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Implications of Breuil-Herzig-Hu-Morra-Schraen's conjectures on Zábrádi's functor

This paper investigates the limitations of Zábrawdi's functor in recovering the Galois representation $\overline{L}^{\boxtimes}(\rho)$ from representations of $\mathrm{GL}_n(\mathbb{Q}_p)$ compatible with a generic $n$ -dimensional Galois representation $\rho$ , demonstrating that such recovery is impossible when $\rho$ is reducible and $n \geq 3$ .

Original authors: Nataniel Marquis

Published 2026-01-15

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CC BY 4.0

Original authors: Nataniel Marquis

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

Imagine you are trying to solve a massive, intricate jigsaw puzzle. In the world of advanced mathematics, specifically a field called number theory, there is a famous puzzle known as the Local Langlands Correspondence. It's a rulebook that tries to connect two very different languages: one spoken by "Galois groups" (which describe the symmetries of numbers) and another spoken by "GLn groups" (which describe symmetries of matrices).

For a long time, mathematicians could only solve this puzzle for very small, simple pieces (specifically for $2 \times 2$ matrices). But for larger, more complex pieces ( $n \times n$ matrices), the picture is still blurry.

This paper, written by Nataniel Marquis, is a "reality check" for a new, very promising tool that mathematicians hope will help solve the big puzzle.

The Two Main Characters

The Target ( $\rho$ ): Think of this as a specific, complex shape in the "Galois" language. It's a representation of dimension $n$ (like a 3D object).
The Candidate ( $\Pi$ ): This is a shape in the "GLn" language that mathematicians think should match the Target.
The Translator (Zábrádi's Functor): This is the new tool. Imagine a machine that takes a shape from the GLn language and translates it back into the Galois language.
- The Dream: The authors of a previous paper ([Bre+21]) hoped that if you fed the correct Candidate ( $\Pi$ ) into this Translator, it would perfectly reconstruct a specific, complex Galois object called $L^{\boxtimes}(\rho)$ . They thought this Translator was a "magic wand" that could reveal the hidden structure of the puzzle.

The Paper's Big Question

Marquis asks: "Does this magic wand actually work as well as we hope?"

Specifically, he investigates a scenario where the Target ( $\rho$ ) is reducible. In plain English, this means the Target isn't a single, solid block; it's made of smaller, simpler blocks glued together.

The Findings: A "Bad Behavior" Report

Marquis uses a mix of counting and logical deduction to show that the Translator fails in specific, predictable ways. Here is the breakdown using everyday analogies:

1. The "Too Many Pieces" Problem

Imagine the Target Galois object ( $L^{\boxtimes}(\rho)$ ) is a massive, complex sculpture made of hundreds of tiny, unique Lego bricks.

The Expectation: If you feed the correct Candidate ( $\Pi$ ) into the Translator, you should get a sculpture with all those hundreds of bricks.
The Reality: Marquis proves that if the Target is made of glued-together blocks (reducible), the Candidate ( $\Pi$ ) simply doesn't have enough "bricks" (mathematical components called Jordan-Hölder factors) to build the full sculpture.
The Result: When the Translator processes the Candidate, the resulting object is too small. It's missing pieces. It cannot possibly look like the Target.
- Analogy: It's like trying to build a full-size castle out of a toy set that only has enough bricks for a small tower. No matter how you arrange them, you can't make the castle.

2. The "Toy Example" (The 3D Case)

To prove his point, Marquis builds a "toy model" using a 3-dimensional case ( $n=3$ ). This is like testing a new car engine on a small, controlled track before taking it on the highway.

He sets up a specific, slightly "messy" (non-split) 3D shape.
He assumes the Candidate ( $\Pi$ ) is "weakly compatible," meaning it matches the Target in some basic ways but maybe not all the strict rules.
The Result: Even with this relaxed definition, the Translator produces an object that is too small (dimensionally) to match the Target.
- The Metaphor: If the Target is a 5-story building, the Translator only produces a 4-story building. It's close, but it's not the same.

