MathMeetings.net

The study of symmetries, starting with Klein and Hilbert in the late 19th century is of central importance to number theory. One is interested in behaviour of points in a space under its group of "generalized symmetries"; this idea is central to theories such as ergodic theory and dynamical systems that originated with problems in physics, but also to number theory where the symmetries are very much related to Diophantine equations. Mostly this study is done on manifolds in the usual Euclidean space that takes its structure from the usual space in which we live; however, number theory provides us with new exotic spaces and new exotic geometries -- p-adic numbers, p-adic analysis, p-adic manifolds -- that in a certain perspective serve as tools to study local behaviour of complicated global symmetries originating in Euclidean spaces. Our interest is precisely in developing such a theory in the context of very particular spaces and symmetries; to be precise, p-adic Shimura varieties and p-adic Hecke operators.