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On p-Jordan constant of Cremona group of rank 2 in odd characteristic

This paper establishes bounds on the indices of normal abelian subgroups within finite subgroups of the Cremona group of rank 2 over a field of odd characteristic.

Original authors: Yifei Chen, Constantin Shramov

Published 2026-03-31

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CC BY 4.0

Original authors: Yifei Chen, Constantin Shramov

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 the "shape-shifting" powers of a geometric universe. This paper is about a specific type of universe: a Projective Plane (think of it as a flat, infinite sheet where parallel lines meet at the horizon, and you can stretch, twist, and bend it in wild ways).

The group of all possible ways to rearrange this plane without tearing it is called the Cremona Group. It's a chaotic, infinite monster of a group. The mathematicians in this paper, Yifei Chen and Constantin Shramov, are trying to answer a very specific question: "If you find a small, finite team of shape-shifters inside this chaotic monster, how 'organized' are they?"

The Core Concept: The "Jordan Property"

To understand their discovery, let's use an analogy.

Imagine a massive, chaotic dance floor (the Cremona Group). Inside, there are many small dance troupes (finite subgroups).

The Old Rule (Jordan's Theorem): In a calm, zero-gravity universe (characteristic zero), every dance troupe, no matter how wild, must contain a "core group" of dancers who are all holding hands in a perfect circle (a normal abelian subgroup). The size of the troupe compared to this core circle is always limited by a specific number.
The New Problem (Positive Characteristic): Now, imagine the dance floor is on a bumpy, vibrating planet (a field with "odd characteristic," like a specific type of math universe). The old rule breaks down. The troupes can get messy. However, the authors prove a weaker but still powerful rule: Even in this bumpy universe, every troupe still has a "core circle" of organized dancers, provided we ignore the dancers who are vibrating in sync with the planet's rhythm (the $p$ -part of the group).

The Main Discovery: The "Safety Net"

The authors prove that for any finite troupe in this bumpy universe, you can always find a "safe zone" (a normal abelian subgroup) that is:

Stable: It doesn't get shaken apart by the rest of the troupe.
Coprime to the Chaos: It doesn't include the dancers vibrating with the planet's rhythm.
Bounded: The size of the whole troupe compared to this safe zone is limited by a formula: $J \times (\text{Chaos Size})^3$ .

Think of it like a safety net. If the "Chaos Size" (the part of the group vibrating with the planet) gets huge, the safe zone might get smaller relative to the whole, but the authors found the exact maximum multiplier ( $J$ ) needed to guarantee the net exists.

The "Magic Numbers" ( $J$ )

The paper calculates the specific "safety factor" ( $J$ ) needed for different types of planets (different prime numbers $p$ ):

If the planet is "7 or higher" ( $p \ge 7$ ): The safety factor is 7,200. This means the troupe can be up to 7,200 times larger than its core circle.
If the planet is "5" ( $p = 5$ ): The safety factor drops to 168.
If the planet is "3" ( $p = 3$ ): The safety factor is just 10.

Why the difference?
Think of the number 3 as a very "sticky" or "resonant" frequency. In this specific math universe, the vibrations are so strong that the groups can't get as messy as they can in the "7 or higher" universes. The rules are tighter, so the safety factor is smaller.

How Did They Solve It? (The Detective Work)

The authors didn't just guess these numbers. They broke the problem down into three main "crime scenes" (geometric objects) where these dance troupes usually hang out:

The Line ( $P^1$ ): The simplest stage. They classified every possible troupe here.
The Plane ( $P^2$ ): The main stage. They looked at how troupes move points and lines around.
The "Exotic" Surfaces (Del Pezzo and Conic Bundles): These are like twisted, folded versions of the plane. The authors had to check if the troupes behaved differently on these folded surfaces.

They used a "divide and conquer" strategy. They looked at the "maximal" troupes (the biggest, most complex ones) and checked if they had a core circle. If the biggest ones had a core, then all the smaller ones inside them did too.

The "Sharpness" of the Result

The paper doesn't just give a loose estimate; it proves these numbers are tight.

