Periodic Reporting for period 3 - LEGS (p-adic Langlands and the Emerton-Gee stack)
Berichtszeitraum: 2023-10-01 bis 2025-03-31
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.
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).
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).
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.