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Computing the local $2$-component of a non-selfdual automorphic representation of $\mathrm{GL}_3$

This paper explicitly determines the local 2-adic component of a non-selfdual automorphic representation of $\mathrm{GL}_3$ constructed by van Geemen and Top by computing Hecke eigenvalues to show it is a parabolically induced representation involving a specific supercuspidal representation of $\mathrm{GL}_2(\mathbb{Q}_2)$ , thereby providing an explicit description of the associated 2-adic Galois representation.

Original authors: Yamamoto Hirofumi

Published 2026-03-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yamamoto Hirofumi

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 a detective trying to solve a mystery about a very complex, invisible object floating in a high-dimensional space. This object is called an automorphic representation (let's call it "The Shape"). Mathematicians know this Shape exists and that it has a special connection to the geometry of a strange, twisted surface.

However, The Shape is too big and too complicated to look at all at once. So, the detective (the author, Yamamoto) decides to zoom in on just one tiny, specific part of it: the part related to the number 2. In the world of numbers, "zooming in on 2" means looking at how things behave when you only care about powers of 2 (like 2, 4, 8, 16...). This is called the 2-adic component.

Here is the story of how the detective cracked the case, explained simply:

1. The Mystery: A Shape That Doesn't Mirror Itself

The Shape in question is "non-selfdual." Imagine holding up a mirror to a normal object; usually, the reflection looks the same (like your face). But this Shape is like a left-handed glove: if you look at its reflection, it looks like a right-handed glove. It's asymmetrical. The mathematicians who built it (van Geemen and Top) knew it existed, but they didn't know exactly what it looked like up close at the number 2.

2. The Clues: The "Hecke" Fingerprint

To identify the Shape, the detective uses a set of tools called Hecke operators. Think of these as special scanners or fingerprint readers. When you run a scanner over the Shape, it gives back a number (an eigenvalue).

If the Shape were a simple, smooth ball, the scanner would give a simple number.
If it's a complex, jagged rock, the scanner gives a weird, complicated number.

The author used a computer to run these scanners on the Shape. The computer spat out a list of strange numbers (like $2\sqrt{-1}$ , $8-8\sqrt{-1}$ , etc.). These numbers are the "fingerprint" of the Shape's local 2-part.

3. The Breakthrough: Taking it Apart

The detective realized the Shape wasn't a single, solid block. It was actually a sandwich made of two different ingredients glued together.

The Bread: A simple, smooth flavor (an unramified character, let's call it "The Seasoning").
The Filling: A very spicy, complex, and wild ingredient (a supercuspidal representation, let's call it "The Wild Spice").

The detective proved that the Shape is built by taking "The Wild Spice," wrapping it in "The Seasoning," and then expanding it out into the full 3-dimensional space.

4. Identifying the "Wild Spice"

The hardest part was figuring out exactly what "The Wild Spice" was. It's a representation of a 2x2 matrix group, which is like a secret code.

The detective knew this code was built using a specific "recipe" involving a quadratic extension (a special number system built on top of the 2-adic numbers).
By comparing the computer-generated fingerprints against a library of known recipes, the detective found the exact match.
The recipe involves a specific matrix $\alpha$ (which looks like a tiny fraction of a matrix) and a character $\Lambda$ (a set of rules for how the code behaves).

The detective calculated that the "Wild Spice" is constructed by taking a small, compact group, applying a specific rule ( $\Lambda$ ), and then "inducing" (expanding) it to cover the whole space.

5. The Final Verdict

The paper concludes with a precise blueprint. It says:

"The local 2-part of this mysterious Shape is exactly a sandwich: The Wild Spice (a specific supercuspidal representation of GL2) mixed with The Seasoning (a specific unramified character), all glued together by a parabolic subgroup."

Why Does This Matter?

You might ask, "Who cares about a 2-adic sandwich?"

The Connection to Galois: In math, there is a famous bridge called the Langlands Program that connects these "Shapes" (automorphic representations) to "Galois Representations" (which describe the symmetries of number systems).
The Payoff: By figuring out exactly what the Shape looks like at the number 2, the author also figured out exactly what the corresponding Galois representation looks like at the number 2.
The Application: This is crucial for understanding deep properties of numbers and geometry. It's like figuring out the engine specifications of a car by listening to the sound it makes at a specific speed. Now that we know the engine (the 2-adic component), we can predict how the car (the Galois representation) will behave in the future.

