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The rationality problem for multinorm one tori

Sumito Hasegawa, Kazuki Kanai, Yasuhiro Oki

TL;DR

This work analyzes the rationality problem for multinorm one tori $T_{\mathbf{K}/k}$, showing that for splitting fields with nilpotent Galois groups stable rationality and retract rationality are equivalent and can be detected via generalized character lattices. By introducing $I_{G/\mathcal{H}}^{(\varphi)}$ and $J_{G/\mathcal{H}}^{(\varphi)}$ and developing reduction techniques, the authors extend Endo’s results from norm one tori to the multinorm setting, establishing necessary and sufficient conditions for stable/retract rationality in the nilpotent and $p$-group cases. They provide explicit constructions of lattices that are quasi-permutation or not quasi-invertible, enabling a systematic approach to the rationality problem through flabby resolutions. As an application to the multinorm principle over global fields, the paper connects retract rationality to the vanishing of Tate–Shafarevich groups, producing new instances where the multinorm principle holds and offering a framework for further combinatorial and cohomological analysis.

Abstract

In this paper, we study the rationality problem for multinorm one tori, a natural generalization of norm one tori. For multinorm one tori that split over finite Galois extensions with nilpotent Galois group, we prove that stable rationality and retract rationality are equivalent, and give a criterion for the validity of the above two conditions. This generalizes the result of Endo (2011) on the rationality problem for norm one tori. To accomplish it, we introduce a generalization of character groups of multinorm one tori. Moreover, we establish systematic reduction methods originating in work of Endo (2001) for an investigation of the rationality problem for arbitrary multinorm one tori. In addition, we provide a new example for which the multinorm principle holds.

The rationality problem for multinorm one tori

TL;DR

This work analyzes the rationality problem for multinorm one tori , showing that for splitting fields with nilpotent Galois groups stable rationality and retract rationality are equivalent and can be detected via generalized character lattices. By introducing and and developing reduction techniques, the authors extend Endo’s results from norm one tori to the multinorm setting, establishing necessary and sufficient conditions for stable/retract rationality in the nilpotent and -group cases. They provide explicit constructions of lattices that are quasi-permutation or not quasi-invertible, enabling a systematic approach to the rationality problem through flabby resolutions. As an application to the multinorm principle over global fields, the paper connects retract rationality to the vanishing of Tate–Shafarevich groups, producing new instances where the multinorm principle holds and offering a framework for further combinatorial and cohomological analysis.

Abstract

In this paper, we study the rationality problem for multinorm one tori, a natural generalization of norm one tori. For multinorm one tori that split over finite Galois extensions with nilpotent Galois group, we prove that stable rationality and retract rationality are equivalent, and give a criterion for the validity of the above two conditions. This generalizes the result of Endo (2011) on the rationality problem for norm one tori. To accomplish it, we introduce a generalization of character groups of multinorm one tori. Moreover, we establish systematic reduction methods originating in work of Endo (2001) for an investigation of the rationality problem for arbitrary multinorm one tori. In addition, we provide a new example for which the multinorm principle holds.

Paper Structure

This paper contains 21 sections, 75 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Main theorems
  3. Application to multinorm principle
  4. Basic facts on the rationality of tori
  5. Multinorm one tori and their character groups
  6. Multinorm one tori
  7. $G$-lattices $I_{G/\mathcal{H}}^{(\varphi)}$ and $J_{G/\mathcal{H}}^{(\varphi)}$
  8. Reduction to smaller $G$-lattices
  9. Reduction to lattices over smaller groups
  10. $p$-groups
  11. Proof of Theorem \ref{['mth1']}
  12. Flabby resolutions of particular lattices
  13. Quasi-permutation lattices
  14. Non-quasi-invertible lattices
  15. Proof of Theorem \ref{['mth2']}
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $p$ be a prime, $k$ a field, $\mathbf{K}=\prod_{i=1}^{r}K_{i}^{h_i}$ a finite étale $k$-algebra with $r\geq 1$ and $h_i\geq 1$, and $L$ the Galois closure of the composite field of $K_{1},\ldots,K_{r}$ over $k$. Assume Then the following hold.

Theorems & Definitions (154)

  • Theorem 1.1: Endo2001
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Definition 2.1: Endo2011
  • ...and 144 more