The point of this talk is to give three examples of derived structures influencing representations that have connections with number theory. These structures arise from the differential graded algebra of group cochains valued in the endomorphism ring of a representation.

Two examples have to do with representations of a Galois group. One of these realizes a number theoretic criterion for the modulo p multiplicity one condition for Jacobians of modular curves at an Eisenstein maximal ideal of a Hecke algebra; this is joint work with Preston Wake. Another furnishes a realization as a derived Galois deformation ring of an exterior algebra considered in works of Galatius--Venkatesh, Hansen--Thorne, and Venkatesh. The third example features smooth modulo p representations of a p-adic Lie group, answering some questions of Sorensen about the relationship between its Iwasawa algebra and the associated derived Hecke algebra.