  We propose a refined version of the Beilinson-Bloch conjecture for the motive
associated with a modular form of even weight. This conjecture relates the
dimension of the image of the relevant p-adic Abel-Jacobi map to certain
combinations of Heegner cycles on Kuga-Sato varieties. We prove theorems in the
direction of the conjecture and, in doing so, obtain higher weight analogues of
results for elliptic curves due to Darmon.
