🔢 mathematics

A conjectural construction of Arthur packets in Fargues-Scholze's categorical local Langlands correspondence

This paper proposes a conjectural construction of Arthur packets within Fargues-Scholze's categorical local Langlands correspondence by generalizing the geometric push-forward of skyscraper sheaves from the regular conormal bundle over $\mathbb{C}$ to an analogous operation on the stack of singularities over $\overline{\mathbb{Q}}_\ell$ .

Original authors: Geo Kam-Fai Tam

Published 2026-08-26

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Geo Kam-Fai Tam

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of modern mathematics, there is a deep and enduring quest to understand the hidden symmetries that govern numbers and shapes. This quest often leads researchers to study groups, which are mathematical structures that describe how things can be moved or transformed without changing their essential nature. For decades, mathematicians have been trying to connect two very different ways of looking at these groups: one that treats them as collections of numbers and equations, and another that views them as geometric landscapes. This connection, known as the local Langlands correspondence, acts like a Rosetta Stone, translating complex algebraic problems into geometric ones and vice versa. While this translation has been successfully completed for some specific types of groups, a major challenge remains: how to handle the most complicated and irregular cases, where the standard rules of symmetry break down. These difficult cases are known as Arthur packets, and they represent the frontier where our current understanding of these mathematical structures begins to fray.

A recent paper by Geo Kam-Fai Tam of Xiamen University Malaysia proposes a new way to navigate this difficult terrain. The work does not claim to have solved the problem with a final, unshakeable proof. Instead, it offers a carefully constructed blueprint, a conjectural map that suggests how these elusive Arthur packets might be built using the latest tools in geometry. The author takes a framework developed by Laurent Fargues and Peter Scholze, which treats the problem as a relationship between two different kinds of mathematical spaces, and extends it to cover these messy, irregular cases. The core idea is to replace the rigid, point-like descriptions of the old theory with a more flexible, geometric approach that can handle the "singularities"—the rough spots and tears in the mathematical fabric where standard methods fail.

The paper begins by setting the stage with two main characters: a space of parameters and a space of bundles. The space of parameters is like a library of all possible ways a group can behave, while the space of bundles is a landscape of geometric objects built from that group. The central achievement of the Fargues-Scholze framework was to show that these two spaces are deeply connected, almost like two sides of the same coin. However, this connection was previously understood clearly only for the "smooth" or "tempered" cases, where the behavior is predictable and well-behaved. The new paper asks what happens when we look at the wilder, more chaotic cases. Here, the behavior is not just a simple point in the library; it is a complex structure with layers of depth and hidden connections.

To tackle this, the author draws on a concept from a different branch of mathematics called the geometric Satake correspondence. This is a powerful tool that allows mathematicians to move information between the library of parameters and the landscape of bundles. The paper explains how to use this tool to define "Hecke operators," which act like lenses that focus on specific parts of the landscape. By using these operators, the author shows how to construct a "spectral action," a mechanism that takes a specific geometric object from the parameter side and pushes it onto the bundle side to reveal the corresponding mathematical representations. This process is the engine that drives the connection between the two worlds.

The heart of the paper lies in its proposal for how to handle the irregular cases, the Arthur packets. In the past, mathematicians like James Arthur had to invent special rules to describe these packets, often treating them as finite sets of representations that could overlap in confusing ways. Tam's paper suggests a different path. It proposes that these packets can be constructed by looking at the "singularities" of the parameter space. Imagine the parameter space as a smooth surface that, in certain places, folds over itself or develops sharp edges. The paper suggests that the information needed to build the Arthur packets is hidden in these folds. By using a specific mathematical operation known as a "vanishing cycle functor," which essentially tracks how shapes change as they pass through these singular points, the author constructs a new type of geometric object called an "Arthur sheaf."

This Arthur sheaf is the key. The paper conjectures that if you take this sheaf and apply the spectral action described earlier, it will automatically generate the correct set of representations for the Arthur packet. This is a significant shift in perspective. Instead of defining the packets by listing their members and checking if they fit certain rules, the new approach builds them from the ground up using the geometry of the singularities. The author tests this idea on simple examples, such as the group of two-by-two matrices, and shows that the construction works, reproducing the known results for these cases. This gives hope that the method is sound and could be applied to more complex groups.

The paper is explicit about its nature: it is a conjecture, a proposal for how things should work, rather than a finished theorem. The author acknowledges that the full machinery required to prove this rigorously is still being developed and relies on advanced tools from derived algebraic geometry. However, the value of the work lies in its clarity and its unification of different ideas. It brings together the geometric insights of Fargues and Scholze with the representation-theoretic insights of Arthur and others, creating a single, coherent picture. By suggesting that the irregular, messy parts of the theory can be understood through the geometry of singularities, the paper opens a new door for mathematicians to explore. It invites the community to look at these difficult problems not as obstacles to be bypassed, but as features to be studied, where the very roughness of the geometry holds the key to the solution.

Ultimately, this work is about finding order in complexity. It suggests that even in the most chaotic corners of the mathematical universe, there is a geometric logic waiting to be uncovered. The author does not claim to have the final answer, but provides a compelling and detailed roadmap for how to find it. By translating the problem into the language of sheaves and singularities, the paper offers a fresh perspective that could reshape how mathematicians think about the fundamental structures of numbers and symmetry. It is a reminder that in the pursuit of mathematical truth, sometimes the most important step is not to force a solution, but to propose a new way of seeing the problem itself.

