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The rationality problem for multinorm one tori, II

Sumito Hasegawa, Kazuki Kanai, Yasuhiro Oki

Abstract

We investigate the stable and retract rationality of multinorm one tori associated to finite {é}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.

The rationality problem for multinorm one tori, II

Abstract

We investigate the stable and retract rationality of multinorm one tori associated to finite {é}tale algebras. Our results are organized according to the greatest common divisor of the degrees of the factors. We show that these tori are stably rational for , and obtain a criterion for retract rationality that can be attributed to our previous results. For , 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.

Paper Structure

This paper contains 8 sections, 49 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Basic facts on the rationality of tori
  3. Multinorm one tori and their character groups
  4. Proof of Theorems \ref{['mtd1']} and \ref{['mth3']}
  5. Proof of Theorems \ref{['mtd2']} and \ref{['mtd3']}
  6. Proof of Theorem \ref{['mth1']}
  7. Coflabby resolution of particular lattices over dihedral groups
  8. Proof of Theorem \ref{['mth2']}

Key Result

Theorem 1.1

Let $k$ be a field, and $\mathbf{K}=\prod_{i=1}^{r}K_{i}$ be a finite étale algebra over $k$, where $r\geq 2$, which satisfies $d_{k}(\mathbf{K})=1$. Then the multinorm one torus $T_{\mathbf{K}/k}$ is stably rational over $k$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (92)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Definition 2.1: Endo2011, Hasegawa
  • ...and 82 more