# Rationality problem for norm one tori for dihedral extensions

Akinari Hoshi  Department of Mathematics, Niigata University, Niigata
950-2181, Japan<EMAIL_ADDRESS>and  Aiichi Yamasaki
Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
<EMAIL_ADDRESS>
###### Abstract.

We give a complete answer to the rationality problem (up to stable
$k$-equivalence) for norm one tori $R^{(1)}_{K/k}(\mathbbm{G}_{m})$ of $K/k$
whose Galois closures $L/k$ are dihedral extensions with the aid of Endo and
Miyata [EM75, Theorem 1.5, Theorem 2.3] and Endo [End11, Theorem 2.1]. By
using a similar technique, we give refinements of the proof of stably rational
cases of Endo and Miyata’s theorems as an appendix of the paper.

###### 2010 Mathematics Subject Classification:

Primary 11E72, 12F20, 13A50, 14E08, 20C10, 20G15.

Key words and phrases. Rationality problem, algebraic tori, norm one tori,
stably rational, retract rational, flabby resolution.
This work was partially supported by JSPS KAKENHI Grant Numbers 19K03418,
20H00115, 20K03511.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Proof of Theorem 1.1
  4. 4 Appendix: refinements of the proof of stably rational cases with $G\simeq D_{n}$ $(n\geq 3:odd)$

##  1\. Introduction

Let $k$ be a field and $X$ be an integral $k$-variety of dimension $n$. The
Lüroth problem asks whether a $k$-unirational variety $X$ is $k$-rational,
i.e. $X$ is $k$-birational to $\mathbbm{P}^{n}_{k}$ if there exists a dominant
rational map $\mathbbm{P}^{n}_{k}\dashrightarrow X$. Over an algebraically
closed field $k$ with char $k=0$, examples of $k$-unirational varieties which
are not $k$-rational are given by Clemens and Griffiths [CG72], Iskovskikh and
Manin [IM71], Artin and Mumford [AM72]. See e.g. Manin [Man86], Manin and
Tsfasman [MT86], Iskovskikh [Isk97], Colliot-Thélène and Sansuc [CTS07,
Section 1], Beauville [Bea16], Voisin [Voi16]. Over an algebraically non-
closed field $k$, Swan [Swa69, Theorem 1] gave the first counterexamples to
Noether’s problem over $\mathbbm{Q}$ which are not $\mathbbm{Q}$-rational but
$\mathbbm{Q}$-unirational varieties whose function fields are isomorphic to
the field of invariants $\mathbbm{Q}(x_{1},\ldots,x_{p})^{C_{p}}$ where
$C_{p}$ is the cyclic group of prime order $p$ with $p=47,113,233$. We see
that $\mathbbm{Q}(x_{1},\ldots,x_{p})^{C_{p}}=L(M)^{C_{p-1}}(t)$ where
$L(M)^{C_{p-1}}$ can be understood as the function field $\mathbbm{Q}(T)$ of
an algebraic $\mathbbm{Q}$-torus $T$ of dimension $p-1$ via Galois descent
where $C_{p-1}={\rm Gal}(L/\mathbbm{Q})$ and $L=\mathbbm{Q}(\zeta_{p})$ is the
$p$-th cyclotomic field (see survey papers Swan [Swa83] and Hoshi [Hos20,
Section 2]). For algebraic group varieties $X$, examples of $k$-unirational
varieties which are not $k$-rational are going back to Chevalley. Indeed,
Chevalley [Che54, Section V] gave examples of norm one tori
$R^{(1)}_{K/k}(\mathbbm{G}_{m})$ of $K/k$ which are not $k$-rational but
$k$-unirational where $k=\mathbbm{Q}_{p}$ is the field of $p$-adic numbers and
$K/k$ are non-cyclic abelian extensions. Later, Endo and Miyata [EM75] gave a
necessary and sufficient condition for stably/retract rationality of norm one
tori $R^{(1)}_{K/k}(\mathbbm{G}_{m})$ of Galois extensions $K/k$ (see Theorem
2.2 and Theorem 2.3). In particular, $R^{(1)}_{K/k}(\mathbbm{G}_{m})$ is not
(retract) $k$-rational but $k$-unirational if some $p$-Sylow subgroup of
$G={\rm Gal}(K/k)$ is not cyclic.

The Zariski problem asks whether a stably $k$-rational variety $X$ is
$k$-rational, i.e. $X$ is $k$-birational to $\mathbbm{P}^{n}_{k}$ if
$X\times_{k}\mathbbm{P}^{m}_{k}$ is $k$-birational to $\mathbbm{P}^{m+n}_{k}$.
By using Galois lattice arguments (algebraic $k$-torus technique) on ${\rm
Pic}\,\overline{X}$ attached to the dihedral group $D_{n}$ of order $2n$
$(n\geq 3:$ odd$)$, for an algebraically non-closed field $k$ (resp. an
algebraically closed field $k$) with ${\rm char}\,k\neq 2$, examples of not
$k$-rational but stably $k$-rational varieties $X$ with
$X\times_{k}\mathbbm{P}^{3}$ $k$-rational are given by Beauville, Colliot-
Thélène, Sansuc and Swinnerton-Dyer [BCTSSD85] as (generalized) Châtelet
surfaces given by $y^{2}-az^{2}=P(x)\in k[x]$ where ${\rm deg}_{x}P(x)=n\geq
3$ and ${\rm Gal}(P(x)/k)\simeq D_{n}$ with $n=3$ (resp. $3$-folds given by
$y^{2}-a(t)z^{2}=P(t,x)\in k(t)[x]$ where ${\rm deg}_{x}P(t,x)=3$). Indeed,
stable $k$-rationality of $X$ is obtained by constructing a $k$-rational
torsor $\mathcal{F}$ on $X$ under $S_{0}$ which is $k$-birational to
$X\times_{k}S_{0}$ where $S_{0}$ is an algebraic $k$-torus with ${\rm
Hom}(S_{0},\mathbbm{G}_{m})\simeq{\rm Pic}\,\overline{X}$ for $n=3$ (see also
Colliot-Thélène and Sansuc [CTS80]) and ${\rm Pic}\,\overline{X}$ is stably
permutation for odd $n\geq 3$ although non-rationality of $X$ needs results of
Iskovskikh [Isk67], [Isk70], [Isk72] over $k\neq\overline{k}$ (resp. Clemens
and Griffiths [CG72] and Beauville [Bea77a], [Bea77b] over $k=\overline{k}$).
See [BCTSSD85] for details and also Shepherd-Barron [She04]. We also refer to
Benoist and Wittenberg [BW20], [BW23] which investigate Clemens-Griffiths
method, intermediate Jacobians and rationality over non-closed field $k$.

