  We bound Selmer groups attached to Grossencharacters of CM elliptic curves by
the appropriate $L$-value. Our method is to use Kato's explicit reciprocity law
and the Main Conjecture as proved by Rubin. These results are then used
together with work of Han and fundamental results of Nekovar to bound the
torsion in the image of the $p$-adic cycle class map for self products of CM
elliptic curves. Our results are valid for Chow groups of arbitrary codimension
for almost all $p$.
