Oberwolfach Reports

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Volume 13, Issue 2, 2016, pp. 1027–1098
DOI: 10.4171/OWR/2016/20

Published online: 2017-02-07

Arbeitsgemeinschaft: The Geometric Langlands Conjecture

Laurent Fargues[1], Dennis Gaitsgory[2], Peter Scholze[3] and Kari Vilonen[4]

(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