p-adic Langlands and the Emerton-Gee stack

Projektinformationen

LEGS

ID Finanzhilfevereinbarung: 884596

DOI

10.3030/884596

EK-Unterschriftsdatum 20 April 2020

Startdatum 1 Oktober 2020

Enddatum 30 September 2026

Finanziert unter

• EXCELLENT SCIENCE - European Research Council (ERC)

Gesamtkosten

€ 2 195 110,00

EU-Beitrag

€ 2 195 110,00

Koordiniert durch

IMPERIAL COLLEGE OF SCIENCE TECHNOLOGY AND MEDICINE
United Kingdom

Periodic Reporting for period 3 - LEGS (p-adic Langlands and the Emerton-Gee stack)

Berichtszeitraum: 2023-10-01 bis 2025-03-31

Zusammenfassung vom Kontext und den Gesamtzielen des Projekts

The project is to understand the relationship between the conjectural p-adic Langlands correspondence and the Emerton--Gee stacks. In more down to earth terms, the objective is to classify certain symmetries occurring in number theory in terms of geometric objects.

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

Jack Sempliner has been working with the PI on constructing moduli stacks of ordinary Galois representations.We believe that can prove that these stacks are lci, and considering whether we can use them to describe the ordinary cohomology of Shimura varieties.

The PI's book with Matthew Emerton on the Emerton-Gee stacks (on which the project is based) has been completely revised for publication.

The PI has with Matthew Emerton and Eugen Hellmann formulated categorical versions of the p-adic local Langlands correspondence, which will be described in lecture notes for a 2022 IHES summer school.

The PI has with Andrea Dotto and Matthew Emerton proved a categorical p-adic local Langlands correspondence for GL_2(Q_p).

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)

The formulation of a general p-adic local Langlands correspondence in categorical terms is completely new. Until the end of the project the plan is to work on proving more cases of this correspondence, and understanding its connections to other parts of number theory. In particular I hope to understand the correspondence explicitly for GL_2(F) for F a general p-adic field.

picture of the EG stack