Summary Analogy

Imagine you have a mysterious, glowing alien artifact. You can't see the whole thing, but you can touch a small corner of it.

You tap the corner with different tools (Hecke operators) and record the sound it makes.
You realize the sound isn't random; it's a specific chord.
You deduce that the artifact is made of a rare, glowing crystal (the supercuspidal part) wrapped in a standard metal casing (the unramified part).
You write down the exact chemical formula for the crystal and the metal.
Because you know the formula, you can now predict how the entire artifact will react to gravity, light, and time, even parts you haven't touched yet.

That is what this paper does: it uses computer calculations to decode the "sound" of a mathematical object at the number 2, revealing its exact internal structure and allowing us to understand its deeper connections to the universe of numbers.

1. Problem Statement

The paper addresses the explicit determination of the local component at the prime $p=2$ ( $\Pi_2$ ) of a specific non-selfdual cuspidal automorphic representation $\Pi$ of $GL_3(\mathbb{A}_{\mathbb{Q}})$ .

Context: This representation $\Pi$ was constructed by van Geemen and Top (referenced as [9]) and is realized in the cuspidal cohomology of the congruence subgroup $\Gamma_0(128) \subset SL_3(\mathbb{Z})$ .
Significance of the Problem: While the local components $\Pi_p$ for odd primes $p \geq 3$ are unramified and computable via standard algorithms, the component at $p=2$ is highly ramified and non-trivial. Determining $\Pi_2$ is crucial for understanding the associated Galois representation $\rho_{vGT, \ell}$ (constructed in [9]) at the prime 2, specifically its restriction to the Weil group $W_{\mathbb{Q}_2}$ .
Goal: To explicitly describe the structure of $\Pi_2$ as a representation of $GL_3(\mathbb{Q}_2)$ , identifying whether it is supercuspidal, a principal series, or a parabolically induced representation, and to determine the specific parameters of its constituents.

2. Methodology

The author employs a hybrid approach combining algebraic number theory, representation theory, and explicit computational algebra.

Cohomological Realization: The representation $\Pi$ is realized via the space of cuspidal cohomology classes $H^3_!(\Gamma_0(128), \mathbb{C})$ . Using an isomorphism described in [1], this space is identified with a space of functions on $P_2(\mathbb{Z}/128)$ satisfying specific symmetry and vanishing conditions.
Computer-Assisted Hecke Eigenvalue Computation:
- The author writes a computer program (available on GitHub) to compute the dimension of the cohomology space and to find a specific Hecke eigenvector $f$ corresponding to $\Pi$ .
- The eigenvector is uniquely identified by its eigenvalue under the Hecke operator $T_{\text{diag}(3,1,1)}$ , which is $1 + 2\sqrt{-1}$ .
- The author computes the eigenvalues of $f$ under several specific Hecke operators associated with matrices involving powers of 2 (e.g., $\text{diag}(2,2,1)$ , $\text{diag}(1,64,1)$ , etc.).
Representation Theoretic Analysis:
- Non-supercuspidality: The author uses the non-zero eigenvalue of a specific Hecke operator to prove $\Pi_2$ is not supercuspidal.
- Induction Structure: By analyzing the action of Hecke operators on the space of vectors fixed by specific compact open subgroups ( $K_{GL_3}$ and $K'_{GL_3}$ ), the author deduces that $\Pi_2$ must be a parabolically induced representation of the form $\text{Ind}_{P}^{GL_3(\mathbb{Q}_2)}(\pi \boxtimes \chi)$ , where $P$ is the standard parabolic with Levi factor $GL_2 \times GL_1$ .
- Explicit Determination of Constituents:
  - The character $\chi$ is determined to be unramified.
  - The supercuspidal representation $\pi$ of $GL_2(\mathbb{Q}_2)$ is analyzed using the theory of types (Bushnell-Kutzko theory). The author determines the specific element $\alpha \in M_2(\mathbb{Q}_2)$ and the character $\Lambda$ on the compact open subgroup $J_\alpha$ that define $\pi$ via compact induction ( $c\text{-Ind}$ ).
- Verification: The theoretical deductions are verified by matching the computed Hecke eigenvalues with the theoretical predictions derived from the proposed structure of $\Pi_2$ .