Let $K/k$ be a separable field extension of degree $m$ and $L/k$ be the Galois
closure of $K/k$. Let $G={\rm Gal}(L/k)$ and $H={\rm Gal}(L/K)$ with
$[G:H]=m$. The Galois group $G$ may be regarded as a transitive subgroup of
the symmetric group $S_{m}$ of degree $m$. We may assume that $H$ is the
stabilizer of one of the letters in $G\leq S_{m}$, i.e.
$L=k(\theta_{1},\ldots,\theta_{m})$ and $K=L^{H}=k(\theta_{i})$ where $1\leq
i\leq m$. Then we have $\bigcap_{\sigma\in G}H^{\sigma}=\\{1\\}$ where
$H^{\sigma}=\sigma^{-1}H\sigma$ and hence $H$ contains no normal subgroup of
$G$ except for $\\{1\\}$.

Let $T=R^{(1)}_{K/k}(\mathbbm{G}_{m})$ be the norm one torus of $K/k$, i.e.
the kernel of the norm map
$R_{K/k}(\mathbbm{G}_{m})\rightarrow\mathbbm{G}_{m}$ where $R_{K/k}$ is the
Weil restriction (see [Vos98, page 37, Section 3.12]). Note that
$R_{K/k}(\mathbbm{G}_{m})$ is $k$-rational with ${\rm
dim}\,R_{K/k}(\mathbbm{G}_{m})=[K:k]=m$ and ${\rm dim}\,T=m-1$. Let $D_{n}$ be
the dihedral group of order $2n$ $(n\geq 3)$ and $Z(D_{n})$ be the center of
$D_{n}$. In this paper, we investigate the rationality problem for
$R^{(1)}_{K/k}(\mathbbm{G}_{m})$ with $G\simeq D_{n}$. For $G\simeq D_{n}$, by
the condition $\bigcap_{\sigma\in G}H^{\sigma}=\\{1\\}$, we obtain that
$H=\\{1\\}$ or $|H|=2$ with $H\neq Z(D_{n})$. When $H=\\{1\\}$, i.e. $K/k$ is
Galois, Endo and Miyata [EM75, Theorem 2.3, Theorem 1.5] proved that
$R^{(1)}_{K/k}(\mathbbm{G}_{m})$ is stably $k$-rational (resp. not retract
$k$-rational) if $n$ is odd (resp. even). When $|H|=2$ with $H\neq Z(D_{n})$,
Endo [End11, Theorem 3.1, Theorem 2.1] showed that
$R^{(1)}_{K/k}(\mathbbm{G}_{m})$ is stably $k$-rational (resp. not retract
$k$-rational) if $n$ is odd (resp. $2$-power). See Section 2 for details.

The main theorem of this paper gives a complete answer to the rationality
problem (up to stable $k$-equivalence) for norm one tori
$R^{(1)}_{K/k}(\mathbbm{G}_{m})$ of $K/k$ whose Galois closures $L/k$ are
dihedral extensions with the aid of Endo and Miyata [EM75, Theorem 2.3,
Theorem 1.5] and Endo [End11, Theorem 3.1]:

######  Theorem 1.1.

Let $K/k$ be a finite non-Galois separable field extension and $L/k$ be the
Galois closure of $K/k$. Let $D_{n}=\langle x,y\mid
x^{n}=y^{2}=1,y^{-1}xy=x^{-1}\rangle$ be the dihedral group of order $2n$
$(n\geq 4:$ even$)$ and $Z(D_{n})=\langle x^{\frac{n}{2}}\rangle$ be the
center of $D_{n}$. Assume that $G={\rm Gal}(L/k)\simeq D_{n}$ and $H={\rm
Gal}(L/K)\leq G$. Then $|H|=2$ with $H\neq Z(D_{n})$ and we have:
(i) if $n\equiv 0\ ({\rm mod}\ 4)$, then $R^{(1)}_{K/k}(\mathbbm{G}_{m})$ is
not retract $k$-rational;
(ii) if $n\equiv 2\ ({\rm mod}\ 4)$, then $R^{(1)}_{K/k}(\mathbbm{G}_{m})$ is
stably $k$-rational. More precisely, $R^{(1)}_{K/k}(\mathbbm{G}_{m})\times
R_{k_{1}/k}(\mathbbm{G}_{m,k_{1}})\times
R_{k_{2}/k}(\mathbbm{G}_{m,k_{2}})\times R_{k_{3}/k}(\mathbbm{G}_{m,k_{3}})$
of dimension $(n-1)+\frac{n}{2}+\frac{n}{2}+2=2n+1$ and
$R_{k_{4}/k}(\mathbbm{G}_{m,k_{4}})\times
R_{k_{5}/k}(\mathbbm{G}_{m,k_{5}})\times\mathbbm{G}_{m,k}$ of dimension
$n+n+1=2n+1$ are birationally $k$-equivalent where
$R_{k_{i}/k}(\mathbbm{G}_{m,k_{i}})$ are $k$-rational with $k_{1}=L^{\langle
x^{\frac{n}{2}},x^{2}y\rangle}$, $k_{2}=L^{\langle
x^{\frac{n}{2}},xy\rangle}$, $k_{3}=L^{\langle x\rangle}$, $k_{4}=L^{\langle
x^{\frac{n}{2}}\rangle}$, $k_{5}=L^{\langle xy\rangle}$, $\langle
x^{\frac{n}{2}},x^{2}y\rangle\simeq\langle x^{\frac{n}{2}},xy\rangle\simeq
C_{2}\times C_{2}$, $\langle x\rangle\simeq C_{n}$, $\langle
x^{\frac{n}{2}}\rangle\simeq\langle xy\rangle\simeq C_{2}$ and
$[k_{1}:k]=[k_{2}:k]=\frac{n}{2}$, $[k_{3}:k]=2$, $[k_{4}:k]=[k_{5}:k]=n$.

We organize this paper as follows. In Section 2, we prepare related materials
and known results including some examples in order to prove Theorem 1.1. In
Section 3, the proof of Theorem 1.1 is given. In Section 4, we give
refinements of the proof of stably rational cases with $G\simeq D_{n}$ $(n\geq
3:$ odd$)$ of Endo and Miyata’s theorems [EM75, Theorem 2.3] with $H=\\{1\\}$
and [End11, Theorem 3.1] with $H\simeq C_{2}$, using a technique similar to
that given in the proof of Theorem 1.1.