They found specific examples of dance troupes where the core circle is exactly as small as their formula predicts.
For example, in the "7 or higher" universe, they found a troupe of 7,200 people that has no non-trivial core circle at all (the core is just one person). This proves you can't lower the safety factor from 7,200 to 7,199.

Why Does This Matter?

In the world of mathematics, knowing the limits of chaos is crucial.

Classification: It helps mathematicians list and categorize all possible symmetries in these geometric worlds.
Predictability: It tells us that even in the most chaotic, vibrating mathematical universes, there is always a hidden layer of order (the abelian subgroup) that we can rely on.
The "Odd" Characteristic: Most previous work focused on "calm" universes (characteristic zero). This paper is a major step forward in understanding the "bumpy" universes (positive characteristic), specifically showing that while they are messier, they aren't too messy.

Summary in a Nutshell

Chen and Shramov proved that in a specific type of chaotic geometric universe (odd characteristic), every finite group of shape-shifters contains a stable, organized core. They calculated the exact "worst-case scenario" ratio between the chaos and the order for different types of universes, showing that the universe is chaotic, but predictably so. They also showed that their predictions are the best possible ones, as nature (mathematics) provides examples that hit these limits exactly.

1. Problem Statement

The paper addresses the Jordan property for the Cremona group of rank 2, denoted as $\text{Cr}_2(k)$ , which is the group of birational automorphisms of the projective plane $\mathbb{P}^2$ over a field $k$ .

Classical Context: In characteristic zero, Jordan's Theorem states that any finite subgroup of $\text{Cr}_2(k)$ contains a normal abelian subgroup of bounded index (independent of the group size).
Positive Characteristic Challenge: In positive characteristic $p$ , the classical Jordan property fails because finite subgroups can contain arbitrarily large $p$ -subgroups (Sylow $p$ -subgroups) without large normal abelian subgroups.
The $p$ -Jordan Property: The authors investigate the $p$ -Jordan property, defined as follows: A group $\Gamma$ is $p$ -Jordan if there exist constants $J(\Gamma)$ and $e(\Gamma)$ such that every finite subgroup $G \subset \Gamma$ contains a normal abelian subgroup $A$ of order coprime to $p$ with index:
$[G:A] \leq J(\Gamma) \cdot |G(p)|^{e(\Gamma)}$
where $G(p)$ is the Sylow $p$ -subgroup of $G$ .
Specific Goal: The paper aims to determine the optimal constants $J_p(\text{Cr}_2)$ and the exponent $e$ for the Cremona group of rank 2 over fields of odd characteristic ( $p \neq 2$ ). Previous work established that $e=3$ is sufficient, but the precise value of the constant $J$ was unknown and depended on $p$ .

2. Methodology

The authors employ a geometric reduction strategy combined with detailed finite group theory.

A. Geometric Reduction

The problem of classifying finite subgroups of $\text{Cr}_2(k)$ is reduced to studying finite groups acting on specific geometric objects via the Minimal Model Program (MMP):

Regularization: Any finite subgroup $G \subset \text{Cr}_2(k)$ can be realized as a group of automorphisms of a smooth rational projective surface $S$ .
MMP Reduction: By running a $G$ $G$ -equivariant MMP, $S$ $S$ is birationally equivalent to either:
- A Del Pezzo surface (of degree $1 \leq d \leq 9$ ).
- A Conic bundle over $\mathbb{P}^1$ .
Case Analysis: The authors analyze the automorphism groups of these surfaces case-by-case based on their degree and the characteristic $p$ .

B. Group-Theoretic Tools

The core of the proof involves bounding the indices of normal abelian subgroups in various extensions and semi-direct products. Key tools include:

Classification of Subgroups: Utilizing classifications of finite subgroups of $\text{PGL}_2(k)$ (acting on $\mathbb{P}^1$ ) and $\text{PGL}_3(k)$ (acting on $\mathbb{P}^2$ ) in odd characteristic.
Extension Lemmas: Developing lemmas (e.g., Lemmas 3.4, 6.1–6.8) to bound indices in groups fitting into exact sequences $1 \to H \to G \to \bar{G} \to 1$ , where $H$ is abelian or a specific finite group (like $A_5$ , $S_4$ , etc.).
Characteristic Subgroups: Proving the existence of characteristic abelian subgroups to ensure normality in the larger group context.
Handling Exceptions: Specifically addressing cases where standard bounds fail (e.g., specific semi-direct products involving $\mu_5$ or Heisenberg groups) by using geometric constraints (e.g., non-existence of certain actions on specific surfaces).

