Observe $\mathbb{Z}\_q^n = \mathbb{Z}\_q \times \cdots \times\mathbb{Z}\_q$ as a module over $\mathbb{Z}\_q\equiv\mathbb{Z}/q\mathbb{Z}$, for general $q$.
I am interested in the following questions:


How many submodules of size $q^k$, $k\leq n$, does it have? How many of them are free? Can something be said about the ratio of these two numbers when $n\to\infty$ and $k=\lambda n$, $\lambda\in(0,1)$?


Can someone give me a reference where such or similar problems have been studied? Or else provide pointers how to tackle them?



The number of submodules of $\mathbb{Z}_q^n$
Aren't you just asking for the number of subgroups of $\mathbb{Z}\_q^n$
of size $q^k$?
There isn't going to be a simple answer. The classification and
enumeration of subgroups of finite abelian groups goes back to Garrett
Birkhoff. See
<http://plms.oxfordjournals.org/content/s2-38/1/385.citation>. Some more
recent work is mentioned at
<http://www.northeastern.edu/iloseu/bhmn/Ringel04.html>. One can give an
answer in terms of symmetric functions. If $G$ is a finite abelian
group of order $p\_1^{a\_1}p\_2^{a\_2}\cdots$ (where the $p\_i$'s are
distinct primes) then the face lattice $L(G)$ is the product of the
lattices for each Sylow subgroup. This reduces the problem to abelian
$p$-groups. Let $P\_\lambda(x;t)$ be a Hall-Littlewood symmetric
function. Let
 $$ P\_\mu(x;t)P\_\nu(x;t) =\sum\_\lambda f^\lambda\_{\mu\nu}(t)
 P\_\lambda(x;t). $$
Let $n(\lambda) =\sum (i-1)\lambda\_i$. Then
$p^{n(\lambda)}f^\lambda\_{\mu\nu}(1/p)$ is the number of subgroups $H$
of
type $\mu$ of an abelian $p$-group of type $\lambda$ such that $G/H$
has type $\nu$ (or something close to this). In particular, one can
deduce that if $h\_i$ is a complete symmetric function and
 $$ h\_mh\_n = \sum\_\lambda c\_{m,n}^\lambda(t)P\_\lambda(x;t), $$
then $p^{n(\lambda)}c\_{m,n}^\lambda(1/p)$ is the number of subgroups of
order $p^m$ in an abelian $p$-group of type $\lambda\vdash m+n$. All
this is explained in Chapter II of Macdonald, *Symmetric functions and
Hall Polynomials*.


