GeoLocLang

Project Information

GeoLocLang

Grant agreement ID: 742608

DOI

10.3030/742608

Project closed

EC signature date 12 May 2017

Start date 1 October 2017

End date 31 March 2023

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 301 863,00

EU contribution

€ 1 301 863,00

Coordinated by

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
France

Periodic Reporting for period 4 - GeoLocLang (GeoLocLang)

Reporting period: 2022-04-01 to 2023-03-31

Summary of the context and overall objectives of the project

We are taking a new point of view on the local Langlands program. Usually, the local Langlands program links two different "classical" types of objects coming from representation theory, a well-known and classical subject of mathematics.
From our new point of view those representations are replaced by new objects: "sheaves on a moduli space of bundles on the Fargues-Fontaine curve". This is what we call a geometrization process: replace a classical algebra object showing up in representation theory by a sheaf on a geometric object.

This new type of geometry involves two new types of objects:
- perfectoid spaces and diamonds introduced by Scholze
- the Fargues Fontaine curve that has completely changed the domain of p-adic Hodge theory
Combined together we can define and study a moduli space of bundles on the Fargues-Fontaine curve, and study "sheaves" on it. Those sheaves on this new geometric object allows us to prove completely new results on the local Langlands program.
The language of algebra from representation theory is replaced by the one of geometry. Geometric intuition allows us to guess and prove new results in this context.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

We have been able to completely understand the points of the moduli space of bundles on the Fargues-Fontaine curve. This is the article "G-torseurs en théorie de Hodge p-adique" (to appear at Compositio Math). This is gives us a complete description of the points of our moduli space of bundles on the curve.
This is a first step in the understanding of the geometry of this moduli space.

We have been able to prove the geometrization conjecture for GL1. This is the article "Simple connexité des fibres d'une application d'Abel-Jacobi et corps de classe local" (Annales de l'ENS). In particular we give a purely geometric proof of local class field theory using our new moduli space.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

We did more than expected in the project. Originally we only intended to construct the semi-simple local Langlands correspondence. This has been done but we have been further by constructing what we call the spectral action, a monoidal action of
the category of perfect complexes on the moduli of Langlands parameters. This implies the construction of the semi-simple Langlands correspondence as a particular case. Right now, this result, at the end of the project, is the best result
in the domain.

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