Let $A$ be a finitely generated $\mathbb{Z}_pG$-module, where $G$ is a finite $p$-group and $\mathbb{Z}_p$ is the ring of $p$-adic integers; assume moreover that $A$ is cohomologically trivial, that is, the Tate cohomology groups $H^n(H,A)$ vanish for all $H\leq G$ and all integers $n$. I have (essentially) three closely related questions.
Firstly, denote by $T(A)$ the torsion submodule of $A$, that is, $T(A)$ is formed by all the $a\in A$ such that $p^n a=0$ for some integer $n\geq 0$; observe that $T(A)$ is $\mathbb{Z}_pG$-submodule of $A$.
Does there exist such a $A$ so that $T(A)$ is not a direct factor of $A$?
Observe that if not, then we would have $A=T(A)\oplus F$ for some free $Z_pG$-module $F$; the latter follows from the fact that the torsion-free $Z_pG$-modules are projective (a result that goes back at least to Nakayama; see e.g., Serre's "Corps Locaux") and that $Z_pG$ is a local ring. It is readily seen that the above question amounts to asking if there exists such a module $A$ with T(A) not cohomologically trivial.
My second question is about the indecomposable $A$'s. Is it true that every such $A$, cohomologically trivial and indecomposable, has rank $1$ over $Z_pG$ (i.e., $A$ is a quotient of $Z_pG$? I think "no" with no strong evidences.
My thrid question is about counting such $A$'s. More precisely, fix a free $Z_pG$-module $L$ of rank $r$, say, and let $a_{r,n}$ denote the number of free submodules of $L$ of index $n$ (requiring that such a submodule, say $M$, is free amounts to saying that $L/M$ is a finite cohomologically trivial module). We can hence define a Dirichlet series $$\zeta_r(s):=\sum_{n\geq 0}a_{r,p^n}p^{-ns}.$$ I cannot detect if such a series is already studied in the literature (though closely related cases are well studied). How $\zeta_r(s)$ depends on $r$? Note that it can be expressed in terms of $\zeta_1(s)$ in an easy manner if the second question has a positive answer. I expect that $\zeta_r(s)$ is a rational function in $p$ and $p^{-s}$. Did the asymptotic behavior of the sequence $a_{n,r}$, for $r$ fixed, already studied? (Consider equally, for the latter question, counting all submodules of $L$ of finite index).
Thanks in advance!