##  2\. Preliminaries

Let $k$ be a field and $K$ be a finitely generated field extension of $k$. A
field $K$ is called rational over $k$ (or $k$-rational for short) if $K$ is
purely transcendental over $k$, i.e. $K$ is isomorphic to
$k(x_{1},\ldots,x_{n})$, the rational function field over $k$ with $n$
variables $x_{1},\ldots,x_{n}$ for some integer $n$. $K$ is called stably
$k$-rational if $K(y_{1},\ldots,y_{m})$ is $k$-rational for some algebraically
independent elements $y_{1},\ldots,y_{m}$ over $K$. Two fields $K$ and
$K^{\prime}$ are called stably $k$-isomorphic if $K(y_{1},\ldots,y_{m})\simeq
K^{\prime}(z_{1},\ldots,z_{n})$ over $k$ for some algebraically independent
elements $y_{1},\ldots,y_{m}$ over $K$ and $z_{1},\ldots,z_{n}$ over
$K^{\prime}$. When $k$ is an infinite field, $K$ is called retract
$k$-rational if there is a $k$-algebra $R$ contained in $K$ such that (i) $K$
is the quotient field of $R$, and (ii) the identity map $1_{R}:R\rightarrow R$
factors through a localized polynomial ring over $k$, i.e. there is an element
$f\in k[x_{1},\ldots,x_{n}]$, which is the polynomial ring over $k$, and there
are $k$-algebra homomorphisms $\varphi:R\rightarrow
k[x_{1},\ldots,x_{n}][1/f]$ and $\psi:k[x_{1},\ldots,x_{n}][1/f]\rightarrow R$
satisfying $\psi\circ\varphi=1_{R}$ (cf. Saltman [Sal84]). $K$ is called
$k$-unirational if $k\subset K\subset k(x_{1},\ldots,x_{n})$ for some integer
$n$. It is not difficult to see that

$k$-rational $\Rightarrow$ stably $k$-rational $\Rightarrow$ retract
$k$-rational $\Rightarrow$ $k$-unirational.

Let $\overline{k}$ be a fixed separable closure of the base field $k$. Let $T$
be an algebraic $k$-torus, i.e. a group $k$-scheme with fiber product (base
change) $T\times_{k}\overline{k}=T\times_{{\rm Spec}\,k}\,{\rm
Spec}\,\overline{k}\simeq(\mathbbm{G}_{m,\overline{k}})^{n}$; $k$-form of the
split torus $(\mathbbm{G}_{m})^{n}$. An algebraic $k$-torus $T$ is said to be
$k$-rational (resp. stably $k$-rational, retract $k$-rational,
$k$-unirational) if the function field $k(T)$ of $T$ over $k$ is $k$-rational
(resp. stably $k$-rational, retract $k$-rational, $k$-unirational). Two
algebraic $k$-tori $T$ and $T^{\prime}$ are said to be birationally
$k$-equivalent $(k$-isomorphic$)$ if their function fields $k(T)$ and
$k(T^{\prime})$ are $k$-isomorphic. For an equivalent definition in the
language of algebraic geometry, see e.g. Manin [Man86], Manin and Tsfasman
[MT86], Colliot-Thélène and Sansuc [CTS07, Section 1], Beauville [Bea16],
Merkurjev [Mer17, Section 3].

Let $L$ be a finite Galois extension of $k$ and $G={\rm Gal}(L/k)$ be the
Galois group of the extension $L/k$. Let $M=\bigoplus_{1\leq i\leq
n}\mathbbm{Z}\cdot u_{i}$ be a $G$-lattice with a $\mathbbm{Z}$-basis
$\\{u_{1},\ldots,u_{n}\\}$, i.e. finitely generated $\mathbbm{Z}[G]$-module
which is $\mathbbm{Z}$-free as an abelian group. Let $G$ act on the rational
function field $L(x_{1},\ldots,x_{n})$ over $L$ with $n$ variables
$x_{1},\ldots,x_{n}$ by

(1) $\displaystyle\sigma(x_{i})=\prod_{j=1}^{n}x_{j}^{a_{i,j}},\quad 1\leq
i\leq n$

for any $\sigma\in G$, when $\sigma(u_{i})=\sum_{j=1}^{n}a_{i,j}u_{j}$,
$a_{i,j}\in\mathbbm{Z}$. The field $L(x_{1},\ldots,x_{n})$ with this action of
$G$ will be denoted by $L(M)$. There is the duality between the category of
$G$-lattices and the category of algebraic $k$-tori which split over $L$ (see
[Ono61, Section 1.2], [Vos98, page 27, Example 6]). In fact, if $T$ is an
algebraic $k$-torus, then the character group $\widehat{T}={\rm
Hom}(T,\mathbbm{G}_{m})$ of $T$ may be regarded as a $G$-lattice. Conversely,
for a given $G$-lattice $M$, there exists an algebraic $k$-torus $T={\rm
Spec}(L[M]^{G})$ which splits over $L$ such that $\widehat{T}\simeq M$ as
$G$-lattices.

The invariant field $L(M)^{G}$ of $L(M)$ under the action of $G$ may be
identified with the function field $k(T)$ of the algebraic $k$-torus $T$ where
$\widehat{T}\simeq M$. Note that the field $L(M)^{G}$ is always
$k$-unirational (see [Vos98, page 40, Example 21]). Isomorphism classes of
tori of dimension $n$ over $k$ correspond bijectively to the elements of the
set $H^{1}(\mathcal{G},{\rm GL}_{n}(\mathbbm{Z}))$ where $\mathcal{G}={\rm
Gal}(\overline{k}/k)$ since ${\rm Aut}(\mathbbm{G}_{m}^{n})={\rm
GL}_{n}(\mathbbm{Z})$. The $k$-torus $T$ of dimension $n$ is determined
uniquely by the integral representation $h:\mathcal{G}\rightarrow{\rm
GL}_{n}(\mathbbm{Z})$ up to conjugacy, and the group $h(\mathcal{G})$ is a
finite subgroup of ${\rm GL}_{n}(\mathbbm{Z})$ (see [Vos98, page 57, Section
4.9])).

A $G$-lattice $M$ is called permutation $G$-lattice if $M$ has a
$\mathbbm{Z}$-basis permuted by $G$, i.e. $M\simeq\oplus_{1\leq i\leq
m}\mathbbm{Z}[G/H_{i}]$ for some subgroups $H_{1},\ldots,H_{m}$ of $G$. $M$ is
called stably permutation $G$-lattice if $M\oplus P\simeq P^{\prime}$ for some
permutation $G$-lattices $P$ and $P^{\prime}$. $M$ is called invertible if it
is a direct summand of a permutation $G$-lattice, i.e. $P\simeq M\oplus
M^{\prime}$ for some permutation $G$-lattice $P$ and a $G$-lattice
$M^{\prime}$. $M$ is called flabby (or flasque) if $\widehat{H}^{-1}(H,M)=0$
for any subgroup $H$ of $G$ where $\widehat{H}$ is the Tate cohomology. $M$ is
called coflabby (or coflasque) if $H^{1}(H,M)=0$ for any subgroup $H$ of $G$.
We say that two $G$-lattices $M_{1}$ and $M_{2}$ are similar if there exist
permutation $G$-lattices $P_{1}$ and $P_{2}$ such that $M_{1}\oplus
P_{1}\simeq M_{2}\oplus P_{2}$. The set of similarity classes becomes a
commutative monoid with respect to the sum $[M_{1}]+[M_{2}]:=[M_{1}\oplus
M_{2}]$ and the zero $0=[P]$ where $P$ is a permutation $G$-lattice. For a
$G$-lattice $M$, there exists a short exact sequence of $G$-lattices
$0\rightarrow M\rightarrow P\rightarrow F\rightarrow 0$ where $P$ is
permutation and $F$ is flabby which is called a flabby resolution of $M$ (see
Endo and Miyata [EM75, Lemma 1.1], Colliot-Thélène and Sansuc [CTS77, Lemma
3], Manin [Man86, Appendix, page 286]). The similarity class $[F]$ of $F$ is
determined uniquely and is called the flabby class of $M$. We denote the
flabby class $[F]$ of $M$ by $[M]^{fl}$. We say that $[M]^{fl}$ is invertible
if $[M]^{fl}=[E]$ for some invertible $G$-lattice $E$. For $G$-lattice $M$, it
is not difficult to see that

