GeoLocLang

Projektinformationen

GeoLocLang

ID Finanzhilfevereinbarung: 742608

DOI

10.3030/742608

Projekt abgeschlossen

EK-Unterschriftsdatum 12 Mai 2017

Startdatum 1 Oktober 2017

Enddatum 31 März 2023

Finanziert unter

• EXCELLENT SCIENCE - European Research Council (ERC)

Gesamtkosten

€ 1 301 863,00

EU-Beitrag

€ 1 301 863,00

Koordiniert durch

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
France

Periodic Reporting for period 4 - GeoLocLang (GeoLocLang)

Berichtszeitraum: 2022-04-01 bis 2023-03-31

Zusammenfassung vom Kontext und den Gesamtzielen des Projekts

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.

Arbeit, die ab Beginn des Projekts bis zum Ende des durch den Bericht erfassten Berichtszeitraums geleistet wurde, und die wichtigsten bis dahin erzielten Ergebnisse

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.

Fortschritte, die über den aktuellen Stand der Technik hinausgehen und voraussichtliche potenzielle Auswirkungen (einschließlich der bis dato erzielten sozioökonomischen Auswirkungen und weiter gefassten gesellschaftlichen Auswirkungen des Projekts)

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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