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Rationality problem for norm one tori of tensor products of étale algebras and Hasse norm principle

This paper establishes that if the degrees of two étale algebras are coprime and their respective norm one tori are stably or retract rational, then the norm one torus of their tensor product shares these rationality properties, thereby ensuring the Hasse norm principle holds for the resulting extension over a global field.

Original authors: Mathieu Florence, Akinari Hoshi, Aiichi Yamasaki

Published 2026-05-19

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Mathieu Florence, Akinari Hoshi, Aiichi Yamasaki

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 master builder working in a world made of invisible, mathematical shapes called algebraic tori. These aren't donuts you can eat; they are complex, multi-dimensional structures defined by rules of symmetry and numbers.

This paper, written by Florence, Hoshi, and Yamasaki, is like a new rulebook for how these shapes behave when you smash them together.

The Core Concept: The "Shape-Shifting" Blocks

Think of an algebraic torus as a special kind of Lego block.

Some blocks are simple and easy to understand (we call these rational).
Some are tricky but can be turned into simple blocks if you add a few extra "helper" pieces (we call these stably rational).
Some are even more stubborn; they can't be simplified easily, but they can be "retracted" or pulled back into a simple shape under specific conditions (we call these retract rational).

The authors are interested in a specific type of block called a Norm One Torus. You can think of these as "balance scales." They represent a rule where the total "weight" (or norm) of a collection of numbers must equal exactly 1.

The Big Question: What happens when you combine them?

The central puzzle the authors solve is this:

"If I have two complex blocks, Block A and Block B, and I know that Block A is 'good' (it can be simplified) and Block B is also 'good,' what happens if I fuse them together into a giant new block, Block A + Block B?"

In the real world, if you glue two complex machines together, the result is usually a mess. But in this mathematical world, the authors found a magical condition where the result stays "good."

The Magic Condition: The "Coprime" Rule

The paper discovers that the fusion works beautifully if the sizes of the two blocks are "coprime."

Analogy: Imagine Block A is made of 3 distinct layers, and Block B is made of 4 distinct layers. The numbers 3 and 4 have no common factors (they are coprime).
The Result: If you glue a 3-layer block to a 4-layer block, the resulting 12-layer block is still "good" (it remains stably or retract rational).
The Failure: If you try to glue a 3-layer block to another 3-layer block (both divisible by 3), the result is a mess. The new block becomes "bad" (it loses its nice properties).

The authors prove that as long as the "layer counts" (degrees of the field extensions) share no common factors, the nice properties of the original blocks are preserved in the new, combined block.

The "Hasse Norm Principle": The Global Check

The paper also tackles a famous problem in number theory called the Hasse Norm Principle.

The Metaphor: Imagine you are trying to verify if a secret code exists. You can't check the whole code at once. Instead, you check it in many different local neighborhoods (local fields).
The Rule: The Hasse Norm Principle asks: "If the code works in every single local neighborhood, does it work for the whole global system?"
The Finding: The authors show that for these specific combined blocks (where the sizes are coprime), the answer is YES. If the rule holds locally everywhere, it holds globally. This is a huge relief for mathematicians because it means they don't have to check the impossible "global" case directly; checking the local pieces is enough.

Why This Matters (According to the Paper)

New Examples: Before this paper, we knew of some "good" blocks, but they were rare. This paper provides a recipe to build many new examples of "good" blocks by combining smaller, known "good" blocks, provided their sizes don't share factors.
Solving the Puzzle: It gives a clear, constructive method (using something called "flabby resolutions," which is like a specific way of unwrapping the blocks) to prove that these new combinations work.
Global Fields: When the base numbers are "global fields" (like the rational numbers or function fields), this work guarantees that the Hasse Norm Principle holds for these new, complex combinations.

Summary

In simple terms, the authors found a compatibility rule for mathematical shapes.

Input: Two "nice" shapes.
Condition: Their sizes must not share any common divisors (like 3 and 4, but not 3 and 6).
Output: The combined shape is also "nice," and it obeys a famous global consistency rule (Hasse Norm Principle).

If you try to combine shapes with shared factors, the magic breaks, and the result is messy. But if you follow the "coprime" rule, the universe of these mathematical shapes stays orderly and predictable.

Technical Summary: Rationality of Norm One Tori of Tensor Products of Étale Algebras and the Hasse Norm Principle

1. Problem Statement
The paper addresses two interconnected problems in the arithmetic geometry of algebraic tori over a field $k$ :

The Rationality Problem: Given two algebraic $k$ -tori $T_1$ and $T_2$ possessing a specific rationality property $P$ (such as stable $k$ -rationality or retract $k$ -rationality), does their tensor product $T_1 \otimes_k T_2$ also possess property $P$ ? Specifically, the authors investigate the norm one torus $T_{A \otimes B} = R^{(1)}_{A \otimes B/k}(\mathbb{G}_m)$ associated with the tensor product of two étale $k$ -algebras $A$ and $B$ .
The Hasse Norm Principle: For global fields $k$ , does the Hasse norm principle hold for the field extension $K/k$ (or the étale algebra $A/k$ ) constructed as a tensor product of extensions of coprime degrees?

While the general answer to the rationality question is negative, the authors identify specific conditions under which the property is preserved, thereby generating new examples of stably and retract rational tori and establishing the validity of the Hasse norm principle for these cases.

