  We prove that the Farrell-Jones assembly map for connective algebraic
K-theory is rationally injective, under mild homological finiteness conditions
on the group and assuming that a weak version of the Leopoldt-Schneider
conjecture holds for cyclotomic fields. This generalizes a result of
B\"okstedt, Hsiang, and Madsen, and leads to a concrete description of a large
direct summand of $K_n(\mathbb{Z}[G])\otimes_{\mathbb{Z}}\mathbb{Q}$ in terms
of group homology. In many cases the number theoretic conjectures are true, so
we obtain rational injectivity results about assembly maps, in particular for
Whitehead groups, under homological finiteness assumptions on the group only.
The proof uses the cyclotomic trace map to topological cyclic homology,
B\"okstedt-Hsiang-Madsen's functor C, and new general isomorphism and
injectivity results about the assembly maps for topological Hochschild homology
and C.
