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

Abstract

Abstract: We study the derived Hecke action at $p$ on the ordinary $p$-adic cohomology of arithmetic subgroups of reductive groups ${\rm G}(\Bbb{Q})$. This is the analog at $\ell=p$ of derived Hecke actions studied by Venkatesh in the tame case, and is the derived analog of Hida's theory for ordinary Hecke algebras. 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.