Indecomposable direct summands of cohomologies of curves

Abstract: Groups with a non-cyclic Sylow $p$-subgroup have too many representations over a field of characteristic~$p$ to describe them fully. A~natural question arises, whether the world of representations coming from algebraic varieties with a group action is as vast as the realm of all modular representations. In this article, we explore the possible ``building blocks'' (the indecomposable direct summands) of cohomologies of smooth projective curves with a group action. We show that usually there are infinitely many such possible summands. To prove this, we study a family of $\mathbb Z/p \times \mathbb Z/p$-covers and describe the cohomologies of the members of this family completely.