  The paper studies the modular representation theory of the cyclotomic Hecke
algebras of type $G(r,p,n)$ with $(\eps,q)$-separated parameters. We show that
the decomposition numbers of these algebras are completely determined by the
decomposition matrices of related cyclotomic Hecke algebras of type $G(s,1,m)$,
where $1\le s\le r$ and $1\le m\le n$. Furthermore, the proof gives an explicit
algorithm for computing these decomposition numbers. Consequently, in
principle, the decomposition matrices of these algebras are now known in
characteristic zero. In proving these results, we develop a Specht module
theory for these algebras, explicitly construct their simple modules and
introduce and study analogues of the cyclotomic Schur algebras of type
$G(r,p,n)$ when the parameters are $(\eps,q)$-separated. The main results of
the paper rest upon two Morita equivalences: the first reduces the calculation
of all decomposition numbers to the case of the \textit{$l$-splittable
decomposition numbers} and the second Morita equivalence allows us to compute
these decomposition numbers using an analogue of the cyclotomic Schur algebras
for the Hecke algebras of type $G(r,p,n)$.
