Laurent Fargues (CNRS, France) and Peter Scholze (University of Bonn, Germany) made a preprint entitled Geometrization of the local Langlands correspondence available to the mathematical community on 26 February. This title announces what had been expected for several years from the collaboration between these two heavyweights of modern mathematics: a substantial advance in the so-called Langlands conjectures (or Langlands programme). Michael Rapoport’s presentation of Scholze’s work at the International Mathematics Congress in Rio de Janeiro – where he was awarded the less unexpected Fields Medal of the century– already predicted a result in this direction. Fargues himself, five years ago, traced the outlines of the proof in a brief 46-page exposition. The full article, in this first version, is 348 pages long.
Surely the preprint summary is self-explanatory:
“Following the idea of [Far16], we develop the foundations of the geometric Langlands program on the Fargues-Fontaine curve. In particular, we define a category of \(\ell\)-adic sheaves on the stack \(\text{Bun}_G\) of \(G\)-bundles on the Fargues-Fontaine curve, prove a geometric Satake equivalence over the Fargues-Fontaine curve, and study the stack of \(L\)-parameters. As applications, we prove finiteness results for the cohomology of local Shimura varieties and general moduli spaces of local shtukas, and define \(L\)-parameters associated with irreducible smooth representations of \(G(E)\), a map from the spectral Bernstein center to the Bernstein center, and the spectral action of the category of perfect complexes on the stack of \(L\)-parameters on the category of \(\ell\)-adic sheaves on \(\text{Bun}_G\).”
Well, maybe not so much…
The truth is that trying to present the conjectures of Robert Langlands (who is still, at 84, active in his chair at the Institute of Advanced Study, Princeton), even at an elementary level, is the task of at least one book. To tell the truth, my fellow blogger Juan Arias de Reyna may be able to do it in the space of a single post – I have seen him do harder tasks. In any case, the implications of these results for Number Theory and its relation with other areas, in particular with Group Theory in the case of the local conjectures referred to in Fargues and Scholze’s paper, would be extraordinary. To give an idea of its relevance, with Scholze (2018) there are now four Fields medalists who have won the prize for their work in this area (Drinfeld (1990), Lafforgue (2002), Châu (2010)). Likewise, Wiles’ proof of the Taniyama-Shimura-Weil conjecture, which contained Fermat’s Last Theorem as a corollary, can be reinterpreted as a particular case of one of Langlands’ conjectures (the reciprocity conjecture).
I leave here only a (probably somewhat literary) metaphor to illustrate the importance of Langlands’ exceptional vision, for which he received the 2018 Abel Prize. If Number Theory were Cartography, we might imagine that before Langlands we were, like the ancient explorers in the time of Magellan and Elcano, patiently and painstakingly describing coastlines. Langlands’ programme can be likened, in this context, to the idea of building a space satellite that would allow us to have a more accurate picture and at the same time a simpler way of obtaining the very thing that was taking so much work and effort to describe (in a questionable way, moreover). In this parallel history, Fargues and Scholze’s work could possibly be likened to the twelve seconds of the Wright brothers’ first flight. A superlative achievement, no doubt, and one that clearly represents a huge step forward. But, as I hope is clear from the parallelism, we still have a long way to go before we reach space.
References
Laurent Fargues, Peter Scholze: Geometrization of the local Langlands correspondence. https://arxiv.org/abs/2102.13459.
Laurent Fargues: Geometrization of the local Langlands correspondence: An overview. https://webusers.imj-prg.fr/~laurent.fargues/Geometrization_review.pdf.
Michael Rapoport: The Work of Peter Scholze. Proc. Int. Cong. of Math. Rio de Janeiro. 1: 71–86.
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