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Published online: 2017-02-07
Arbeitsgemeinschaft: The Geometric Langlands ConjectureLaurent Fargues, Dennis Gaitsgory, Peter Scholze and Kari Vilonen (1) Université de Paris VI, France
(2) Harvard University, Cambridge, USA
(3) Universität Bonn, Germany
(4) Northwestern University, Evanston, USA
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.
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Fargues Laurent, Gaitsgory Dennis, Scholze Peter, Vilonen Kari: Arbeitsgemeinschaft: The Geometric Langlands Conjecture. Oberwolfach Rep. 13 (2016), 1027-1098. doi: 10.4171/OWR/2016/20