Let $p$ be prime. Let $\mathbb{Z}_p$ be the $p$-adic integers.
I'm intersted in $\mathbb{Z}_p[\text{PGL}_2(\mathbb{F}_{p^n})]$-modules of low rank over $\mathbb{Z}_p$ (rank $2$ is already very interesting).
Is there a characterization of such modules? Where can I find examples? What are good books or other resources to learn about them?
I'm familiar with complex representations of finite groups, with the $p$-adic numbers, and somewhat familiar with group cohomology (if that's at all relevant).