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pith:LKKZ5TJB

pith:2026:LKKZ5TJBXMDHD7I4OS6JLEE3P2

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Rationality problem for norm one tori of tensor products of \'etale algebras and Hasse norm principle

Aiichi Yamasaki, Akinari Hoshi, Mathieu Florence

When degrees of two étale algebras over k are coprime, stable or retract rationality of their norm one tori passes to the tensor product torus and the norm one torus of the tensor product algebra.

arxiv:2605.17427 v1 · 2026-05-17 · math.AG · math.NT

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Claims

C1strongest claim

If gcd(m_i, n_j | 1≤i≤r, 1≤j≤s)=1 and T_A and T_B are stably (resp. retract) k-rational, then the algebraic k-torus T_A ⊗ T_B and the norm one torus T_{A⊗B} are stably (resp. retract) k-rational. In particular, if k is a global field, then the Hasse norm principle holds for (A⊗B)/k.

C2weakest assumption

The coprimeness condition gcd of all degrees m_i and n_j equals 1 is required for the rationality preservation to hold under tensor product; the abstract presents this as a necessary hypothesis for the stated theorem.

C3one line summary

Under a coprimeness condition on extension degrees, stable or retract rationality of norm one tori is preserved under tensor product, implying the Hasse norm principle holds for the combined extension over global fields.

References

20 extracted · 20 resolved · 1 Pith anchors

[1] [Arn84] J. E. Arnold, Jr, Groups of permutation projective dimension two , Proc. Amer. Math. Soc. 91 (1984) 505–509. [Azu50] G. Azumaya, Corrections and supplementaries to my paper concerning Kru ll-R 1984

[2] Beauville, The L¨ uroth problem, Rationality problems in algebraic geometry, 1–27, Lectur e Notes in Math., 2172, Fond 2016

[3] v ii+422 pp. [Bou90] N. Bourbaki, Algebra II, Chapters 4–7 , Translated from the French by P. M. Cohn and J. Howie, Elem. M ath. (Berlin), Springer-Verlag, Berlin, 1990, vii+461 pp. NORM ONE TORI OF T 1990

[4] Colliot-Th´ el` ene, A 2021

[5] [DW14] C. Demarche, D. W ei, Hasse principle and weak approximation for multinorm equat ions, Israel J. Math. 202 (2014) 275–293. [DP87] Yu. A. Drakokhrust, V. P. Platonov, The Hasse norm principle fo 2014

Receipt and verification

First computed	2026-05-20T00:03:57.988067Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

5a959ecd21bb0671fd1c74bc95909b7eb1313805cc9574fc6be9fe39f08ed2fc

Aliases

arxiv: 2605.17427 · arxiv_version: 2605.17427v1 · doi: 10.48550/arxiv.2605.17427 · pith_short_12: LKKZ5TJBXMDH · pith_short_16: LKKZ5TJBXMDHD7I4 · pith_short_8: LKKZ5TJB

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curl -sH 'Accept: application/ld+json' https://pith.science/pith/LKKZ5TJBXMDHD7I4OS6JLEE3P2 \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 5a959ecd21bb0671fd1c74bc95909b7eb1313805cc9574fc6be9fe39f08ed2fc

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "430dd87f0928ccded7a5ea1528ef8dd8c6a0c6675109ba8ac4ee18584eb05656",
    "cross_cats_sorted": [
      "math.NT"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.AG",
    "submitted_at": "2026-05-17T12:48:44Z",
    "title_canon_sha256": "d3dc4cdfca003f9d01d6b53b351417db14367a1cde5c78ccfe51c105c00b0d39"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.17427",
    "kind": "arxiv",
    "version": 1
  }
}