The p-adic Langlands program

Thursday, November 15, 2018 - 11:00am - 12:00pm
Keller 3-180
Matthew Emerton (University of Chicago)
I will describe some of the history of, progress in, and future prospects for the p-adic Langlands program. This is an aspect of the Langlands program that grew out of the successful proof of Langlands reciprocity in various important cases (in particular, the modularity of elliptic curves over Q) twenty or so years ago. It relates the deformation theory of Galois representations to the representation-theoretic aspects of the theory of automorphic forms, for example via the investigation of representations of p-adic groups on p-adic vector spaces. As I will explain, while there has been significant progress in the p-adic Langlands program, a large amount remains to be done --- indeed, even the basic conjectural framework of the program remains unsettled. In the talk I hope to indicate some possibly fruitful directions for future research.