Finiteness of Ш over totally real fields

Kolyvagin's method for the proof of the finiteness of Ш is extended to abelian varieties with real multiplication, of $L$-rank 0, defined over totally real fields, if they are factors of the Jacobians of Shimura curves. The finiteness of Ш for such a variety is proved, starting from the conditions that a Heegner point on it is not a torsion point.