p-adic Langlands and the Emerton-Gee stack

Project Information

LEGS

Grant agreement ID: 884596

DOI

10.3030/884596

EC signature date 20 April 2020

Start date 1 October 2020

End date 30 September 2026

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 2 195 110,00

EU contribution

€ 2 195 110,00

Coordinated by

IMPERIAL COLLEGE OF SCIENCE TECHNOLOGY AND MEDICINE
United Kingdom

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

Reporting period: 2023-10-01 to 2025-03-31

Summary of the context and overall objectives of the project

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.

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

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).

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)

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