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The rationality problem for multinorm one tori, II

This paper investigates the stable and retract rationality of multinorm one tori associated with finite étale algebras, establishing stable rationality for degree greatest common divisor $d=1$ , providing criteria for retract rationality and conditions for its failure when $d>1$ , and generalizing previous results by Endo and Miyata to cases involving Galois extensions with cyclic Sylow subgroups or dihedral groups.

Original authors: Sumito Hasegawa, Kazuki Kanai, Yasuhiro Oki

Published 2026-04-06

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Sumito Hasegawa, Kazuki Kanai, Yasuhiro Oki

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 an architect trying to design a building. In the world of mathematics, specifically in a field called algebraic geometry, these "buildings" are shapes called varieties. Some of these shapes are very simple and easy to understand, like a perfect sphere or a flat sheet of paper. Mathematicians call these "rational" shapes because you can map them perfectly to a standard grid (like a projective space) without tearing or gluing.

However, many shapes are too complex to be perfectly mapped to a grid. So, mathematicians invented a hierarchy of "good enough" shapes:

Rational: Perfectly simple (the gold standard).
Stably Rational: You can't simplify it directly, but if you add a few extra dimensions (like adding a floor to a building), it suddenly becomes simple.
Retract Rational: It's a bit messier. You can't simplify it directly, but you can "retract" it (pull it back) from a simple shape without losing its essential identity.
Unirational: It's covered by a simple shape, but you might have to squish parts of it together.

The paper you provided, "The Rationality Problem for Multinorm One Tori, II," is about a specific type of mathematical building called a Multinorm One Torus.

What is a "Multinorm One Torus"?

Think of a Torus as a donut shape. In math, it's a shape with a specific kind of symmetry. A Multinorm One Torus is a complex donut built by combining several different "field extensions."

To visualize this, imagine you have several different keys (let's call them $K_1, K_2, \dots$ ). Each key opens a specific lock. A "Norm One Torus" is like a machine that only accepts keys where the "total weight" of the key is exactly 1. A Multinorm one is a machine that checks the weight of multiple keys at once and only accepts them if all of them have a weight of 1.

The authors are asking: Is this complex machine (the torus) simple enough to be considered "rational"?

The Main Character: The "Greatest Common Divisor" ( $d$ )

The authors discovered that the answer depends heavily on a single number: $d$ .
This number is the Greatest Common Divisor (GCD) of the sizes (degrees) of the keys involved.

Example: If you have keys of size 3 and size 5, the GCD is 1.
Example: If you have keys of size 4 and size 6, the GCD is 2.
Example: If you have keys of size 6 and size 9, the GCD is 3.

The paper splits the problem into two main scenarios based on this number $d$ .

Scenario 1: The "Easy" Case ( $d = 1$ )

The Analogy: Imagine you have a set of keys with sizes 3 and 5. They are "coprime" (they share no common factors other than 1).

The Result: The authors prove that if the keys have no common factor ( $d=1$ ), the resulting machine is Stably Rational.

Translation: Even though the machine looks complicated, if you add a little bit of extra space (a few extra dimensions), it becomes perfectly simple. It's like taking a twisted knot and realizing that if you just stretch it out a bit, it untangles into a straight line.

Scenario 2: The "Tricky" Case ( $d > 1$ )

The Analogy: Now imagine your keys are sizes 4 and 6. They share a common factor of 2. The machine is now "stuck" with a shared rhythm that makes it harder to untangle.

The Result: This is where the paper gets technical.

When it fails: The authors provide a checklist. If the keys are too similar in a specific way (for example, if the "Galois closures"—the ultimate versions of these keys—don't overlap in a specific pattern), the machine is NOT Retract Rational. It's too twisted to ever be simplified, even with extra space.
The "Dihedral" Twist: The paper also looks at a special group of symmetries called Dihedral groups (think of the symmetries of a regular polygon, like a hexagon: you can rotate it or flip it over).
- If the shared factor $d$ is odd, the machine is simple (Stably Rational).
- If the shared factor $d$ is even, it's a coin toss. It is simple if and only if it can be broken down into a specific, simpler combination of keys. The authors give a precise recipe for what that combination must look like.

The "Secret Weapon": Character Groups

How did they figure this out? They didn't just look at the donuts; they looked at the blueprints (called Character Groups).

