On the finiteness of Sha for motives associated to modular forms

Doc.Math.J.DMV 2 (1997) 31-46 get the paper from Documenta Mathematica

Let f be a modular form of even weight on \Gamma_0(N) with associated motive M_f. Let K be a quadratic imaginary field satisfying certain standard conditions. We improve a result of Nekovár 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 M_f is not divisible by p, then the p-part of the Tate-Shafarevic group of 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.