  For a smooth projective geometrically uniruled threefold defined over a
perfect field we show that there exists a canonical abelian variety over the
field, namely the second algebraic representative, whose rational Tate modules
model canonically the third l-adic cohomology groups of the variety for all
primes l. In addition, there exists a rational correspondence inducing these
identifications. In the case of a geometrically rationally chain connected
variety, one obtains canonical identifications between the integral Tate
modules of the second algebraic representative and the third l-adic cohomology
groups of the variety, and if the variety is a geometrically stably rational
threefold, these identifications are induced by an integral correspondence. Our
overall strategy consists in studying -- for arbitrary smooth projective
varieties -- the image of the second ell-adic Bloch map restricted to the Tate
module of algebraically trivial cycle classes in terms of the "correspondence
(co)niveau filtration". This complements results with rational coefficients due
to Suwa. In the appendix, we review the construction of the Bloch map and its
basic properties.