2. Methodology
The authors employ a constructive approach based on the cohomology of finite groups and the theory of $G$ -lattices (where $G$ is the Galois group of the splitting field). The core methodology involves:

Flabby Resolutions: Utilizing the correspondence between algebraic tori and $G$ -lattices, the rationality of a torus $T$ is determined by the "flabby class" $[M]_{fl}$ of its character module $M$ . $T$ is stably rational if $[M]_{fl} = 0$ and retract rational if $[M]_{fl}$ is invertible.
Permutation Order ( $p$ -ord): The paper introduces a new invariant for a $G$ -lattice $M$ , denoted $p\text{-ord}(M)$ . This is defined as the smallest integer $a \geq 1$ such that $a \cdot \text{Id}_M$ factors through a permutation $G$ -lattice. This invariant serves as a measure of how far a lattice is from being a direct summand of a permutation lattice.
Tensor Products of Extensions: The authors develop explicit cohomological tools to handle tensor products of extensions of $G$ -lattices. By analyzing the exact sequences associated with augmentation maps (e.g., $0 \to I_X \to \mathbb{Z}[X] \to \mathbb{Z} \to 0$ ), they construct resolutions for tensor products of lattices.
Coprimality Conditions: The central technical mechanism relies on the condition that the permutation orders of the character modules of the constituent tori are coprime. This allows the splitting of certain exact sequences involving tensor products, ensuring that the flabby class of the product behaves predictably.

3. Key Contributions and Results

Main Theorem on Tori Rationality (Theorem 6.15):
Let $T_1$ and $T_2$ be algebraic $k$ -tori with character modules $M_1$ and $M_2$ . If the permutation orders $d_1 = p\text{-ord}(M_1)$ and $d_2 = p\text{-ord}(M_2)$ satisfy $\gcd(d_1, d_2) = 1$ , then:
- If $T_1$ and $T_2$ are stably $k$ -rational, then $T_1 \otimes_k T_2$ is stably $k$ -rational.
- If $T_1$ and $T_2$ are retract $k$ -rational, then $T_1 \otimes_k T_2$ is retract $k$ -rational.
Application to Norm One Tori (Theorem 7.4):
Let $A = \prod K_i$ and $B = \prod E_j$ be étale $k$ -algebras with degrees $m_i = [K_i:k]$ and $n_j = [E_j:k]$ . If $\gcd(m_i, n_j) = 1$ for all $i, j$ , and if the norm one tori $T_A$ and $T_B$ are stably (resp. retract) $k$ -rational, then:
- The algebraic torus $T_A \otimes_k T_B$ is stably (resp. retract) $k$ -rational.
- The norm one torus $T_{A \otimes B}$ of the tensor product algebra is stably (resp. retract) $k$ -rational.
- In the specific case where $A$ and $B$ are field extensions of coprime degrees, $A \otimes_k B$ is a field extension, and its norm one torus inherits the rationality property.
Hasse Norm Principle (Theorem 8.1 and Corollaries):
For a global field $k$ , if $K = K_1 \otimes_k \cdots \otimes_k K_t$ is a composite of finite separable extensions with pairwise coprime degrees $[K_i:k]$ , and if the individual norm one tori $T_{K_i}$ are stably or retract rational, then:
- The Hasse norm principle holds for $K/k$ .
- The weak approximation property holds for $T_K$ .
- This is derived from the vanishing of the unramified Brauer group (or equivalently, the vanishing of the Shafarevich-Tate group) for the resulting torus, which follows from the rationality results.
Explicit Classifications (Sections 8.1–8.3):
The authors provide detailed tables classifying the Galois groups $G = \text{Gal}(L/k)$ for which the norm one torus of a composite extension $K/k$ (with degree $n \leq 30$ ) is stably or retract rational. These include cases where $G$ is a subdirect product of groups associated with the individual extensions (e.g., $G \leq S_{n_1} \times S_{n_2}$ ). They explicitly list cases for $n=6, 10, 12, 14, 15$ and extend the classification to $18 \leq n \leq 30$ .

4. Significance and Claims
The paper claims to provide a systematic method for generating new examples of stably and retract rational norm one tori by combining existing ones via tensor products, provided their degrees (or more precisely, their permutation orders) are coprime.

Novelty of Invariant: The introduction of the "permutation order" $p\text{-ord}(M)$ is highlighted as a systematic tool that had not been previously studied in this context, allowing for a precise criterion (coprimality) for the preservation of rationality under tensor products.
Constructive Proofs: The proofs are described as constructive, relying on explicit manipulations of flabby resolutions of $G$ -lattices rather than abstract existence arguments.
Arithmetic Implications: The results directly link the geometric property of rationality to the arithmetic Hasse norm principle. By establishing rationality for composite extensions of coprime degrees, the authors confirm the Hasse norm principle for these specific field extensions over global fields.
Limitations: The authors note that the coprimality condition is necessary; they provide examples (Example 7.6) where the tensor product of two stably rational tori fails to be retract rational if the degrees share a common factor. They also acknowledge that the general problem of $k$ -rationality (as opposed to stable or retract) for tensor products with $r \geq 3$ remains open.

In summary, the paper establishes a robust framework for understanding the behavior of rationality properties under tensor products of algebraic tori, specifically leveraging the coprimality of degrees to ensure the preservation of these properties and the validity of the Hasse norm principle for the resulting field extensions.