Speaker: Payman Kassaei
Title: P-adic Variation of Modular Forms
Affiliation: King's College, London.
Date: Friday, September 7, 2007.
Place and Time: Room 101, Love Building, 3:35-4:30 pm.
Refreshments: Room 204, Love Building, 3:00 pm.

Abstract. Modular forms are certain holomorphic functions on the complex upper half-plane which are infinitely symmetric. Number theorists are interested in the systems of eigenvalues obtained from the action of Hecke operators on modular forms. These (a priori complex) numbers are algebraic integers which are often of arithmetic significance. One is interested in studying congruences modulo (powers of) a prime p between these eigenvalues. This is most efficiently done through a systematic study of p-adic analytic variation of modular forms.
In my talk I will survey the progress in this area from the conception of the notion of a p-adic analytic family of modular forms to Coleman-Mazur' construction of the eigencurve which is in some sense the universal such family. If time allows I will venture into the connections between the theme of p-adic variation and the Langlands program.