Derived Hecke action at $p$ and the ordinary $p$-adic cohomology of arithmetic manifolds

Abstract

We study the derived Hecke action at $p$ on the ordinary $p$-adic cohomology of arithmetic subgroups of semisimple groups $\mathrm{G}(\mathbb{Q})$, i.e., we study the derived version of Hida's theory for ordinary Hecke algebras. This is the analog at $\ell =p$ of derived Hecke actions studied by Venkatesh in the tame case. We show that properties of the derived Hecke action at $p$ are related to deep conjectures in Galois cohomology which are higher analogs of the classical Leopoldt conjecture.