TITLE:p-adic variations of automorphic sheaves

ABSTRACT: Elliptic modular forms are have played an increasingly important rolein modern Number Theory.  Starting with the  works of J.P. Serre andN. Katz more than 30 years ago, it was discovered that, given a prime number p,such modular forms have also a p-adic nature and, especially, live in p-adicfamilies. This phenomenon is the  counterpart of the theory of p-adicdeformations of Galois representations and has become a basic tool for numbertheorists. I will present joint work with A. Iovita and V. Pilloni providing ageometric explanation of this, purely p-adic, phenomenon.