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Periodic Reporting for period 3 - GeoLocLang (GeoLocLang)

Reporting period: 2020-10-01 to 2022-03-31

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.
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.
We hope to construct the local Langlands correspondence in the direction [representations of a p-adic group] --> [local Langlands parameters] in an ongoing joint work with Peter Scholze. For this we are studying thoroughly the geometry of the stack of bundles on the Fargues-Fontaine curve.
At the end of our article we hope to state a more general geometrization conjecture than the one announced in the ERC project (a categorical version).