The Conclusion: "It's Not a Magic Wand"

The paper concludes that for these specific types of complex, reducible shapes:

The Dream is False: You cannot recover the full, complex Galois object ( $L^{\boxtimes}(\rho)$ ) just by using Zábrádi's functor on a compatible Candidate.
The "Bad Behavior": The functor "behaves badly" in the sense that it loses information. It acts like a filter that drops some of the essential details of the puzzle.
The Takeaway: While Zábrádi's functor is a powerful tool and keeps more information than older tools, it is not the perfect solution everyone hoped for. It cannot simply "reverse engineer" the full Galois representation from the Candidate in these cases.

Summary in One Sentence

Nataniel Marquis shows that a new mathematical translator (Zábrádi's functor) is not powerful enough to perfectly reconstruct complex, multi-part number puzzles from their matrix counterparts, because it inevitably loses some of the necessary pieces along the way.

Technical Summary of "Implications of Breuil-Herzig-Hu-Morra-Schraen's conjectures on Zábrádi's functor"

Problem Statement
The paper addresses a central gap in the mod $p$ local Langlands correspondence for $\text{GL}_n(\mathbb{Q}_p)$ where $n \geq 3$ . While the correspondence is well-understood for $\text{GL}_2(\mathbb{Q}_p)$ , the classification of smooth admissible representations of $\text{GL}_n(\mathbb{Q}_p)$ modulo $p$ remains incomplete, particularly regarding supercuspidal representations and extensions.

Breuil, Herzig, Hu, Morra, and Schraen (in [Bre+21]) formulated conjectures describing the shape of representations $\Pi$ associated to a generic $n$ -dimensional Galois representation $\rho$ via global methods. A key component of this framework is the representation $L^{\boxtimes}(\rho) := \bigotimes_{1 \leq i < n} (\Lambda^i \rho)$ , which is constructed from $\rho$ and is conjectured to be recoverable from $\Pi$ via Zábrádi's functor $V_\Delta$ . The authors of [Bre+21] posed the question of whether $V_\Delta(\Pi) \simeq L^{\boxtimes}(\rho)$ holds.

The specific problem investigated in this note is whether this "best scenario" is possible, particularly when $\rho$ is reducible or when $\Pi$ satisfies only "weak" compatibility conditions. The author, Nataniel Marquis, aims to demonstrate that $V_\Delta(\Pi)$ generally fails to recover $L^{\boxtimes}(\rho)$ in terms of dimension and structure, even under relaxed hypotheses.

Methodology
The paper employs a combination of representation-theoretic counting arguments, the structural properties of Zábrádi's functor, and explicit computations in a "toy" case ( $n=3$ ).

Zábrádi's Functor and Fontaine Equivalence: The author utilizes Zábrádi's functor $V_\Delta$ , which maps smooth representations of $\text{GL}_n(\mathbb{Q}_p)$ to finite-dimensional representations of $(G_{\mathbb{Q}_p, \Delta} \times \mathbb{Q}_p^\times)$ . This functor relies on a multivariable Fontaine equivalence involving étale $(\phi, \Gamma)$ -modules over a ring $E_\Delta$ . Key properties used include the right exactness of the functor and its behavior on principal series and supersingular representations (Theorems 2.10, 2.11, 2.12).
Jordan-Hölder Component Counting: The core strategy involves comparing the number of Jordan-Hölder components (composition factors) of the source representation $\Pi$ $Π$ (or its subquotients) with those of the target $L^{\boxtimes}(\rho)$ $L^{⊠} (ρ)$ .
- The author establishes that if $\Pi$ is compatible with a specific parabolic subgroup $\tilde{P}_\rho$ , the number of its Jordan-Hölder components is bounded by the number of isotypic components of a related tensor product restricted to the center of the Levi subgroup.
- Conversely, the author proves that $L^{\boxtimes}(\rho)$ possesses a significantly larger number of Jordan-Hölder components than the bound imposed on $\Pi$ when $\rho$ is reducible.
Explicit "Toy" Case ( $n=3$ ): To investigate "weak compatibility" (where $\Pi$ $Π$ is compatible with $\tilde{P}_\rho$ $\tilde{P}_{ρ}$ and $V_\xi(\Pi) \simeq L^{\otimes}(\rho)$ $V_{ξ} (Π) ≃ L^{\otimes} (ρ)$ ), the author analyzes a 3-dimensional representation $\rho$ $ρ$ of Loewy length 3.
- The paper constructs the specific structure of $L^{\boxtimes}(\rho)$ and its subrepresentation $L^{\boxtimes}(\rho)_{ord}$ .
- It analyzes the extension groups in $\text{GL}_3(\mathbb{Q}_p)$ and their images under $V_\Delta$ using results from [JZ25] and [Hau16].
- The author uses cohomological arguments (inflation-restriction sequences) and Schur's Lemma to bound the dimension of subrepresentations of $V_\Delta(\Pi)$ .