$\displaystyle\textrm{permutation}\ \ \Rightarrow\ \ $
$\displaystyle\textrm{stably\ permutation}\ \ \Rightarrow\ \
\textrm{invertible}\ \ \Rightarrow\ \ \textrm{flabby\ and\ coflabby}$
$\displaystyle\hskip 22.76219pt\Downarrow\hskip 96.73936pt\Downarrow$
$\displaystyle\hskip 19.91692pt[M]^{fl}=0\hskip 28.45274pt\Rightarrow\hskip
14.22636pt[M]^{fl}\ \textrm{is\ invertible}.$

For algebraic $k$-tori $T$, we see that $[\widehat{T}]^{fl}=[{\rm
Pic}\,\overline{X}]$ where $X$ is a smooth $k$-compactification of $T$, i.e.
smooth projective $k$-variety $X$ containing $T$ as a dense open subvariety,
and $\overline{X}=X\times_{k}\overline{k}$ (see Voskresenskii [Vos69, Section
4, page 1213], [Vos70, Section 3, page 7], [Vos74], [Vos98, Section 4.6],
Kunyavskii [Kun07, Theorem 1.9] and Colliot-Thélène [CT07, Theorem 5.1, page
19] for any field $k$).

The flabby class $[M]^{fl}=[\widehat{T}]^{fl}$ plays crucial role in the
rationality problem for $L(M)^{G}\simeq k(T)$ as follows (see Colliot-Thélène
and Sansuc [CTS77, Section 2], [CTS87, Proposition 7.4], Voskresenskii [Vos98,
Section 4.6], Kunyavskii [Kun07, Theorem 1.7], Colliot-Thélène [CT07, Theorem
5.4], Hoshi and Yamasaki [HY17, Section 1]):

######  Theorem 2.1.

Let $L/k$ be a finite Galois extension with Galois group $G={\rm Gal}(L/k)$
and $M$, $M^{\prime}$ be $G$-lattices.
(i) $($Endo and Miyata [EM73, Theorem 1.6]$)$ $[M]^{fl}=0$ if and only if
$L(M)^{G}$ is stably $k$-rational.
(ii) $($Voskresenskii [Vos74, Theorem 2]$)$ $[M]^{fl}=[M^{\prime}]^{fl}$ if
and only if $L(M)^{G}$ and $L(M^{\prime})^{G}$ are stably $k$-isomorphic.
(iii) $($Saltman [Sal84, Theorem 3.14]$)$ $[M]^{fl}$ is invertible if and only
if $L(M)^{G}$ is retract $k$-rational.

Let $K/k$ be a separable field extension of degree $m$ and $L/k$ be the Galois
closure of $K/k$. Let $G={\rm Gal}(L/k)$ and $H={\rm Gal}(L/K)$ with
$[G:H]=m$. The norm one torus $R^{(1)}_{K/k}(\mathbbm{G}_{m})$ has the
Chevalley module $J_{G/H}$ as its character module and the field
$L(J_{G/H})^{G}$ as its function field where $J_{G/H}=(I_{G/H})^{\circ}={\rm
Hom}_{\mathbbm{Z}}(I_{G/H},\mathbbm{Z})$ is the dual lattice of $I_{G/H}={\rm
Ker}\ \varepsilon$ and $\varepsilon:\mathbbm{Z}[G/H]\rightarrow\mathbbm{Z}$ is
the augmentation map (see [Vos98, Section 4.8]). We have the exact sequence
$0\rightarrow\mathbbm{Z}\rightarrow\mathbbm{Z}[G/H]\rightarrow
J_{G/H}\rightarrow 0$ and ${\rm rank}_{\mathbbm{Z}}J_{G/H}=m-1$. Write
$J_{G/H}=\oplus_{1\leq i\leq m-1}\mathbbm{Z}x_{i}$. Then the action of $G$ on
$L(J_{G/H})=L(x_{1},\ldots,x_{m-1})$ is of the form (1).

The rationality problem for norm one tori is investigated by many authors, see
e.g. [HY17], [HHY20], [HY21] and the references therein. Let $C_{n}$ be the
cyclic group of order $n$.

######  Theorem 2.2 (Endo and Miyata [EM75, Theorem 1.5], Saltman [Sal84,
Theorem 3.14]).

Let $L/k$ be a finite Galois field extension and $G={\rm Gal}(L/k)$. Then the
following conditions are equivalent:
(i) $R^{(1)}_{L/k}(\mathbbm{G}_{m})$ is retract $k$-rational;
(ii) all the Sylow subgroups of $G$ are cyclic.

######  Theorem 2.3 (Endo and Miyata [EM75, Theorem 2.3], Colliot-Thélène and
Sansuc [CTS77, Proposition 3]).

Let $L/k$ be a finite Galois field extension and $G={\rm Gal}(L/k)$. Then the
following conditions are equivalent:
(i) $R^{(1)}_{L/k}(\mathbbm{G}_{m})$ is stably $k$-rational;
(ii) $G\simeq C_{m}$ $(m\geq 1)$ or $G\simeq C_{m}\times\langle x,y\mid
x^{n}=y^{2^{d}}=1,yxy^{-1}=x^{-1}\rangle$ $(m\geq 1:odd,n\geq 3:odd,d\geq 1)$
with ${\rm gcd}\\{m,n\\}=1$;
(iii) $G\simeq\langle s,t\mid s^{m}=t^{2^{d}}=1,tst^{-1}=s^{r}\rangle$ $(m\geq
1:odd,d\geq 0)$ with $r^{2}\equiv 1\pmod{m}$;
(iv) all the Sylow subgroups of $G$ are cyclic and
$H^{4}(G,\mathbbm{Z})\simeq\widehat{H}^{0}(G,\mathbbm{Z})$ where $\widehat{H}$
is the Tate cohomology.

######  Theorem 2.4 (Endo [End11, Theorem 2.1]).

Let $K/k$ be a finite non-Galois, separable field extension and $L/k$ be the
Galois closure of $K/k$. Assume that the Galois group of $L/k$ is nilpotent.
Then $R^{(1)}_{K/k}(\mathbbm{G}_{m})$ is not retract $k$-rational.

######  Theorem 2.5 (Endo [End11, Theorem 3.1]).

Let $K/k$ be a finite non-Galois, separable field extension and $L/k$ be the
Galois closure of $K/k$. Let $G={\rm Gal}(L/k)$ and $H={\rm Gal}(L/K)\leq G$.
Assume that all the Sylow subgroups of $G$ are cyclic. Then
$R^{(1)}_{K/k}(\mathbbm{G}_{m})$ is retract $k$-rational, and the following
conditions are equivalent:
(i) $R^{(1)}_{K/k}(\mathbbm{G}_{m})$ is stably $k$-rational;
(ii) $G\simeq C_{m}\times D_{n}$ $(m\geq 1:odd,n\geq 3:odd)$ with ${\rm
gcd}\\{m,n\\}=1$ and $H\simeq C_{2}$;
(iii) $H\simeq C_{2}$ and $G\simeq C_{r}\rtimes H$ $(r\geq 3:odd)$ where $H$
acts non-trivially on $C_{r}$.

