Peter Scholze - 2/6 On the local Langlands conjectures for reductive groups over p-adic fields

Hadamard Lectures 2017

Abstract: Consider a reductive group G over a p-adic field F. The local Langlands conjecture relates the irreducible smooth representations of G(F) with the set of (local) L-parameters, which are maps from the Weil group of F to the L-group of G; refinements of the conjecture relate the fibres of this map with the automorphism group of the L-parameter. Based on ideas from V. Lafforgue's work in the global function field case, I outlined a strategy for attaching (semisimple) L-parameters to irreducible smooth representations of G(F) in my 2014 Berkeley course. At the same time and place, L. Fargues formulated a conjecture relating the local Langlands conjecture with a geometric Langlands conjecture on the Fargues-Fontaine curve. The goal of this course will be to discuss some of the developments since then. On the foundational side, this concerns basics on the etale cohomology of diamonds including smooth and proper base change and Poincare duality, leading up to a good notion of "constructible" sheaves on the stack of G-bundles on the Fargues-Fontaine curve. On the applied side, this concerns the construction of (semisimple) L-parameters, the conjecture of Harris (as modified by Viehmann) on the cohomology of non-basic Rapoport-Zink spaces, and the conjecture of Kottwitz on the cohomology of basic Rapoport-Zink spaces.

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