The rationality problem for multinorm one tori, II

Kazuki Kanai; Sumito Hasegawa; Yasuhiro Oki

arxiv: 2604.02848 · v1 · submitted 2026-04-03 · 🧮 math.AG · math.RA

The rationality problem for multinorm one tori, II

Sumito Hasegawa , Kazuki Kanai , Yasuhiro Oki This is my paper

Pith reviewed 2026-05-13 18:27 UTC · model grok-4.3

classification 🧮 math.AG math.RA

keywords multinorm one toristable rationalityretract rationalityfinite etale algebrasGalois extensionsalgebraic torirationality criteriaSylow subgroups

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The pith

Multinorm one tori associated to finite étale algebras are stably rational when the gcd of the factor degrees equals one.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that multinorm one tori coming from finite étale algebras are stably rational exactly when d, the greatest common divisor of the degrees of the algebra factors, is 1. For d greater than 1 it supplies sufficient conditions under which the tori fail to be retract rational. It also generalizes theorems of Endo-Miyata and Endo by giving equivalent conditions for stable rationality when the tori split over Galois extensions whose groups have all Sylow subgroups cyclic, with an analogous statement for dihedral groups. These results clarify when such tori admit rational parametrizations over the base field.

Core claim

We show that these tori are stably rational for d=1, and obtain a criterion for retract rationality that can be attributed to our previous results. For d>1, we provide sufficient conditions for the failure of retract rationality. We further generalize results of Endo-Miyata (1975) and Endo (2011) by giving an equivalent condition for multinorm one tori to be stably rational under the assumption that they split over Galois extensions with Galois groups in which all Sylow subgroups are cyclic. A similar result also holds when they split over dihedral Galois extensions.

What carries the argument

Multinorm one tori, the kernels of the multinorm maps from the multiplicative group of a finite étale algebra to the base field multiplicative group, together with their splitting behavior over restricted Galois extensions.

If this is right

When d equals 1 every such torus is stably rational over the base field.
For d greater than 1 certain splitting conditions suffice to show the torus is not retract rational.
Stable rationality is equivalent to a stated Galois-cohomological condition when the splitting group has cyclic Sylow subgroups.
The same equivalence holds when the splitting group is dihedral.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The retract-rationality criterion inherited from prior work may apply to broader classes of tori beyond the multinorm-one case.
One could search computationally for small-degree examples with d greater than 1 that satisfy the failure conditions and thereby produce new explicit non-rational tori.
The results suggest that rationality questions for norm tori in number fields may reduce to similar Galois-group restrictions.

Load-bearing premise

The tori must split over Galois extensions whose groups have all Sylow subgroups cyclic or are dihedral in order for the stable-rationality equivalences to hold.

What would settle it

An explicit multinorm one torus that splits over a Galois extension with all Sylow subgroups cyclic yet fails to be stably rational would disprove the claimed equivalence condition.

read the original abstract

We investigate the stable and retract rationality of multinorm one tori associated to finite {\'e}tale algebras. Our results are organized according to the greatest common divisor $d$ of the degrees of the factors. We show that these tori are stably rational for $d=1$, and obtain a criterion for retract rationality that can be attributed to our previous results. For $d>1$, we provide sufficient conditions for the failure of retract rationality. We further generalize results of Endo--Miyata (1975) and Endo (2011) by giving an equivalent condition for multinorm one tori to be stably rational under the assumption that they split over Galois extensions with Galois groups in which all Sylow subgroups are cyclic. A similar result also holds when they split over dihedral Galois extensions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper adds sufficient conditions for multinorm one tori to fail retract rationality when d>1 and gives equivalent stable rationality criteria under cyclic-Sylow and dihedral Galois assumptions.

read the letter

The main thing to know about this paper is that it supplies sufficient conditions for the failure of retract rationality for multinorm one tori when d>1, and equivalent conditions for stable rationality when the splitting Galois group has all Sylow subgroups cyclic or is dihedral. These build on the 1975 and 2011 papers in a direct way. What the paper does well is to organize results according to the gcd d of the degrees. For d=1 it establishes stable rationality and a retract criterion from prior work. The d>1 failure conditions and the Galois generalizations are new and stated precisely. The soft spots are that everything for d>1 and the generalizations requires the tori to split over Galois extensions with the listed group structures. This scopes the results but leaves the general case open. The retract criterion is explicitly from the authors' earlier paper, so this manuscript applies and extends rather than re-derives it from scratch. This is a paper for specialists working on birational geometry of tori and Galois cohomology. Someone in that area will get value from the new conditions for classifying more examples. The thinking is clear and the citations are handled honestly. It deserves peer review to verify the details of the new arguments.