##  3\. Proof of Theorem 1.1

Let $K/k$ be a finite non-Galois separable field extension and $L/k$ be the
Galois closure of $K/k$. Let $D_{n}$ be the dihedral group of order $2n$
$(n\geq 3)$ and $Z(D_{n})$ be the center of $D_{n}$. Assume that $G={\rm
Gal}(L/k)\simeq D_{n}=\langle x,y\mid x^{n}=y^{2}=1,y^{-1}xy=x^{-1}\rangle$
$(n\geq 4:$ even$)$ and $H={\rm Gal}(L/K)\leq G$. Then it follows from the
condition $\bigcap_{\sigma\in G}H^{\sigma}=\\{1\\}$ that $|H|=2$ with $H\neq
Z(D_{n})$. We prepare the following lemma:

######  Lemma 3.1.

Let $K/k$ be a finite separable field extension and $L/k$ be the Galois
closure of $K/k$. Let $G={\rm Gal}(L/k)$ and $H={\rm Gal}(L/K)\leq G$. Let
${\rm Aut}(G)$ be the group of automorphisms of $G$ and
$K^{\varphi}=L^{\varphi(H)}$ $(\varphi\in{\rm Aut}(G))$. Then
$R^{(1)}_{K/k}(\mathbbm{G}_{m})$ is stably $k$-rational $($resp. retract
$k$-rational$)$ if and only if $R^{(1)}_{K^{\varphi}/k}(\mathbbm{G}_{m})$ is
stably $k$-rational $($resp. retract $k$-rational$)$ for any $\varphi\in{\rm
Aut}(G)$.

###### Proof.

We should just consider $L^{\varphi(H)}=K^{\varphi}=k(\theta^{\prime}_{i})$
instead of $L^{H}=K=k(\theta_{i})$. ∎

Proof of Theorem 1.1. Assume that $G\simeq D_{n}=\langle x,y\mid
x^{n}=y^{2}=1,y^{-1}xy=x^{-1}\rangle$ and $n=2m$ $(m\geq 2)$. The problem is
determined up to conjugacy in $G$ because $J_{G/H}\simeq J_{G/H^{\sigma}}$
$(\sigma\in G)$ as $G$-lattices (conjugate just corresponds to base change).
Then we see that $H$ is conjugate to $\langle y\rangle$ or $\langle xy\rangle$
in $G$. The group ${\rm Aut}(G)$ permutes $\langle y\rangle$ and $\langle
xy\rangle$. Hence, by Lemma 3.1, we may assume that $H\simeq C_{2}=\langle
xy\rangle$.

We may assume that $G\simeq D_{n}=\langle x,y\mid
x^{n}=y^{2}=1,y^{-1}xy=x^{-1}\rangle\leq S_{n}$ where $x=(1\ 2\ \cdots\ n)$,
$y=(1\ n)(2\ n-1)\cdots(m\ m+1)$, $xy=(2\ n)(3\ n-1)\cdots(m\ m+2)$ with
$m=n/2$.

(i) Case $n\equiv 0\ ({\rm mod}\ 4)$. Write $n=2m$ $(m\geq 2:$ even$)$. We
take $G=\langle x,y\mid x^{n}=y^{2}=1,y^{-1}xy=x^{-1}\rangle\simeq D_{n}$ and
$H=\langle xy\rangle\simeq C_{2}$. We consider $[J_{G/H}]^{fl}|_{G^{\prime}}$
which is obtained by the restriction of the action of $G$ on $[J_{G/H}]^{fl}$
to the subgroup $G^{\prime}=\langle x^{2},y\rangle\simeq D_{m}$ with
$H^{\prime}=H\cap G^{\prime}=1$. Then $G^{\prime}$ becomes a transitive
subgroup of $S_{n}$ and $[J_{G/H}]^{fl}|_{G^{\prime}}=[J_{G^{\prime}}]^{fl}$
is not invertible by Theorem 2.1 and Theorem 2.2. This implies that
$[J_{G/H}]^{fl}$ is not invertible (see [CTS77, Remarque R2], [HY17, Lemma
2.17]) and hence $R^{(1)}_{K/k}(\mathbbm{G}_{m})$ is not retract $k$-rational
(by Theorem 2.1).

(ii) Case $n\equiv 2\ ({\rm mod}\ 4)$. Write $n=2m$ $(m\geq 3:$ odd$)$. We
take $G=\langle x,y\mid x^{n}=y^{2}=1,y^{-1}xy=x^{-1}\rangle\simeq D_{n}$ and
$H=\langle xy\rangle\simeq C_{2}$. We have the exact sequence

$\displaystyle 0\to
I_{G/H}\to\mathbbm{Z}[G/H]\xrightarrow{\varepsilon}\mathbbm{Z}\to 0$

where $\varepsilon$ is the augmentation map. We will construct an exact
sequence of $G$-lattices

$\displaystyle 0\to C\to P\to I_{G/H}\to 0$

with $P$ permutation and $C$ stably permutation, i.e. $[C]=0$. In particular,
$C$ and the dual $(C)^{\circ}={\rm Hom}_{\mathbbm{Z}}(C,\mathbbm{Z})$ are
invertible. Then we get a flabby resolution

$\displaystyle 0\to J_{G/H}\to(P)^{\circ}\to(C)^{\circ}\to 0$

of $J_{G/H}$. This implies that $[J_{G/H}]^{fl}=[(C)^{\circ}]=0$ and hence
$R^{(1)}_{K/k}(\mathbbm{G}_{m})$ is stably $k$-rational (by Theorem 2.1).

