The categorical form of Fargues' conjecture

Peter Scholze, Max Planck Institute, University of Bonn

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For a reductive group G over the p-adic numbers Q_p, the local Langlands conjecture relates the category of representations of G(Q_p) with the space of L-parameters, i.e. maps from the Weil group of Q_p to the Langlands dual (L-)group of G. In a different direction, motivated by p-adic Hodge theory, Fargues and Fontaine defined a curve that behaves like a smooth projective curve over an algebraically closed field, yet also behaves like Q_p itself; for example, Poincare duality on the curve is local Tate duality. In 2014, Fargues conjectured that geometric Langlands works for the Fargues-Fontaine curve, and gives an extension of the local Langlands conjecture. This makes it possible to use the techniques of geometric representation theory to approach the local Langlands correspondence.

As a partial progress report on our joint work with Fargues, the goal of this talk will be to formulate a categorical form of his conjecture, relating the category of l-adic sheaves on the stack Bun_G of G-bundles on the Fargues-Fontaine curve -- which includes the category of representations of G(Q_p) -- with the category of coherent sheaves on the stack of L-parameters. In classical terms, such a conjecture includes the expected description of (elliptic) L-packets in terms of representations of the centralizer group of the L-parameter. On the principal block, it is related to a recent conjecture of Hellmann.

Defining the relevant category of l-adic sheaves on Bun_G requires a development of the l-adic formalism for perfectoid spaces and diamonds, along with some further innovations related to the theory of condensed mathematics and solid abelian groups recently developed with Clausen.