3. Key Contributions and Results

A. The Main Theorem (Theorem 1.9)

The paper establishes the precise $p$ -Jordan constant for $\text{Cr}_2(k)$ in odd characteristic:

Theorem: For every field $k$ of odd characteristic $p$ , every finite subgroup $G \subset \text{Cr}_2(k)$ contains a normal abelian subgroup of order coprime to $p$ and index at most:
$[G:A] \leq J_p(\text{Cr}_2) \cdot |G(p)|^3$
where the constant $J_p(\text{Cr}_2)$ is:

$J_p = 7200$ if $p \geq 7$ .
$J_p = 168$ if $p = 5$ .
$J_p = 10$ if $p = 3$ .

Furthermore, the exponent $e=3$ is optimal; no bound exists with $e < 3$ .

B. Sharpness of the Bounds

The authors prove these bounds are sharp when $k$ is algebraically closed by constructing specific counter-examples for any smaller constant:

For $p \geq 7$ : The group $G = (A_5 \times A_5) \rtimes \mu_2$ acting on $\mathbb{P}^1 \times \mathbb{P}^1$ (embedded in $\text{Cr}_2$ ) has no non-trivial normal abelian subgroups. Its order is $7200$, and since $|G(p)|=1$ , the index is exactly $7200$.
For $p = 5$ : The group $G = \text{PSL}_2(\mathbb{F}_7)$ acts on $\mathbb{P}^2$ . It is simple, so the only normal abelian subgroup is trivial. The index is $168$.
For $p = 3$ : A specific group $G \cong \mu_2^4 \rtimes D_{10}$ acting on a Del Pezzo surface of degree 4 yields a minimal index of $10$.

C. Intermediate Results

Projective Plane ( $\mathbb{P}^2$ ): The authors refine the bounds for finite subgroups of $\text{PGL}_3(k)$ in odd characteristic, distinguishing cases based on whether the group preserves a point, line, triangle, or conic.
Del Pezzo Surfaces: They provide explicit bounds for automorphism groups of Del Pezzo surfaces of all degrees ( $d=1$ to $9$), avoiding reliance on the full classification of automorphism groups in positive characteristic by using geometric properties (e.g., anticanonical linear systems).
Conic Bundles: They establish bounds for groups acting on conic bundles, reducing the problem to subgroups of $\text{PGL}_2(k) \times \text{PGL}_2(k)$ and their semi-direct products.

4. Significance

Precision in Positive Characteristic: While the $p$ -Jordan property was known to hold for $\text{Cr}_2(k)$ , this paper provides the exact optimal constants for odd characteristics. This moves the field from qualitative existence to quantitative precision.
Resolution of the Exponent: The paper confirms that the exponent $e=3$ is the best possible, matching the behavior of $\text{GL}_n(k)$ and $\text{PGL}_3(k)$ , and demonstrating that the Cremona group does not require a higher exponent.
Methodological Advancement: The paper introduces a robust framework for analyzing finite subgroups of birational automorphism groups by combining the Minimal Model Program with intricate group-theoretic estimates. This approach avoids the need for the full, complex classification of automorphism groups of Del Pezzo surfaces in positive characteristic.
Limitations and Future Work: The authors explicitly note that the case of characteristic 2 remains open. The lack of a complete classification of finite subgroups of $\text{PGL}_3(k)$ in characteristic 2 prevents them from determining the exact constant $J_2$ , though they conjecture it to be 3 based on preliminary data.

Conclusion

Chen and Shramov successfully characterize the $p$ -Jordan property for the rank 2 Cremona group in odd characteristic. They prove that the index of the normal abelian subgroup is bounded by $J_p \cdot |G(p)|^3$ , with $J_p$ taking specific values (7200, 168, 10) depending on the prime $p$ . These bounds are shown to be sharp, providing a complete answer to the problem for all odd characteristics.