The Metaphor: Imagine the donut is a musical instrument. The "Character Group" is the sheet music that tells you how the instrument vibrates.
Instead of trying to untangle the donut directly, the authors translated the problem into music theory. They asked: "Is this sheet music a simple, repeating pattern (Permutation)?"
If the music is a simple pattern, the donut is rational. If the music is a chaotic mess, the donut is not.

They used a technique called Coflabby Resolution.

The Metaphor: Imagine you have a messy pile of yarn (the complex shape). You want to know if it's a simple ball. You try to wrap it in a perfect, pre-wound ball of yarn (a "Permutation Lattice"). If you can wrap the messy yarn inside the perfect ball and have some perfect yarn left over, then your messy yarn was actually just a slightly twisted version of the perfect ball. This process of wrapping and unwrapping is the "Coflabby Resolution."

Why Does This Matter?

You might ask, "Who cares about rational donuts?"

Number Theory: These shapes are deeply connected to equations involving whole numbers (like finding integer solutions to complex equations). Understanding the shape helps solve these equations.
Classification: Mathematicians love to sort things. They want to know: "What does the universe of all possible shapes look like?" This paper helps fill in the map, telling us exactly which shapes are "simple" and which are "chaotic."
Building on History: This paper is the sequel ("Part II") to previous work. It takes the rules established by giants in the field (like Endo and Miyata) and applies them to more complex, multi-key scenarios that hadn't been solved before.

Summary in One Sentence

The authors figured out that if you combine several mathematical "keys" to build a complex shape, the shape is easy to understand (rational) if the keys don't share a common factor, but if they do share a factor, the shape is usually a tangled mess—unless the keys follow a very specific, strict pattern of symmetry.

1. Problem Statement

The paper investigates the stable rationality and retract rationality of multinorm one tori.

Context: Let $k$ be a field and $K = \prod_{i=1}^r K_i$ be a finite étale algebra over $k$ (a product of finite separable field extensions). The multinorm one torus $T_{K/k}$ is defined as the kernel of the norm map from the Weil restriction of scalars:
$T_{K/k} := \text{Ker}\left( N_{K/k} : \text{Res}_{K/k} \mathbb{G}_m \to \mathbb{G}_m \right).$
Goal: Determine conditions under which $T_{K/k}$ $T_{K / k}$ is:
- Stably rational: Birational to a projective space after a product with some $\mathbb{P}^m$ .
- Retract rational: A retract of a rational variety (a weaker condition than stable rationality).
Key Parameter: The authors organize their results based on the greatest common divisor of the degrees of the factors:
$d_k(K) := \gcd([K_1:k], \dots, [K_r:k]).$
The behavior of the torus differs significantly depending on whether $d_k(K) = 1$ or $d_k(K) > 1$ .

2. Methodology

The paper employs the standard algebraic approach to the rationality of tori, translating geometric properties into the language of Galois lattices.

Character Groups: For a torus $T$ $T$ splitting over a Galois extension $L/k$ $L / k$ with group $G = \text{Gal}(L/k)$ $G = Gal (L / k)$ , the rationality of $T$ $T$ is determined by the properties of its character module $X^*(T)$ $X^{*} (T)$ , a $\mathbb{Z}[G]$ $Z [G]$ -lattice.
- $T$ is stably rational $\iff$ $X^*(T)$ is quasi-permutation.
- $T$ is retract rational $\iff$ $X^*(T)$ is quasi-invertible.
Lattice Construction: The character group of the multinorm one torus is identified with a specific lattice $J_{G/H}$ (the dual of the kernel of a sum of augmentation maps), where $H = \{ \text{Gal}(L/K_i) \}_{i=1}^r$ is a multiset of subgroups of $G$ .
Reduction Techniques:
- Sylow Subgroups: Using theorems by Endo and Miyata, the problem is often reduced to checking properties over $p$ -Sylow subgroups of $G$ .
- Weight Functions: The authors utilize "weight functions" on multisets of subgroups to simplify the lattice structure (removing redundant subgroups via reduction to "strongly reduced" sets).
- Coflabby Resolutions: To prove stable rationality in complex cases (specifically for dihedral groups), the authors construct explicit coflabby resolutions of the relevant lattices. A lattice is stably permutation if it fits into a specific exact sequence involving permutation lattices and coflabby kernels.