Let $e_{1},\ldots,e_{n}$ be the standard $\mathbbm{Z}$-basis of
$\mathbbm{Z}[G/H]$ via the left regular representation of $G/H$:

$\displaystyle x$ $\displaystyle:e_{1}\mapsto\cdots\mapsto e_{n}\mapsto
e_{1},$ $\displaystyle y$ $\displaystyle:e_{i}\leftrightarrow e_{n+1-i}\
(1\leq i\leq m).$

Define $f_{i}:=e_{i}-e_{i+1}$ $(1\leq i\leq n-1)$ and $f_{n}:=e_{n}-e_{1}$.
Then $f_{1},\ldots,f_{n-1}$ becomes a $\mathbbm{Z}$-basis of $I_{G/H}$ with
$\sum_{i=1}^{n}f_{i}=0$, i.e. $f_{n}=-\sum_{i=1}^{n-1}f_{i}$:

$\displaystyle x$ $\displaystyle:f_{1}\mapsto\cdots\mapsto
f_{n-1}\mapsto-\sum_{i=1}^{n-1}f_{i},$ $\displaystyle y$
$\displaystyle:f_{i}\leftrightarrow-f_{n-i}\ (1\leq i\leq m).$

We consider a permutation $G$-lattice
$P\simeq\mathbbm{Z}[G/Z(G)]\oplus\mathbbm{Z}[G/H]\simeq\mathbbm{Z}[G/\langle
x^{m}\rangle]\oplus\mathbbm{Z}[G/\langle
xy\rangle]\simeq\mathbbm{Z}[D_{m}]\oplus\mathbbm{Z}[D_{n}/C_{2}]$ $(n=2m)$
where $H={\rm Stab}_{1}(G)$ with $\mathbbm{Z}$-basis
$\alpha_{1},\ldots,\alpha_{m}$, $\beta_{1},\ldots,\beta_{m}$,
$\gamma_{1},\ldots,\gamma_{n}$ on which $G$ acts by

$\displaystyle x$
$\displaystyle:\alpha_{1}\mapsto\cdots\mapsto\alpha_{m}\mapsto\alpha_{1},\beta_{1}\mapsto\cdots\mapsto\beta_{m}\mapsto\beta_{1},\gamma_{1}\mapsto\cdots\mapsto\gamma_{n}\mapsto\gamma_{1},$
$\displaystyle y$ $\displaystyle:\alpha_{i}\leftrightarrow\beta_{m-i}\ (1\leq
i\leq
m-1),\alpha_{m}\leftrightarrow\beta_{m},\gamma_{j}\leftrightarrow\gamma_{n+1-j}\
(1\leq j\leq n).$

Note that ${\rm rank}_{\mathbbm{Z}}P=2m+n=2n$.

We define a $G$-homomorphism

$\displaystyle\varphi:P\rightarrow I_{G/H},\ $ $\displaystyle\alpha_{i}\mapsto
f_{i}+f_{m+i},\ \beta_{i}\mapsto-(f_{i}+f_{m+i})\ (1\leq i\leq m),$
$\displaystyle\gamma_{j}\mapsto
x^{j-1}\left(\sum_{l=1}^{m-1}f_{l}+f_{m+1}\right)\ (1\leq j\leq n).$

We claim that $\varphi$ is surjective. Indeed, we see that
$\sum_{l=2}^{m-1}f_{l}=\varphi(\gamma_{1})-\varphi(\alpha_{1})$,
$f_{m}+f_{m+2}=-\varphi(\gamma_{1})+\varphi(\alpha_{1})+\varphi(\gamma_{2})$,
$f_{i}+f_{i+2}=x^{i-m}(f_{m}+f_{m+2})\in{\rm Image}(\varphi)$ and
$\sum_{l=i}^{i+3}f_{l}\in{\rm Image}(\varphi)$. Thus it follows from
$\sum_{l=2}^{m-1}f_{l},\sum_{l=i}^{i+3}f_{l}\in{\rm Image}(\varphi)$ that
$f_{1}$ (resp. $f_{2}$) $\in{\rm Image}(\varphi)$ when $m\equiv 1$ $({\rm
mod}\ 4)$ (resp. $m\equiv 3$ $({\rm mod}\ 4)$) and hence
$f_{i}=x^{i-1}(f_{1})$ (resp. $f_{i}=x^{i-2}(f_{2})$) $\in{\rm
Image}(\varphi)$. This implies that $\varphi$ is surjective.

We get an exact sequence of $G$-lattices

$\displaystyle 0\to C\to P\xrightarrow{\varphi}I_{G/H}\to 0$

where $C={\rm Ker}(\varphi)$ with ${\rm rank}_{\mathbbm{Z}}C=n+1=2m+1$. We
find a $\mathbbm{Z}$-basis $a_{i}=\alpha_{i}+\beta_{i}$,
$b_{i}=x^{i-1}(\alpha_{1}+\beta_{m}-\gamma_{1}-\gamma_{m+1})$ $(1\leq i\leq
m)$, $c_{1}=\sum_{i=1}^{m}\alpha_{i}$ of $C$ and the action of $G$ on
$C=\langle a_{1},\ldots,a_{m},b_{1},\ldots,b_{m},c_{1}\rangle_{\mathbbm{Z}}$
is given by

$\displaystyle x$ $\displaystyle:a_{1}\mapsto\cdots\mapsto a_{m}\mapsto
a_{1},b_{1}\mapsto\cdots\mapsto b_{m}\mapsto b_{1},c_{1}\mapsto c_{1},$
$\displaystyle y$ $\displaystyle:a_{i}\leftrightarrow
a_{m-i},b_{i}\leftrightarrow b_{m+1-i}\ (1\leq
i\leq\tfrac{m-1}{2}),a_{m}\mapsto a_{m},b_{\frac{m+1}{2}}\mapsto
b_{\frac{m+1}{2}},c_{1}\mapsto\sum_{i=1}^{m}a_{i}-c_{1}.$

We remark that the action of $G$ on $C$ is not faithful. Indeed, the subgroup
$Z(G)=\langle x^{m}\rangle\simeq C_{2}$ acts on $C$ trivially and hence we can
regard $C$ as a $D_{m}$-lattice where $D_{m}=G/Z(G)=G/\langle x^{m}\rangle$
$(n=2m)$.

Take an element $c_{2}=\sum_{i=1}^{m}\beta_{i}\in P$. Then we have

$\displaystyle x$ $\displaystyle:c_{1}\mapsto c_{1},c_{2}\mapsto c_{2},$
$\displaystyle y$ $\displaystyle:c_{1}\mapsto c_{2},c_{2}\mapsto c_{1}.$

Then we consider the trivial $G$-lattice $\langle
z_{0}\rangle_{\mathbbm{Z}}\simeq\mathbbm{Z}$ and extend the map $\varphi$ from
$P$ to $P\oplus\langle z_{0}\rangle_{\mathbbm{Z}}$ by
$\widetilde{\varphi}(z_{0})=0$:

$\displaystyle 0\to C\oplus\langle z_{0}\rangle_{\mathbbm{Z}}\to
P\oplus\langle
z_{0}\rangle_{\mathbbm{Z}}\xrightarrow{\widetilde{\varphi}}I_{G/H}\to 0$

where $C\oplus\langle z_{0}\rangle_{\mathbbm{Z}}={\rm
Ker}(\widetilde{\varphi})$ with ${\rm rank}_{\mathbbm{Z}}(C\oplus\langle
z_{0}\rangle_{\mathbbm{Z}})=(n+1)+1=2m+2$.

