Given the group algebra of a p-group over a field of characteristic p. Can the 2-periodic indecomposable modules $M$ ($M$ with $\Omega^{2}(M)=M$) be classified? I am not experienced much with modular representation theory, but my feeling is that apart from the tame case, there might not exist many such modules.
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This question is addressed by the paper Wild categories of periodic modules (Illinois J. Math. 32 (1988), no.3, 557-561) by Jon Carlson and Alfredo Jones. As suggested by the title, the answer to your question is that the $2$-periodic modules can usually not be classified. In particular, the authors consider the very simple case of $G = (\mathbb{Z}/p)^2$ over a field $K$ of characteristic $p\geq 7$. For such $G$, they construct a wild subcategory of $2$-periodic modules.
For larger $p$-groups $G$ of wild representation type, one can induce the $2$-periodic modules of a rank 2 elementary abelian subgroup $E$ up to $G$. Induction is exact and preserves $2$-periodicity. Technically this doesn't prove that the $2$-periodic $KG$-modules form a wild subcategory, since induction does not preserve indecomposability; but it does suggest as much, and at least shows that one should expect many $2$-periodic modules.
