**Date & Time:** Monday, April 29, 2013, 15:30-16:30.

**Venue:** Ramanujan Hall

**Title:** Differential Modular forms on Shimura curves over totally real fields

**Speaker:** Debargha Banerjee, MPIM Bonn

**Abstract:** In this talk, we will introduce the theory of differential modular forms for compact Shimura curves over totally real fields. These are the modular forms obtained by applying the arithmetic jet space functor (adjoint to the Witt vector functor) to the ring of modular forms (global sections of certain line bundles). We will show that these differential modular forms help us to detect the ordinary quaternionic abelian schemes and schemes with Frobenius lifts (local analogue of CM abelian schemes). These are also useful to understand the categorical quotients of Shimura curves modulo Hecke correspondences/isogeny. If time permits, we also construct the Serre-Tate expansions of such differential modular forms as a possible alternative to the Fourier expansion maps.