Because ${\rm gcd}\\{2,m\\}=1$, there exist $u,v\in\mathbbm{Z}$ such that
$2u+mv=1$. Then we can get a $\mathbbm{Z}$-basis
$a_{i}^{\prime}:=a_{i}+vz_{0}$, $b_{i}$ $(1\leq i\leq m)$,
$c_{l}^{\prime}:=c_{l}-uz_{0}$ $(1\leq l\leq 2)$ of $C\oplus\langle
z_{0}\rangle_{\mathbbm{Z}}\simeq\mathbbm{Z}[D_{m}/\langle
x^{2}y\rangle]\oplus\mathbbm{Z}[D_{m}/\langle
xy\rangle]\oplus\mathbbm{Z}[D_{m}/\langle x\rangle]$ with $\mathbbm{Z}$-rank
$(2m+1)+1=m+m+2$ on which $D_{m}=G/\langle x^{m}\rangle$ acts by

$\displaystyle x$ $\displaystyle:a_{1}^{\prime}\mapsto\cdots\mapsto
a_{m}^{\prime}\mapsto a_{1}^{\prime},b_{1}\mapsto\cdots\mapsto b_{m}\mapsto
b_{1},c_{1}^{\prime}\mapsto c_{1}^{\prime},c_{2}^{\prime}\mapsto
c_{2}^{\prime},$ $\displaystyle y$
$\displaystyle:a_{i}^{\prime}\leftrightarrow
a_{m-i}^{\prime},b_{i}\leftrightarrow b_{m+1-i}\ (1\leq
i\leq\tfrac{m-1}{2}),a_{m}^{\prime}\mapsto
a_{m}^{\prime},b_{\frac{m+1}{2}}\mapsto
b_{\frac{m+1}{2}},c_{1}^{\prime}\leftrightarrow c_{2}^{\prime}.$

Indeed, we can confirm that

$\displaystyle\ \ \ \begin{array}[]{ccc|cc}\lx@intercol\hfil\overbrace{\hskip
42.67912pt}^{m}\hfil\lx@intercol&\lx@intercol\hfil\,\overbrace{\hskip
34.1433pt}^{2}\hfil\lx@intercol\end{array}$ $\displaystyle\det$
$\displaystyle\left(\begin{array}[]{ccc|cc}1&&&0&v\\\
&\ddots&&\vdots&\vdots\\\ &&1&0&v\\\ \hline\cr 0&\cdots&0&1&-u\\\
1&\cdots&1&-1&-u\\\ \end{array}\right)=-2u-mv=-1.$

Then we find that $C\oplus\langle
z_{0}\rangle_{\mathbbm{Z}}\simeq\mathbbm{Z}[D_{m}/\langle
x^{2}y\rangle]\oplus\mathbbm{Z}[D_{m}/\langle
xy\rangle]\oplus\mathbbm{Z}[D_{m}/\langle x\rangle]$ is a permutation
$G$-lattice ($D_{m}$-lattice) and hence $[C]=0$.

The last statement follows from the exact sequence

$\displaystyle 0\to
J_{G/H}\to(P)^{\circ}\oplus\mathbbm{Z}\to(C)^{\circ}\oplus\mathbbm{Z}\to 0$

with $(P)^{\circ}\oplus\mathbbm{Z}\simeq\mathbbm{Z}[G/\langle
x^{m}\rangle]\oplus\mathbbm{Z}[G/\langle xy\rangle]\oplus\mathbbm{Z}$,
$(C^{\circ})\oplus\mathbbm{Z}\simeq\mathbbm{Z}[D_{m}/\langle
x^{2}y\rangle]\oplus\mathbbm{Z}[D_{m}/\langle
xy\rangle]\oplus\mathbbm{Z}[D_{m}/\langle x\rangle]$ because it follows from
Endo and Miyata [EM73, Proposition 1.10] and Lenstra [Len74, Proposition 1.5]
(see also Ono [Ono63, Proposition 1.2.2], Swan [Swa10, Lemma 3.1], Hoshi, Kang
and Kitayama [HKK14, Proof of Theorem 6.5]) that
$L(J_{G/H}\oplus(C)^{\circ}\oplus\mathbbm{Z})^{G}\simeq
L((P)^{\circ}\oplus\mathbbm{Z})^{G}$.∎

As a consequence of the proof of Theorem 1.1 above, we get:

######  Corollary 3.2.

Let $C={\rm Ker(\varphi)}$ be a $D_{m}$-lattice with ${\rm
rank}_{\mathbbm{Z}}C=2m+1$ $(m\geq 3:odd)$ given as in (ii) Case $n\equiv 2\
({\rm mod}\ 4)$ in the proof of Theorem 1.1. Then we have
$\widehat{H}^{0}(D_{m},C)$ $\simeq\mathbbm{Z}/2\mathbbm{Z}$ where
$\widehat{H}^{0}(G,M)=M^{G}/\\{\sum_{g\in G}g\cdot m\mid m\in M\\}$ is the 0th
Tate cohomology. In particular, the $D_{m}$-lattice $C$ is not permutation but
stably permutation which satisfies
$C\oplus\mathbbm{Z}\simeq\mathbbm{Z}[D_{m}/\langle
x^{2}y\rangle]\oplus\mathbbm{Z}[D_{m}/\langle
xy\rangle]\oplus\mathbbm{Z}[D_{m}/\langle x\rangle]$ with $\mathbbm{Z}$-rank
$(2m+1)+1=m+m+2$.

###### Proof.

Recall that we can regard $D_{n}$-lattice $C$ as $D_{m}$-lattice via
$D_{m}=D_{n}/Z(D_{n})=D_{n}/\langle x^{m}\rangle$. By Shapiro’s lemma, we have
$\widehat{H}^{0}(D_{m},\mathbbm{Z}[D_{m}/H])\simeq\widehat{H}^{0}(H,\mathbbm{Z})\simeq\mathbbm{Z}/|H|\mathbbm{Z}$
for any subgroup $H\leq D_{m}$. Hence it follows from the isomorphism
$C\oplus\mathbbm{Z}\simeq\mathbbm{Z}[D_{m}/\langle
x^{2}y\rangle]\oplus\mathbbm{Z}[D_{m}/\langle
xy\rangle]\oplus\mathbbm{Z}[D_{m}/\langle x\rangle]$ given in the proof of
Theorem 1.1 that
$\widehat{H}^{0}(D_{m},C\oplus\mathbbm{Z})\simeq\mathbbm{Z}/2\mathbbm{Z}\oplus\mathbbm{Z}/2\mathbbm{Z}\oplus\mathbbm{Z}/m\mathbbm{Z}$.
Because
$\widehat{H}^{0}(D_{m},\mathbbm{Z})\simeq\mathbbm{Z}/2m\mathbbm{Z}\simeq\mathbbm{Z}/2\mathbbm{Z}\oplus\mathbbm{Z}/m\mathbbm{Z}$
($m:$ odd), we get that
$\widehat{H}^{0}(D_{m},C)\simeq\mathbbm{Z}/2\mathbbm{Z}$. Suppose that
$C=\oplus_{i=1}^{r}\mathbbm{Z}[D_{m}/H_{i}]$ is permutation. Then
$\widehat{H}^{0}(D_{m},C)\simeq\oplus_{i=1}^{r}\mathbbm{Z}/|H_{i}|\mathbbm{Z}$.
This implies that the $\mathbbm{Z}$-rank of $C$ is a multiple of $m$ because
$|H_{i}|=1$ or $2$. Contradiction. ∎

##  4\. Appendix: refinements of the proof of stably rational cases with
$G\simeq D_{n}$ $(n\geq 3:odd)$

We give refinements of the proof of stably rational cases with $G\simeq D_{n}$
$(n\geq 3:$ odd$)$ of Endo and Miyata’s theorems [EM75, Theorem 2.3] with
$H=\\{1\\}$ and [End11, Theorem 3.1] with $H\simeq C_{2}$, using a technique
similar to that given in the proof of Theorem 1.1. We remark that the original
proofs given as in $(1)\Rightarrow(2)$ of [EM75, Theorem 2.3] and
$(1)\Rightarrow(2)$ of [End11, Theorem 3.1] use an induction argument on the
number of prime divisors of $n$ and our refinements can be obtained by
constructing explicit flabby resolutions of $J_{G/H}$ for $H=\\{1\\}$ and
$H\simeq C_{2}$ respectively.