Technical Summary: A Conjectural Construction of Arthur Packets in Fargues-Scholze's Categorical Local Langlands Correspondence

Problem Statement
The paper addresses the challenge of constructing Arthur packets within the framework of the Categorical Local Langlands Correspondence (CLLC) developed by Fargues and Scholze. While the classical Local Langlands Correspondence (LLC) establishes a bijection (or finite-to-one map) between irreducible smooth representations of a reductive $p$ -adic group $G(F)$ and Langlands parameters, it does not fully account for non-tempered representations. Arthur introduced "Arthur packets"—finite sets of representations (potentially with non-empty intersections) parameterized by Arthur parameters—to extend the theory to the unitary dual.

The central problem is to realize these Arthur packets geometrically within the CLLC. The CLLC posits an equivalence between the derived category of lisse-étale sheaves on the moduli stack of $G$ -bundles ( $\text{Bun}_G$ ) and the derived category of coherent sheaves on the moduli stack of Langlands parameters ( $\text{Par}_G$ ). While Fargues' conjecture provides a mechanism to construct $L$ -packets (for elliptic parameters) via the spectral action of skyscraper sheaves on $\text{Par}_G$ , a corresponding geometric construction for Arthur packets (involving singular parameters and non-tempered representations) remains to be fully formulated in this categorical setting, particularly over the coefficient field $\Lambda = \mathbb{Q}_\ell$ .

Methodology
The paper synthesizes techniques from the geometric Langlands program, derived algebraic geometry, and the representation theory of $p$ -adic groups. The methodology proceeds in three logical stages:

Framework Overview: The author reviews the CLLC, defining the moduli stacks $\text{Par}_G$ (parameters) and $\text{Bun}_G$ (bundles). It establishes the "spectral action," where sheaves on $\text{Par}_G$ act on sheaves on $\text{Bun}_G$ via Hecke operators. This action is mediated by the Whittaker sheaf $W_f$ . The paper also reviews the construction of semisimple parameters using excursion algebras and the geometric Bernstein center.
Geometric Analogy (Microlocal Geometry): The paper draws on the work of Cunningham, Fargues, and others ([CFM+22]) who constructed Arthur packets in the archimedean case ( $\Lambda = \mathbb{C}$ ) using microlocal geometry. In that setting, Arthur packets are associated with the regular conormal bundles of orbits in the Vogan variety. The representations in a packet correspond to equivariant local systems on these conormal bundles. The author translates this to the $p$ -adic setting by identifying the conormal bundle with a "singularity stack" ( $\text{Sing}_{\text{Par}_G}$ ) over $\text{Par}_G$ .
Generalization to $\mathbb{Q}_\ell$ : The core technical contribution is the generalization of the construction from $\Lambda = \mathbb{C}$ to $\Lambda = \mathbb{Q}_\ell$ .
- Instead of Vogan varieties and conormal bundles, the paper utilizes the singularity stack $\text{Sing}_{\text{Par}_G/\Lambda}$ , defined via the cotangent complex of $\text{Par}_G$ .
- It defines Arthur parameters as morphisms $\psi: W_F \times SL_2^D \times SL_2^A \to {}^L G$ (for $\Lambda=\mathbb{C}$ ) or via pairs $(\lambda, N)$ involving a nilpotent element $N$ (for $\Lambda=\mathbb{Q}_\ell$ ).
- The construction employs vanishing cycle functors ( $\text{Ev}_C$ ) to map sheaves supported on the parameter space to sheaves supported on the regular part of the singularity stack (analogous to the regular conormal bundle).
- The Arthur sheaf $A_\psi$ is defined as the right adjoint of the vanishing cycle functor applied to the skyscraper sheaf on the regular singularity stack.

Key Contributions and Results

Conjectural Construction of Arthur Sheaves: The paper proposes a definition of the Arthur sheaf $A_\psi$ for a general parameter $\psi$ (or its infinitesimal counterpart) in the $\mathbb{Q}_\ell$ setting. This is achieved by pushing forward the skyscraper sheaf on the regular part of the singularity stack $\text{Sing}_{\text{Par}_G}$ via the adjoint of the vanishing cycle functor.
Conjecture 4.8: The main result is formulated as Conjecture 4.8, which states:
1. The constructed Arthur sheaf $A_\psi$ is a perfect complex (a requirement for the spectral action in the CLLC framework).
2. The Arthur packet $\Pi_\psi$ is obtained by applying the spectral action of $A_\psi$ to the Whittaker sheaf $W_f$ on $\text{Bun}_G$ .
Unification of Cases: The paper demonstrates that for tori, this construction reduces to the standard Fargues conjecture for elliptic parameters. For $PGL_2$ , it explicitly calculates the vanishing cycles and Arthur sheaves, recovering the known Arthur packets (including the Steinberg representation) and showing consistency with the geometric multiplicity matrices established in [CFM+22].
Singular Support: The paper utilizes the theory of singular support (Proposition 4.9) to argue for the perfectness of the constructed sheaves, linking the geometry of the singularity stack to the homological properties of the sheaves.

Significance and Claims
The author explicitly states that this note offers "essentially no new theorems." Its primary significance lies in synthesis and conjectural formulation:

It connects the microlocal geometric construction of Arthur packets (previously developed for $\Lambda=\mathbb{C}$ in [CFM+22]) with the categorical framework of Fargues-Scholze (valid for $\Lambda=\mathbb{Q}_\ell$ ).
It provides a concrete, albeit conjectural, recipe for constructing Arthur packets in the $\ell$ -adic setting, suggesting that the "rational" or "integral" version of Arthur's theory can be accessed via the singularity stack of $\text{Par}_G$ .
It highlights the necessity of derived algebraic geometry (specifically the cotangent complex and singularity stacks) to handle the non-tempered, singular cases of the Langlands correspondence.

The paper serves as a bridge, suggesting that the machinery of vanishing cycles and singularity stacks can replace the classical conormal bundle approach to define Arthur packets categorically, thereby extending the scope of the CLLC beyond tempered representations.

More like this