3. Key Contributions and Results

A. Structure of $\Pi_2$

The main theorem (Theorem 1.1) establishes that $\Pi_2$ is a parabolically induced representation:
$\Pi_2 = \text{Ind}_{P}^{GL_3(\mathbb{Q}_2)}(\pi \boxtimes \chi)$
where:

$P$ is the standard parabolic subgroup with Levi $GL_2 \times GL_1$ .
$\chi$ is an unramified character of $\mathbb{Q}_2^\times$ satisfying $\chi(2) = -2\sqrt{-1}$ .
$\pi$ is a supercuspidal representation of $GL_2(\mathbb{Q}_2)$ with conductor $2^7$ .

B. Explicit Description of $\pi$

The supercuspidal representation $\pi$ is explicitly constructed as a compactly induced representation:
$\pi = c\text{-Ind}_{J_\alpha}^{GL_2(\mathbb{Q}_2)} \Lambda$
The parameters are determined as follows:

The Element $\alpha$ : $\alpha = \frac{1}{8} \begin{pmatrix} 1 & -2 \\ 0 & 0 \end{pmatrix}$ (Note: The paper notation $\alpha = \frac{1}{8} \begin{pmatrix} 1 & -2 \end{pmatrix}$ implies a specific matrix form in $M_2(\mathbb{Q}_2)$ generating a quadratic extension $E = \mathbb{Q}_2(\sqrt{-2})$ ).
The Subgroup $J_\alpha$ : $J_\alpha = E^\times U^3_U$ , where $U^3_U = 1 + P^3$ (a specific congruence subgroup).
The Character $\Lambda$ : The character $\Lambda$ $Λ$ on $J_\alpha$ $J_{α}$ is explicitly defined by:
1. $\Lambda(\text{diag}(a, a)) = \chi^{-1}(a)$ for $a \in \mathbb{Q}_2^\times$ .
2. $\Lambda\left(\begin{smallmatrix} 1 & -2 \\ 0 & 1 \end{smallmatrix}\right) = \frac{1+\sqrt{-1}}{2}$ .
3. $\Lambda\left(\begin{smallmatrix} 1 & 1 \\ -2 & 1 \end{smallmatrix}\right) = \psi(1/2)$ , where $\psi$ is a specific additive character.
4. $\Lambda|_{U^3_U} = \psi_\alpha$ , a character derived from the trace of $\alpha$ .

C. Galois Representation Consequence

As a direct application, the paper provides an explicit description of the local component of the associated Galois representation $\rho_{vGT, \ell}$ at $p=2$ :
$\rho_{vGT, \ell}|_{W_{\mathbb{Q}_2}} \cong \rho_\pi \otimes |\cdot|^{1/2} \oplus \chi \otimes |\cdot|^{-1}$
where $\rho_\pi$ is the 2-dimensional representation of the Weil group corresponding to $\pi$ via the Local Langlands Correspondence for $GL_2$ .

4. Significance

Resolution of a Specific Case: This work provides the first explicit description of the local 2-component for this specific non-selfdual automorphic representation, filling a gap left by previous work which only handled odd primes.
Methodological Advancement: The paper demonstrates the power of combining explicit cohomological computations (using computer algebra) with deep representation-theoretic arguments (types and Hecke algebras) to solve problems in the local Langlands program where theoretical classification alone is insufficient.
Galois Representation Insights: By determining the local structure at 2, the paper clarifies the behavior of the associated Galois representation at the prime of bad reduction, which is essential for understanding the arithmetic properties of the underlying algebraic surface $S_2$ .
Verification of Conjectures: The results serve as a concrete test case for the compatibility between automorphic forms and Galois representations in the context of $GL_3$ over $\mathbb{Q}$ .

In summary, Yamamoto successfully decodes the complex local structure of a specific $GL_3$ automorphic representation at the prime 2, proving it is a parabolically induced representation and providing the exact parameters of its supercuspidal and character components through rigorous computation and theoretical deduction.