I will present a technique for deforming coherent cohomology classes over modular curves. Applied to different sheaves one obtains in this way families of overconvergent modular forms and nearly overconvergent modular forms. I will provide applications to the construction of $p$-adic $L$-functions. This is based on several joint works with A. Iovita, V. Pilloni and G. Stevens.
In this course we will introduce and study $p$-adic modular forms and their higher dimensional generalizations, $p$-adic automorphic forms. We will start by introducing $p$-adic modular forms as $p$-adic limits of classical modular forms. These $p$-adic limits naturally vary in families, so called eigenvarieties. These are geometric objects over the $p$-adic numbers whose points parametrize $p$-adic modular forms. The problem how to decide whether a given $p$-adic modular form is in fact a classical form has been studies extensively over the last decades starting with Coleman’s criterion that small slope implies classical.
We will study such classicality criteria, ending with my recent joint work with C. Breuil and B. Schraen on classicality of $p$-adic automorphic forms on define unitary groups, which connects classicality criteria with geometric properties of eigenvarieties.
Patrick Allen, University of Illinois at Urbana-Champaign
Charlotte Chan, University of Michigan
Valentin Hernandez, Institut de Mathématiques de Jussieu/CNRS
Cong Xue, University of Cambridge
We will also have some questions and answers sessions for the courses. Further details will be announced later.