Referee Report

1 major / 2 minor

Summary. The manuscript investigates the stable and retract rationality of multinorm one tori associated to finite étale algebras. Results are organized according to d, the greatest common divisor of the degrees of the factors. For d=1 the tori are shown to be stably rational and a criterion for retract rationality is obtained by direct appeal to the authors' prior work. For d>1 sufficient conditions are given for the failure of retract rationality. The paper further generalizes Endo-Miyata (1975) and Endo (2011) by supplying equivalent conditions for stable rationality when the splitting Galois extension has group whose Sylow subgroups are all cyclic, and likewise when the group is dihedral.

Significance. If the derivations hold, the work supplies a useful refinement of the rationality theory for multinorm one tori. The case distinction by d organizes the landscape cleanly, the d=1 stable-rationality statement and the d>1 non-retract-rationality criteria are concrete, and the Galois-group generalizations extend the classical Endo-Miyata/Endo theorems to a wider but still explicitly described class of splitting extensions. Explicit attribution of the retract-rationality criterion to the authors' earlier paper is a strength, provided the reference is precise.

major comments (1)

[Section on retract rationality (d=1 case)] The section presenting the retract-rationality criterion for d=1: the criterion is invoked by direct citation of the authors' previous paper without restating its precise Galois-cohomology statement or the exact hypotheses under which it applies. Because this criterion is load-bearing for the main d=1 theorem, a short self-contained recall (one paragraph or a displayed statement) would eliminate any ambiguity about the precise conditions being used.

minor comments (2)

[Introduction] Notation for the finite étale algebra and its factors should be fixed at the beginning of the introduction and used consistently; the current abstract-to-body transition leaves the precise meaning of 'factors' and 'degrees' slightly implicit for a reader who has not yet reached the definitions.
[Section on Galois-group generalizations] The statement of the generalization of Endo-Miyata/Endo would benefit from an explicit comparison table or sentence listing which hypotheses of the 1975/2011 theorems are relaxed and which are retained.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the constructive suggestion for improving clarity. We will incorporate the requested change in the revised version.

read point-by-point responses

Referee: [Section on retract rationality (d=1 case)] The section presenting the retract-rationality criterion for d=1: the criterion is invoked by direct citation of the authors' previous paper without restating its precise Galois-cohomology statement or the exact hypotheses under which it applies. Because this criterion is load-bearing for the main d=1 theorem, a short self-contained recall (one paragraph or a displayed statement) would eliminate any ambiguity about the precise conditions being used.

Authors: We agree that including a brief self-contained recall would enhance readability and remove any potential ambiguity. In the revised manuscript we will insert a short paragraph (or displayed statement) that summarizes the precise Galois-cohomology formulation and the exact hypotheses under which the retract-rationality criterion from our earlier work applies, placed immediately before its invocation in the d=1 case. revision: yes

Circularity Check

1 steps flagged

Modest self-citation load-bearing for retract rationality criterion

specific steps

self citation load bearing [Abstract]
"We show that these tori are stably rational for d=1, and obtain a criterion for retract rationality that can be attributed to our previous results."

The retract-rationality criterion is taken directly from the authors' prior paper and invoked as a load-bearing component of the present claims without fresh proof or external verification supplied in this manuscript.

full rationale

The paper's central retract-rationality criterion is explicitly attributed to the authors' prior work without new verification or independent derivation here. The d=1 stable rationality result and the d>1 failure conditions plus Galois-group generalizations rest on separate cohomology arguments extending Endo-Miyata and Endo, so the overall derivation chain is not forced by definition or fitted inputs. This produces a minor self-citation burden but leaves independent mathematical content.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard facts from Galois cohomology, étale algebras, and birational geometry of tori; no free parameters or newly invented entities are introduced. The retract-rationality criterion is imported from prior work.

axioms (2)

domain assumption Standard properties of finite étale algebras and their associated multinorm tori hold over fields of characteristic zero.
Invoked throughout the organization by degree gcd and the splitting assumptions.
standard math Galois cohomology and birational invariants behave as in the cited Endo-Miyata and Endo papers.
Used to obtain the retract-rationality criterion and the equivalent conditions for cyclic-Sylow and dihedral cases.

pith-pipeline@v0.9.0 · 5440 in / 1665 out tokens · 48473 ms · 2026-05-13T18:27:32.748799+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

T_K/k is stably rational over k iff X^*(T) is quasi-permutation (Prop. 2.6); J_{G/H} quasi-permutation when d_G(H)=1 (Thm 4.2) or all Sylow cyclic with D_G(H) ∩ P^*(G)=∅ (Thm 6.7)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Coflabby/flabby resolutions and similarity classes S(G) for lattices over dihedral/cyclic groups (Sections 2,7)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

Colliot--Th \'e l \`e ne, J-J

J.-L. Colliot--Th \'e l \`e ne, J-J. Sansuc, La R - \'e quivalence sur les tores , Ann. sci. de l' \'E .N.S. 4^ e s \'e rie, 10 no. 2 (1977), 175--229

work page 1977
[2]

