Overconvergent modular forms and perfectoid Shimura curves

We show a new approach to overconvergent modular forms and overconvergent Eichler-Shimura map which uses crucially the recent work of Scholze on perfectoid Shimura varieties. This gives a non-archimedean analogue of "cz+d" approach to classical modular forms. This is a joint work with David Hansen (Columbia) and Christian Johansson (IAS).