## Project Details

### Description

Overview. This proposal addresses the following fundamental question: what are the underlying geometric structures relating p-adic Galois representations to the mod p representation theory of p-adic groups? Some of these geometric structures have recently begun to come into focus thanks to three recent major developments, of which the PIs are among the architects. For this reason we believe the time is right for us to come together in this collaborative effort aimed at solving a number of the most important problems in the p-adic Langlands program.

The first two of these developments are the construction by Emerton and Gee of moduli stacks

parametrizing p-adic representations of the Galois groups of p-adic local fields, and the discovery by Le, Le Hung, Levin, and Morra (in the course of their work on the Breuil–M´ezard conjecture in higher rank) of “monodromy local models”. The Emerton–Gee stacks globalize Mazur’s classical deformation theory of Galois representations; for example, they allow for the formulation of a universal geometric version (now established in the tamely potentially Barsotti–Tate case by Caraiani, Emerton, Gee, and Savitt) of the Breuil–M´ezard conjecture. The monodromy local models turn out to be algebraizations of the Emerton–Gee stacks, and provide access to the geometry of the Emerton–Gee stacks via the methods of geometric representation theory.

The third development to which we refer consists of various extensions of the Taylor–Wiles–Kisin

patching argument, and in particular the rapidly expanding relevance of derived methods. Of particular significance to this project is the derived Hecke algebra, in whose development Harris has been a key contributor. The derived Hecke algebra is likely to play a key role on the spectral side of a possible categorical Langlands correspondence, in which the Galois side is given by an appropriate category of sheaves on the Emerton–Gee stack.

Intellectual Merit. We expect that a geometric perspective is required to make progress on the major outstanding questions relating p-adic Galois representations and p-adic groups. Some of the specific questions that we will address are as follows.

(1) The problem of potentially crystalline lifts: for which weights and types does a given mod p representation of the absolute Galois group of a p-adic local field K admit a potentially crystalline lift of that weight and type? This is one of the most important outstanding questions in local Galois theory. We will study it as the conjunction of two other problems: the geometric Breuil–M´ezard conjecture and the problem of weight entailment.

(2) The construction of a p-adic local Langlands correspondence for GLn(K): outside the case of GL2(Qp), no such correspondence has been constructed; there is not even a generally held conjecture as to what form it might take. The construction of [CEG+] gives a candidate correspondence, constructed from a global setting via patching. Determining whether or not this a priori global construction is in fact purely local is a fundamental problem.

(3) It seems possible that p-adic local Langlands correspondence has to be investigated by more

categorical means, ones that are closer to the methods of the geometric Langlands program than to those of the classical arithmetic Langlands program. This leads to the consideration of derived categories of coherent sheaves on the Emerton–Gee stacks and on the LLLM local models.

(4) A sufficiently strong statement on the local nature of the patched representation of [CEG+] leads directly to an automorphy lifti

The first two of these developments are the construction by Emerton and Gee of moduli stacks

parametrizing p-adic representations of the Galois groups of p-adic local fields, and the discovery by Le, Le Hung, Levin, and Morra (in the course of their work on the Breuil–M´ezard conjecture in higher rank) of “monodromy local models”. The Emerton–Gee stacks globalize Mazur’s classical deformation theory of Galois representations; for example, they allow for the formulation of a universal geometric version (now established in the tamely potentially Barsotti–Tate case by Caraiani, Emerton, Gee, and Savitt) of the Breuil–M´ezard conjecture. The monodromy local models turn out to be algebraizations of the Emerton–Gee stacks, and provide access to the geometry of the Emerton–Gee stacks via the methods of geometric representation theory.

The third development to which we refer consists of various extensions of the Taylor–Wiles–Kisin

patching argument, and in particular the rapidly expanding relevance of derived methods. Of particular significance to this project is the derived Hecke algebra, in whose development Harris has been a key contributor. The derived Hecke algebra is likely to play a key role on the spectral side of a possible categorical Langlands correspondence, in which the Galois side is given by an appropriate category of sheaves on the Emerton–Gee stack.

Intellectual Merit. We expect that a geometric perspective is required to make progress on the major outstanding questions relating p-adic Galois representations and p-adic groups. Some of the specific questions that we will address are as follows.

(1) The problem of potentially crystalline lifts: for which weights and types does a given mod p representation of the absolute Galois group of a p-adic local field K admit a potentially crystalline lift of that weight and type? This is one of the most important outstanding questions in local Galois theory. We will study it as the conjunction of two other problems: the geometric Breuil–M´ezard conjecture and the problem of weight entailment.

(2) The construction of a p-adic local Langlands correspondence for GLn(K): outside the case of GL2(Qp), no such correspondence has been constructed; there is not even a generally held conjecture as to what form it might take. The construction of [CEG+] gives a candidate correspondence, constructed from a global setting via patching. Determining whether or not this a priori global construction is in fact purely local is a fundamental problem.

(3) It seems possible that p-adic local Langlands correspondence has to be investigated by more

categorical means, ones that are closer to the methods of the geometric Langlands program than to those of the classical arithmetic Langlands program. This leads to the consideration of derived categories of coherent sheaves on the Emerton–Gee stacks and on the LLLM local models.

(4) A sufficiently strong statement on the local nature of the patched representation of [CEG+] leads directly to an automorphy lifti

Status | Active |
---|---|

Effective start/end date | 7/1/20 → 6/30/23 |

### Funding

- National Science Foundation (DMS-1952678)

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