Demarche, D

C. Demarche, D. Wei, Hasse principle and weak approximation for multinorm equations, Israel J. Math. 202 (2014), 275--293

work page 2014
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S. Endo, T. Miyata, On a classification of the function fields of algebraic tori, Nagoya Math. J. 56 (1975), 85--104

work page 1975
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Endo, On the rationality of algebraic tori of norm type, J

S. Endo, On the rationality of algebraic tori of norm type, J. Alg. 235 (2001), 27--35

work page 2001
[5]

Endo, The rationality problem for norm one tori, Nagoya Math

S. Endo, The rationality problem for norm one tori, Nagoya Math. J. 202 (2011), 83--106

work page 2011
[6]

Hall, The theory of groups, McMillam Comp., 1959

M. Hall, The theory of groups, McMillam Comp., 1959

work page 1959
[7]

Hasegawa, K

S. Hasegawa, K. Kanai, Y. Oki, The rationality problem for multinorm one tori, preprint, arXiv:2504.04078, 2025

work page arXiv 2025
[8]

Hoshi, A

A. Hoshi, A. Yamasaki, Rationality problem for norm one tori for dihedral extensions, J. Alg. 640 (2024), 368--384

work page 2024
[9]

Kang, Retract rational fields, J

M. Kang, Retract rational fields, J. Algebra, 349 (2012), 22–-37

work page 2012
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M. Lorenz, Multiplicative Invariant Theory, Encyclopaedia of Mathematical Sciences, vol. 135, Springer, 2005

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A. S. Merkurjev, Invariants of algebraic groups and retract rationality of classifying spaces, Algebraic groups: structure and actions, 277--294, Proc. Sympos. Pure Math., 94, Amer. Math. Soc., Providence, RI, 2017

work page 2017
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Oki, The Hasse norm principle for some extensions of degree having square-free prime factors, preprint, arXiv:2504.19453, 2025

Y. Oki, The Hasse norm principle for some extensions of degree having square-free prime factors, preprint, arXiv:2504.19453, 2025

work page arXiv 2025
[14]

D. J. Saltman, Retract rational fields and cyclic Galois extensions, Israel J. Math., 47 (1984), 165--215

work page 1984

[1] [1]

Colliot--Th \'e l \`e ne, J-J

J.-L. Colliot--Th \'e l \`e ne, J-J. Sansuc, La R - \'e quivalence sur les tores , Ann. sci. de l' \'E .N.S. 4^ e s \'e rie, 10 no. 2 (1977), 175--229

work page 1977

[2] [2]

Demarche, D

C. Demarche, D. Wei, Hasse principle and weak approximation for multinorm equations, Israel J. Math. 202 (2014), 275--293

work page 2014

[3] [3]

S. Endo, T. Miyata, On a classification of the function fields of algebraic tori, Nagoya Math. J. 56 (1975), 85--104

work page 1975

[4] [4]

Endo, On the rationality of algebraic tori of norm type, J

S. Endo, On the rationality of algebraic tori of norm type, J. Alg. 235 (2001), 27--35

work page 2001

[5] [5]

Endo, The rationality problem for norm one tori, Nagoya Math

S. Endo, The rationality problem for norm one tori, Nagoya Math. J. 202 (2011), 83--106

work page 2011

[6] [6]

Hall, The theory of groups, McMillam Comp., 1959

M. Hall, The theory of groups, McMillam Comp., 1959

work page 1959

[7] [7]

Hasegawa, K

S. Hasegawa, K. Kanai, Y. Oki, The rationality problem for multinorm one tori, preprint, arXiv:2504.04078, 2025

work page arXiv 2025

[8] [8]

Hoshi, A

A. Hoshi, A. Yamasaki, Rationality problem for norm one tori for dihedral extensions, J. Alg. 640 (2024), 368--384

work page 2024

[9] [9]

Kang, Retract rational fields, J

M. Kang, Retract rational fields, J. Algebra, 349 (2012), 22–-37

work page 2012

[10] [10]

H. W. Lenstra Jr., Rational functions invariant under a finite abelian group, Invent. Math. 25 (1974), 299--325

work page 1974

[11] [11]

Lorenz, Multiplicative Invariant Theory, Encyclopaedia of Mathematical Sciences, vol

M. Lorenz, Multiplicative Invariant Theory, Encyclopaedia of Mathematical Sciences, vol. 135, Springer, 2005

work page 2005

[12] [12]

A. S. Merkurjev, Invariants of algebraic groups and retract rationality of classifying spaces, Algebraic groups: structure and actions, 277--294, Proc. Sympos. Pure Math., 94, Amer. Math. Soc., Providence, RI, 2017

work page 2017

[13] [13]

Oki, The Hasse norm principle for some extensions of degree having square-free prime factors, preprint, arXiv:2504.19453, 2025

Y. Oki, The Hasse norm principle for some extensions of degree having square-free prime factors, preprint, arXiv:2504.19453, 2025

work page arXiv 2025

[14] [14]

D. J. Saltman, Retract rational fields and cyclic Galois extensions, Israel J. Math., 47 (1984), 165--215

work page 1984