Ian Kiming: Modular forms modulo prime powers

Abstract: I will give a survey of results obtained in recent years in the theory of modular forms modulo prime powers: results obtained together with Imin Chen, Nadim Rustom, and Gabor Wiese. I will talk about: the various notions of eigenforms modulo higher prime powers, attaching Galois representations to eigenforms, finiteness questions in the theory and their connection to a question by K. Buzzard, the problem of lifting eigenforms to characteristic 0, and, if time permits, a little on level optimization.

As I go along in the survey, I will address the important question of motivations for this whole area.