Samanda Zhang: Action of derived local-at-p Hecke algebra on cohomology of modular curve

Abstract: In recent years, there has been growing interests in using derived enhancements in understanding structures/phenomena in the Langlands programme. Such leads to interest in studying derived Hecke algebra and its action (e.g. Venkatesh, Schneider, etc.) Contrary to Venkatesh, which is a more global approach, we are interested in local information. Specifically, using Emerton's completed cohomology, we will give a canonical action of Schneider's differential-graded (DG) Hecke algebra on cohomology of modular curve. As this DG algebra is much larger than the classical Hecke algebra, meaning there is additional information in its action, we will explain some general methods in analyzing this DG-action and how one can use such additional DG structure. Surprisingly, this purely local derived structure gives us global Galois-theoretic information.