Prof. Shanwen Wang，Renmin University of China

In this talk, we will explain a refinement of Emerton's factorization of completed cohomology of modular curves, which is determined by the factorization at the classical points. The main ingredient is the q-expansion of the elements of the completed cohomology. This is a part of the joint work with Pierre Colmez on the factorization of Beilinson-Kato system.