  We prove a ``positive'' Jantzen sum formula for the Specht modules of the
cyclotomic Hecke algebras of type~$A$. That is, in the Grothendieck group, we
show that the sum of the pieces of the Jantzen filtration is equal to an
explicit non-negative linear combination of modules $E^\nu_{f,e}$, which are
modular reductions of simple modules for closely connected Hecke algebras in
characteristic zero. The coefficient of $E^\nu_{f,e}$ in the sum formula is
determined by the graded decomposition numbers in characteristic zero, which
are known, and the characteristic of the field. As a consequence we see that
the decomposition numbers of a cyclotomic Hecke algebra at an $e$th root of
unity in characteristic $p$ depend on the decomposition numbers of related
cyclotomic Hecke algebras at $ep^r$th roots of unity in characteristic zero,
for $r\ge0$.
