  As a continuation of \lianyaufour, we study modular properties of the
periods, the mirror maps and Yukawa couplings for multi-moduli Calabi-Yau
varieties. In Part A of this paper, motivated by the recent work of
Kachru-Vafa, we degenerate a three-moduli family of Calabi-Yau toric varieties
along a codimension one subfamily which can be described by the vanishing of
certain Mori coordinate, corresponding to going to the ``large volume limit''
in a certain direction. Then we see that the deformation space of the subfamily
is the same as a certain family of K3 toric surfaces. This family can in turn
be studied by further degeneration along a subfamily which in the end is
described by a family of elliptic curves. The periods of the K3 family (and
hence the original Calabi-Yau family) can be described by the squares of the
periods of the elliptic curves. The consequences include: (1) proofs of various
conjectural formulas of physicists \vk\lkm~ involving mirror maps and modular
functions; (2) new identities involving multi-variable hypergeometric series
and modular functions -- generalizing \lianyaufour. In Part B, we study for
two-moduli families the perturbation series of the mirror map and the type A
Yukawa couplings near certain large volume limits. Our main tool is a new class
of polynomial PDEs associated with Fuchsian PDE systems. We derive the first
few terms in the perturbation series. For the case of degree 12 hypersurfaces
in $\P^4[6,2,2,1,1]$, in one limit the series of the couplings are expressed in
terms of the $j$ function. In another limit, they are expressed in terms of
rational functions. The latter give explicit formulas for infinite sequences of
``instanton numbers'' $n_d$.
