  For certain product varieties, Murre's conjecture on Chow groups is
investigated. In particular, it is proved that Murre's conjecture (B) is true
for two kinds of four-folds. Precisely, if $C$ is a curve and $X$ is an
elliptic modular threefold over $k$ (an algebraically closed field of
characteristic 0) or an abelian variety of dimension 3, then Murre's conjecture
(B) is true for the fourfold $X\times C.$
