Core Concepts

This paper investigates the geometric properties of non-generic components in the Emerton-Gee stack for GL2, providing insights into their smoothness, normality, and singularities, and discusses the implications for the conjectural categorical p-adic Langlands correspondence.

Abstract

Kansal, K., & Savoie, B. (2024). Non-Generic Components of the Emerton-Gee Stack for GL2. arXiv preprint arXiv:2407.07883v2.

This research paper aims to analyze the geometric properties of non-generic components within the Emerton-Gee stack for GL2, specifically focusing on their smoothness, normality, and the nature of their singularities. The authors seek to refine the understanding of these components and explore the implications of their findings for the conjectural categorical p-adic Langlands correspondence.

The authors utilize a combination of algebraic geometry and representation theory techniques. They employ smooth-local charts and auxiliary schemes to study the structure of the Emerton-Gee stack. By analyzing the properties of these charts, they deduce information about the smoothness, normality, and singular loci of the non-generic components.

- The paper provides precise conditions for the smoothness and normality of non-generic components in the Emerton-Gee stack.
- It demonstrates that the normalizations of these components admit smooth-local covers by resolution-rational schemes.
- The authors determine the codimension of the singular loci within these components.
- The research establishes a connection between the singularities of the components and the ramification properties of associated Galois representations.

The geometric properties of non-generic components in the Emerton-Gee stack, particularly their singularities, are intricately linked to the ramification behavior of corresponding Galois representations. These findings contribute to a deeper understanding of the stack's structure and offer valuable insights for the ongoing development of the categorical p-adic Langlands correspondence.

This research significantly advances the understanding of the Emerton-Gee stack, a central object in the p-adic Langlands program. By elucidating the geometry of its non-generic components, the authors provide crucial information for constructing and understanding the conjectural categorical p-adic Langlands correspondence, a profound connection between representation theory and number theory.

The study primarily focuses on non-generic components in the case of GL2. Further research could explore the geometry of generic components and extend the analysis to higher-rank general linear groups. Investigating the implications of these geometric findings for the explicit construction of the categorical p-adic Langlands correspondence remains a promising avenue for future investigation.

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p > 3
K is an unramified extension of Qp of degree f.

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by Kalyani Kans... at **arxiv.org** 10-24-2024

Deeper Inquiries

Answer:
While this paper focuses specifically on the Emerton-Gee stack for $\mathrm{GL}_2$, generalizing these geometric results to other reductive groups presents significant challenges and is a subject of active research. Here's a breakdown of potential generalizations and difficulties:
Potential Generalizations:
Irreducible Components: The labeling of irreducible components of the Emerton-Gee stack by Serre weights should generalize to other groups, using appropriate notions of Serre weights for those groups.
Smoothness and Normality: The criteria for smoothness and normality of these components likely involve intricate relationships between the root data of the group, the shape of the inertial type τ, and the chosen Serre weight.
Resolution of Singularities: The existence of smooth-local covers by resolution-rational schemes might extend, but explicit constructions would be much more involved. The nature of these resolutions would likely depend heavily on the specific reductive group.
Difficulties:
Increased Complexity: The representation theory and geometry associated with general reductive groups are significantly more intricate than those for $\mathrm{GL}_2$.
Lack of Explicit Charts: Constructing explicit smooth-local charts, like those using Breuil-Kisin modules for $\mathrm{GL}_2$, becomes much harder.
Understanding Shapes: The "shape" conditions imposed on Breuil-Kisin modules to cut out irreducible components would need to be generalized and would likely be more complicated.
Current Research:
Researchers are actively exploring these generalizations. Some progress has been made for groups like $\mathrm{GSp}_4$, but a complete understanding for general reductive groups remains a significant open problem.

Answer:
Yes, exploring alternative geometric interpretations of the singularities in the non-generic components of the Emerton-Gee stack is a promising avenue for deeper insights into the p-adic Langlands correspondence. Here are some potential approaches:
Derived Geometry: Instead of focusing solely on the reduced part of the stack, studying the full derived structure of the Emerton-Gee stack might provide a more natural framework for understanding the singularities. Derived algebraic geometry could offer tools to resolve these singularities in a more refined way.
Intersection Theory: Investigating the intersection theory of the irreducible components, particularly near the singular loci, could reveal connections between the geometry of the stack and the representation theory of the relevant groups.
Moduli of Langlands Parameters: The Emerton-Gee stack is expected to be related to a moduli stack of Langlands parameters. Understanding this relationship and how singularities on one side correspond to geometric features on the other could be illuminating.
Relationship to p-adic Hodge Theory: The singularities might reflect deeper phenomena in p-adic Hodge theory. For instance, they could be linked to the structure of period rings or the geometry of certain p-adic period domains.
Potential Benefits:
Refined Conjectures: Alternative geometric interpretations could lead to more refined conjectures about the p-adic Langlands correspondence, potentially suggesting new directions for research.
Connections to Other Areas: These interpretations might reveal unexpected connections between the p-adic Langlands program and other areas of mathematics, such as derived algebraic geometry, p-adic Hodge theory, or the theory of singularities.

Answer:
This research, particularly the results on the geometry of the Emerton-Gee stack, has significant implications for explicitly constructing the functor A in the categorical p-adic Langlands correspondence. Here's how:
Characterizing the Sheaves: The paper provides a concrete description of the support of the conjectural sheaves L(σm,n) and shows that, under certain conditions, they are pushforwards of invertible sheaves on smooth stacks. This characterization is a crucial step towards explicitly constructing these sheaves.
Understanding Self-Duality: The results on the codimension of singular loci and the existence of smooth-local covers allow for a better understanding of the self-duality of L(σm,n). This is essential, as self-duality is a key property expected of the functor A.
Smoothness and Gluing: The existence of smooth-local charts on the Emerton-Gee stack suggests a strategy for constructing A locally on these charts and then gluing these local constructions. The explicit nature of these charts, using Breuil-Kisin modules, provides a concrete setting for performing these local constructions.
Challenges and Future Directions:
Non-Generic Components: The singularities in the non-generic components pose a challenge. The paper's results on resolutions of singularities provide a starting point, but more work is needed to understand how to construct A over these singular components.
Higher Dimensions: Generalizing these constructions to GLd for d > 2 will be significantly more difficult due to the increased complexity of the geometry and representation theory involved.
Overall Impact:
This research provides a concrete framework and essential ingredients for constructing the functor A in the categorical p-adic Langlands correspondence. The explicit nature of the results, particularly the use of Breuil-Kisin modules and the analysis of singularities, offers a promising path towards making the p-adic Langlands program more explicit and computationally accessible.

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