Arbeitsgemeinschaft: The Geometric Langlands Conjecture

Abstract

The Langlands program is a vast, loosely connected, collection of theorems and conjectures. At quite different ends, there is the geometric Langlands program, which deals with perverse sheaves on the stack of $G$-bundles on a smooth projective curve, and the local Langlands program over $p$-adic fields, which deals with the representation theory of $p$-adic groups. Recently, inspired by applications to p-adic Hodge theory, Fargues and Fontaine have associated with any $p$-adic field an object that behaves like a smooth projective curve. Fargues then suggested that one can interpret the geometric Langlands conjecture on this curve, to give a new approach towards the local Langlands program over $p$-adic fields.