Scholze discovered that modular curves with infinite level at p are nice p-adic analytic spaces and admit a period map (the Hodge-Tate period map) towards the flag variety of GL2, which can be thought as a p-adic analogue of the anti-holomorphic Borel embedding in the complex case. This makes it possible to define differential operators in the "anti-holomorphic direction".

In this talk, I will construct some differential operators on modular curves with infinite level at p and discuss its p-adic Hodge-theoretic meaning. As an application, we give a new proof of a classicality result of Emerton which says that regular de Rham Galois representations appearing in the completed cohomology of modular curves come from modular forms.