular forms

Abstract

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.