Key Contributions and Results

Failure in the Reducible Case (Theorem 3.5): The paper proves that for any $n \geq 3$ $n \geq 3$ , if $\rho$ $ρ$ is a reducible, generic, and good conjugate representation, then for any $\Pi$ $Π$ compatible with $\tilde{P}_\rho$ $\tilde{P}_{ρ}$ , the isomorphism $V_\Delta(\Pi) \simeq L^{\boxtimes}(\rho)$ $V_{Δ} (Π) ≃ L^{⊠} (ρ)$ cannot hold.
- Reasoning: The compatibility condition imposes a "shortage" of Jordan-Hölder components in $V_\Delta(\Pi)$ compared to the "profusion" of components in $L^{\boxtimes}(\rho)$ . Specifically, the number of components in $V_\Delta(\Pi)$ is strictly less than the number of irreducible components in $L^{\boxtimes}(\rho)$ .
Failure in the Irreducible Case (Theorem 3.6): The author demonstrates that there exist infinitely many integers $n$ for which there is an irreducible $\rho$ such that no supersingular $\Pi$ satisfies $V_\Delta(\Pi) \simeq L^{\boxtimes}(\rho)$ . This is shown by constructing representations where $\Lambda^2 \rho$ is reducible, while $V_\Delta(\Pi)$ for supersingular $\Pi$ must be either zero or irreducible.
Dimensional Bounds in the $n=3$ Case (Proposition 5.5): For a 3-dimensional $\rho$ $ρ$ of Loewy length 3 that is weakly compatible with $\Pi$ $Π$ :
- The dimension of $V_\Delta(\Pi)$ is bounded by 5 (and by 4 if $p \neq 2$ ).
- Crucially, $V_\Delta(\Pi)$ cannot be isomorphic to $L^{\boxtimes}(\rho)$ , nor can its restriction to the diagonal embedding be isomorphic to $L^{\otimes}(\rho)$ .
- The paper shows that even if one relaxes the compatibility conditions to require only $V_\xi(\Pi) \simeq L^{\otimes}(\rho)$ , the functor $V_\Delta$ still fails to recover the full structure of $L^{\boxtimes}(\rho)$ due to dimensional constraints and the semisimplicity of certain subextensions.

Significance and Claims
The paper claims to provide a "range of results" demonstrating how "badly" Zábrádi's functor behaves when applied to representations compatible with $\rho$ in the sense of [Bre+21].

Refutation of the Best Scenario: The primary claim is that the optimistic conjecture—that Zábrádi's functor recovers $L^{\boxtimes}(\rho)$ from a compatible $\Pi$ —is false in the reducible case for $n \geq 3$ .
Limitations of Weak Compatibility: The results suggest that even relaxing the compatibility conditions (e.g., requiring only $V_\xi(\Pi) \simeq L^{\otimes}(\rho)$ ) does not resolve the discrepancy. The functor $V_\Delta$ inherently loses information or fails to capture the full complexity of $L^{\boxtimes}(\rho)$ in these settings.
Modest Scope: The author explicitly frames the work as an investigation into specific counter-examples and failures rather than a complete classification. The paper notes that while Zábrádi's functor retains more information than Breuil's functor (via multivariable $(\phi, \Gamma)$ -modules), it still does not recover non-vanishing results beyond what was known for Breuil's functor, and in the specific context of recovering $L^{\boxtimes}(\rho)$ , it fails to produce the expected isomorphism.

The paper concludes that the combinatorial structure of $\Pi$ 's subquotients, dictated by compatibility, is insufficient to generate the rich structure of $L^{\boxtimes}(\rho)$ via Zábrádi's functor, necessitating a re-evaluation of the conjectural link between global representations and their local mod $p$ counterparts in dimensions $n \geq 3$ .