The categorical form of Fargues' conjecture

The categorical form of Fargues' conjecture

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Peter Scholze, Max Planck Institute, University of Bonn

Please note the date and time change

 

Zoom link: 

https://princeton.zoom.us/j/99892230441

Password:

(The colloquium password will be distributed to Princeton University and IAS members. We ask that you do not share this password. If you would like to be included in the colloquium and are not a member of either institution, please email the organizer Casey Kelleher (caseyk@princeton.edu) with an email requesting to participate which introduces yourself, your current affiliation and stage in your career.)

 

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.