3. Key Contributions and Results

A. The Case $d_k(K) = 1$

Theorem 1.1: If $d_k(K) = 1$ $d_{k} (K) = 1$ (and $r \ge 2$ $r \geq 2$ ), the multinorm one torus $T_{K/k}$ $T_{K / k}$ is stably rational over $k$ $k$ .
- Proof Sketch: The authors show that the associated lattice $J_{G/H} \oplus \mathbb{Z}$ is a permutation lattice. This is achieved by finding integer coefficients that sum the indices of the subgroups to 1, splitting the defining exact sequence.
Theorem 1.8 (Criterion for Retract Rationality): The torus is retract rational over $k$ if and only if it is retract rational over $L^{(p)}$ for every prime divisor $p$ of $d_k(K)$ , where $L^{(p)}$ is a maximal subextension of degree coprime to $p$ . This reduces the general problem to cases where the degree is a prime power.

B. The Case $d_k(K) > 1$ (Failure of Rationality)

The authors provide sufficient conditions for the failure of retract rationality.

Theorem 1.2 ( $d_k(K) = 2$ ):
- If $r=2$ and specific intersection conditions hold (e.g., $L_1 \cap K_2 = k$ ) with degrees divisible by 4, the torus is not retract rational.
- If $r \ge 3$ and there exists a subset of 3 indices where pairwise intersections are trivial, the torus is not retract rational.
Theorem 1.3 ( $d_k(K) \ge 3$ ): If there exist $i \neq j$ $i \neq = j$ such that $L_i \cap K_j = k$ $L_{i} \cap K_{j} = k$ , the torus is not retract rational.
- Significance: These results generalize previous work by Demarche and Wei, showing that the failure of rationality occurs under much weaker intersection assumptions when $d_k(K) \ge 3$ .

C. Generalization of Endo–Miyata and Endo Results

The paper extends classical results regarding tori splitting over Galois extensions with specific group structures.

Theorem 1.5 (Cyclic Sylow Subgroups): Assuming all Sylow subgroups of the Galois group $G$ $G$ are cyclic, the authors give a necessary and sufficient condition for stable rationality.
- $T_{K/k}$ is stably rational $\iff$ $D_k(K) \cap P^*(G) = \emptyset$ .
- Here, $P^*(G)$ is the set of primes $p$ where the $p$ -Sylow subgroup is normal and the conjugation action has an image of order $\ge 3$ .
Theorem 1.7 (Dihedral Galois Groups): Assuming $G \cong D_n$ $G ≅ D_{n}$ (dihedral group of order $2n$ $2 n$ ):
- If $d_k(K)$ is odd, $T_{K/k}$ is stably rational.
- If $d_k(K)$ is even, stable rationality is equivalent to retract rationality. The authors provide a precise classification of when this holds based on the existence of specific intermediate fields $K'$ and the structure of their Galois groups (involving subgroups generated by $\sigma$ and $\tau$ ).

4. Technical Highlights

Section 7 (Coflabby Resolution): This is the most technical part of the paper. To prove the stable rationality of tori associated with dihedral groups (Theorem 1.7, case (c)), the authors explicitly construct a coflabby resolution for the lattice $J_{D_{2m}/H_m}$ . They define specific homomorphisms and kernels ( $E_m$ ) to show that the lattice is quasi-permutation. This involves intricate calculations with group rings and specific generators of dihedral groups.
Reduction to Field Case: For the case where $K$ is not a field (i.e., $r \ge 2$ ), the authors use techniques from their previous work ([HKO]) to reduce the problem to the case where $K$ is a field, leveraging the structure of the Galois group and the properties of the weight functions.

5. Significance

Completeness: The paper provides a comprehensive classification of multinorm one tori based on the GCD of the degrees of the factors ( $d_k(K)$ ).
Generalization: It significantly generalizes the work of Endo, Miyata, and Endo (2011) from the case where $K$ is a single field to the case where $K$ is a product of fields (étale algebras).
New Criteria: It establishes new, weaker conditions for the failure of retract rationality (Theorems 1.2 and 1.3) and provides a unified criterion for stable rationality in the presence of cyclic or dihedral Galois groups.
Methodological Advancement: The explicit construction of coflabby resolutions for dihedral lattices (Section 7) offers a powerful tool for analyzing rationality problems in non-abelian Galois settings.

In summary, this paper resolves the rationality problem for multinorm one tori in a wide range of cases, distinguishing clearly between the "easy" case ( $d_k(K)=1$ ) and the "hard" cases ( $d_k(K)>1$ ), and providing sharp criteria for when these tori fail to be rational.