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Automorphic functions for square-zero extensions of curves over finite fields

This paper generalizes results on the cuspidality and Hecke-finiteness of automorphic functions for square-zero extensions of curves over finite fields from $\mathrm{PGL}_2$ to arbitrary split connected reductive groups, proving a new case of a finite-dimensionality conjecture for $\mathrm{PGL}_3$ and establishing optimal support bounds via Harder-Narasimhan stratifications and twisted $G$ -Higgs bundles.

Original authors: Ka Fai Wong

Published 2026-08-04

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CC BY 4.0

Original authors: Ka Fai Wong

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 the universe of mathematics as a vast, invisible city built on patterns. In one neighborhood called "Number Theory," mathematicians study the hidden rhythms of numbers, while in another called "Geometry," they map the shapes of spaces. For decades, a brilliant bridge has connected these two neighborhoods: the Langlands Program. Think of it as a universal translator that claims if you know the secret code of a shape (geometry), you can instantly decode the behavior of numbers (arithmetic), and vice versa. Usually, this translator works on "smooth" shapes, like perfect spheres or gentle hills. But what happens if the shape gets a little wobbly? What if it's a shape that has been squashed or stretched in a very specific, tiny way, creating a "square-zero extension"? This is a bit like taking a perfect clay model and pressing a tiny, invisible dimple into it. The rules of the game change slightly, and the old translator might get confused.

The big question mathematicians are asking is: Can we still find the "perfect" patterns (called automorphic functions) in these wobbly, dimpled shapes? Specifically, they want to know if these patterns are finite in number and if they follow strict rules, or if they spiral out of control into infinity. This isn't just a puzzle for its own sake; solving it helps us understand the deep, fundamental structure of how numbers and shapes talk to each other, potentially unlocking secrets about the very fabric of mathematics.

Enter the work of Ka Fai Wong, who decided to tackle this problem for a specific, slightly more complex shape: a curve with a tiny dimple, but this time for a group of symmetries called $PGL_3$ (which is like a 3D version of the symmetries you see in a triangle, but much more complex). Wong's paper is like a master detective breaking down a massive, chaotic crime scene into manageable clues.

The paper's main finding is that for this specific 3D symmetry group ( $PGL_3$ ), the "perfect" patterns do exist, and they are finite in number. Wong proved that if a pattern is "Hecke-finite" (meaning it doesn't run off into an infinite loop of variations), it is automatically "cuspidal" (meaning it's a pure, fundamental pattern that doesn't just come from simpler, broken-down pieces). This confirms a major guess made by other mathematicians for this specific case. Furthermore, Wong didn't just prove they exist; he drew a map showing exactly where these patterns can live. He showed that these patterns are confined to a specific region of the shape's geometry, bounded by a number related to the shape's complexity (specifically, the bound is 3 in a certain mathematical scale).

However, the paper also rules out some possibilities. It explicitly shows that for certain types of "mixed" patterns (where the shape has both a stable part and a wobbly part), the patterns are never finite; they always spiral into infinity. If you try to find a finite pattern in these mixed zones, you will find nothing but an endless, chaotic mess. The paper also clarifies that while we know the patterns are finite for this specific group, the exact "bound" (the size of the map) for other, more complex groups is still a mystery.

Wong's confidence in these results is high because they are mathematically proven, not just guessed or simulated. He used a clever technique called "orbit decomposition," which is like sorting a messy pile of laundry by separating socks from shirts, then shirts from pants, to see which items are actually unique. By sorting the patterns this way, he could prove that the "socks" (the finite, fundamental patterns) are indeed finite in number and stay within the boundaries he calculated. For the "shirts" (the infinite, chaotic patterns), he proved they are indeed infinite.

The paper also introduces a new way to measure how "wobbly" these patterns can get. Imagine the shape is a landscape, and the patterns are hikers. Wong calculated that for the $PGL_3$ case, the hikers can't wander too far from the center; they are restricted to a zone where the difference in their elevation is limited by a specific number (3). This is a significant step forward because, in the past, mathematicians could only guess these limits. Now, for this specific case, they have a precise, proven boundary.

In short, this paper takes a chaotic, wobbly mathematical world and shows that even there, order exists. It proves that for a specific type of symmetry, the "good" patterns are finite, predictable, and confined to a known territory, while the "bad" patterns are infinite and chaotic. It's a victory for the idea that even in the most twisted corners of math, there are rules waiting to be found.

Technical Summary: Automorphic Functions for Square-Zero Extensions of Curves over Finite Fields

Problem Statement
This paper investigates the theory of automorphic functions for square-zero extensions $C$ of smooth projective curves $\bar{C}$ over finite fields. This setting, initiated by Braverman, Kazhdan, and Polishchuk in [BKP23], serves as a bridge between the classical Langlands program over finite fields and the analytic Langlands program over local fields. Specifically, the author studies the space of unramified automorphic functions $S(G(F)\backslash G(A))$ for a split connected reductive group $G$ , focusing on two central properties:

Cuspidality vs. Hecke-finiteness: The conjecture (Braverman-Kazhdan [BK23]) that an unramified function is cuspidal if and only if it is Hecke-finite (i.e., generates a finite-dimensional Hecke orbit).
Finite-dimensionality: The conjecture that the space of unramified Hecke-finite functions is finite-dimensional when $G$ is semisimple.
Support Bounds: Determining the optimal bounds for the support of these functions in terms of the Harder-Narasimhan stratification of $G$ -bundles on the reduced curve $\bar{C}$ .

