  The main result of this paper is the proof for elliptic modular threefolds of
conjectures on the existence and structure of a filtration on the Chow groups
of smooth projective varieties. In the form we prove them these conjectures
were formulated by the second-named author, but as Jannsen has shown, these
conjectures are equivalent to a conjecture of Beilinson. These conjectures have
previously been proved for surfaces in general, but for elliptic modular
surfaces we obtain more precise results which are then used in proving the
conjectures for elliptic modular threefolds.
