Together with David Hansen and Christian Johansson we have uploaded our paper “Overconvergent modular forms and perfectoid Shimura curves” to arXiv (http://arxiv.org/abs/1507.04875).
I mention it here as we use adic spaces and perfectoid spaces in a crucial way. We basically re-work the theory of overconvergent modular forms by using the pro-etale site and the prior work of Scholze on the modular curves at the infinite level. This allows us to be explicit – we construct an analogue of the complex coordinate “z” in the p-adic setting and use it to construct sheaves of overconvergent modular forms. We introduce the overconvergent Eichler-Shimura map after Andreatta-Iovita-Stevens and analyse its properties using our description of the sheaves of overconvergent modular forms. We collect some general results about adic spaces in the appendix.