While the case $G = \text{PGL}_2$ was well-understood in [BKP23], this work aims to generalize these results to higher-rank groups, with a specific focus on $G = \text{PGL}_3$ .

Methodology
The paper employs a combination of representation-theoretic constructions and geometric interpretations:

Global Mackey Theory and Orbit Decomposition:
The core technical tool is a "global Mackey theory" adapted from [BKP23]. The automorphic representation $S(G(F)\backslash G(A))$ is decomposed into a direct sum of subrepresentations $S_\Omega$ indexed by coadjoint orbits $\Omega$ of the Lie algebra $\mathfrak{g} \otimes N^{-1}\omega_{\bar{C}}(F)$ .
- The author generalizes the machinery of [BKP23] to handle more complex nilpotent orbits beyond the regular and subregular cases found in $\text{PGL}_2$ .
- For each orbit $\Omega$ , the representation is analyzed to determine if it contains cuspidal or Hecke-finite functions. This involves studying the stabilizer groups $St_\eta$ of elements $\eta \in \Omega$ and the induced representations from these stabilizers.
Geometric Interpretation via Hitchin Fibers:
To compute the support bounds, the paper utilizes a geometric interpretation of the representation-theoretic construction. Following [KP25], the subrepresentations $S_\eta$ are identified with spaces of functions on Hitchin fibers (specifically, $N^{-1}$ -twisted $G$ -Higgs bundles).
- The embedding of these spaces into the space of automorphic functions is realized via a Fourier transform.
- This allows the author to translate algebraic conditions on the orbits (e.g., properties of characters on stabilizer groups) into geometric constraints on the underlying $G$ -bundles on $\bar{C}$ .
Harder-Narasimhan Stratification:
The support of the functions is analyzed using the Harder-Narasimhan stratification of $\text{Bun}_G(\bar{C})$ . The author introduces a formulation of support bounds using the highest roots of $G$ rather than just simple roots, which is essential for capturing optimal bounds in the square-zero setting.

Key Contributions and Results

Generalization of Orbit Decomposition: The paper provides a detailed classification of cuspidality and Hecke-finiteness for various orbits in general split reductive groups:
- Almost Regular Elliptic Orbits: The direct sum over these orbits constitutes the space of strongly cuspidal functions. For semisimple $G$ , the space of spherical vectors in this sum is finite-dimensional.
- Non-Elliptic Semisimple/Mixed Orbits: Functions in these orbits are shown to be Hecke-infinite.
- Regular Nilpotent Orbits: For semisimple groups of adjoint type (with simple factors of types A, B, C, or $G_2$ ), the space decomposes based on characters of the highest root subgroup. Non-trivial characters yield cuspidal, finite-dimensional spaces of spherical vectors, while trivial characters yield Hecke-infinite spaces.
- Subregular Nilpotent Orbits: For $G = \text{PGL}_n$ , the subregular orbit is related to an extension of $\mathbb{G}_m$ by a Heisenberg group. The author explicitly decomposes this into Hecke-finite and Hecke-infinite components, proving the finite-dimensionality of the spherical Hecke-finite subspace.
Proof for $\text{PGL}_3$ :
The main theorem establishes the following for $G = \text{PGL}_3$ (assuming $\text{char}(k) > 3$ ):
1. The space of unramified Hecke-finite functions is finite-dimensional.
2. Every unramified Hecke-finite function is cuspidal (confirming Conjecture 1.3(1) for this case).
3. The support of these functions is contained in $r^{-1}(U_3)$ , where $U_3$ is a specific stratum in the Harder-Narasimhan filtration defined by the Coxeter number and genus.
Optimal Support Bounds:
The paper determines the optimal constant $b_h(C, G)$ for the support bound.
- For the regular nilpotent orbit, the bound is shown to be sharp at $c_G - 1$ (where $c_G$ is the Coxeter number).
- For the subregular nilpotent orbit in $\text{PGL}_n$ , the bound is sharp at $2n - 3$ .
- Combining these, the author proves that for $\text{PGL}_3$ , the optimal bound is exactly 3. This contrasts with the classical theory where the exact value of the bound is often unknown.

Significance and Claims
The paper claims to extend the foundational work of Braverman-Kazhdan-Polishchuk from $\text{PGL}_2$ to higher-rank groups, specifically resolving the conjectures regarding finite-dimensionality and the equivalence of cuspidality and Hecke-finiteness for $\text{PGL}_3$ .

A significant contribution is the determination of optimal support bounds. The author notes that in the square-zero extension setting, representation-theoretic constructions allow for the exact determination of these bounds (e.g., proving the bound $n-1$ for regular orbits is sharp), whereas in the classical theory over finite fields, the exact value remains conjectural. The paper posits that the invariant $b_h(C, G)$ (related to Hecke-finiteness) is more accessible than its conjecturally equivalent counterpart $b_c(C, G)$ (related to cuspidality) because the orbit decomposition provides finer control over the support of Hecke-finite functions.

The work also introduces a formulation of support bounds using the highest roots of $G$ , arguing that this is essential for capturing the optimal bounds in the square-zero setting, unlike the classical formulation which relies on simple roots.

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