Refinement of the proof of Theorem 2.3 (Endo and Miyata [EM75, Theorem 2.3]):
stably rational cases with $G\simeq D_{n}$ $(n\geq 3:$ odd$)$ and $H=\\{1\\}$.

Let $L/k$ be a finite Galois field extension and $G={\rm Gal}(L/k)$. We
consider the case where $G\simeq D_{n}=\langle x,y\mid
x^{n}=y^{2}=1,y^{-1}xy=x^{-1}\rangle\leq S_{2n}$ $(n\geq 3:$ odd$)$ and
$H=\\{1\\}$ where $x=(1\ 3\ \cdots\ 2n-1)(2\ 4\ \cdots\ 2n)$, $y=(1\ 2n)(2\
2n-1)\cdots(n\ n+1)$.

Take the exact sequence

$\displaystyle 0\to
I_{G}\to\mathbbm{Z}[G]\xrightarrow{\varepsilon}\mathbbm{Z}\to 0$

where $\varepsilon$ is the augmentation map. We will construct an exact
sequence of $G$-lattices

$\displaystyle 0\to C\to P\to I_{G}\to 0$

with $P$ permutation and $C$ stably permutation, i.e. $[C]=0$. Then we get a
flabby resolution

$\displaystyle 0\to J_{G}\to(P)^{\circ}\to(C)^{\circ}\to 0$

of $J_{G}$. This implies that $[J_{G}]^{fl}=[(C)^{\circ}]=0$ and hence
$R^{(1)}_{K/k}(\mathbbm{G}_{m})$ is stably $k$-rational (by Theorem 2.1).

Let $e_{1},\ldots,e_{2n}$ be the standard $\mathbbm{Z}$-basis of
$\mathbbm{Z}[G]$:

$\displaystyle x$ $\displaystyle:e_{1}\mapsto e_{3}\mapsto\cdots\mapsto
e_{2n-1}\mapsto e_{1},e_{2}\mapsto e_{4}\mapsto\cdots\mapsto e_{2n},$
$\displaystyle y$ $\displaystyle:e_{i}\leftrightarrow e_{2n+1-i}\ (1\leq i\leq
n).$

Define $f_{i}:=e_{i}-e_{i+1}$ $(1\leq i\leq 2n-1)$ and $f_{2n}:=e_{2n}-e_{1}$.
Then $f_{1},\ldots,f_{2n-1}$ becomes a $\mathbbm{Z}$-basis of $I_{G}$ with
$\sum_{i=1}^{2n}f_{i}=0$, i.e. $f_{2n}=-\sum_{i=1}^{2n-1}f_{i}$:

$\displaystyle x$ $\displaystyle:f_{1}\mapsto\cdots\mapsto
f_{2n-1}\mapsto-\sum_{i=1}^{2n-1}f_{i},$ $\displaystyle y$
$\displaystyle:f_{i}\leftrightarrow-f_{2n-i}\ (1\leq i\leq n).$

We consider a permutation $G$-lattice
$P\simeq\mathbbm{Z}[G]\oplus\mathbbm{Z}[G]$ with $\mathbbm{Z}$-basis
$\alpha_{1},\alpha_{3},\ldots,\alpha_{2n-1}$,
$\beta_{1},\beta_{3},\ldots,\beta_{2n-1}$,
$\alpha_{2},\alpha_{4},\ldots,\alpha_{2n}$,
$\beta_{2},\beta_{4},\ldots,\beta_{2n}$ on which $G$ acts by

$\displaystyle x$
$\displaystyle:\alpha_{1}\mapsto\alpha_{3}\mapsto\cdots\mapsto\alpha_{2n-1}\mapsto\alpha_{1},\beta_{1}\mapsto\beta_{3}\mapsto\cdots\mapsto\beta_{2n-1}\mapsto\beta_{1},$
$\displaystyle\ \ \
\alpha_{2}\mapsto\alpha_{4}\mapsto\cdots\mapsto\alpha_{2n}\mapsto\alpha_{2},\beta_{2}\mapsto\beta_{4}\mapsto\cdots\mapsto\beta_{2n}\mapsto\beta_{2},$
$\displaystyle y$ $\displaystyle:\alpha_{i}\leftrightarrow\beta_{2n-i}\ (1\leq
i\leq 2n-1),\alpha_{2n}\leftrightarrow\beta_{2n}.$

We define a surjective $G$-homomorphism

$\displaystyle\varphi:P\rightarrow I_{G},\ $ $\displaystyle\alpha_{i}\mapsto
f_{i},\ \beta_{i}\mapsto-f_{i}\ (1\leq i\leq 2n).$

Then we get an exact sequence of $G$-lattices

$\displaystyle 0\to C\to P\xrightarrow{\varphi}I_{G}\to 0$

where $C={\rm Ker}(\varphi)$ with ${\rm rank}_{\mathbbm{Z}}C=2n+1$. We find a
$\mathbbm{Z}$-basis $a_{i}=\alpha_{i}+\beta_{i}$ $(1\leq i\leq n)$,
$b_{1}=\sum_{i=1}^{2n}\alpha_{i}$ of $C$ and the action of $G$ on $C=\langle
a_{1},\ldots,a_{2n},b_{1}\rangle_{\mathbbm{Z}}$ is given by

$\displaystyle x$ $\displaystyle:a_{1}\mapsto a_{3}\mapsto\cdots\mapsto
a_{2n-1}\mapsto a_{1},a_{2}\mapsto a_{4}\mapsto\cdots\mapsto a_{2n}\mapsto
a_{2},b_{1}\mapsto b_{1},$ $\displaystyle y$
$\displaystyle:a_{i}\leftrightarrow a_{2n-i}\ (1\leq i\leq n-1),a_{2n}\mapsto
a_{2n},b_{1}\mapsto\sum_{i=1}^{2n}a_{i}-b_{1}.$

As in the proof of Theorem 1.1, we consider the trivial $G$-lattice $\langle
z_{0}\rangle_{\mathbbm{Z}}\simeq\mathbbm{Z}$ and extend the map $\varphi$ from
$P$ to $P\oplus\langle z_{0}\rangle_{\mathbbm{Z}}$ by
$\widetilde{\varphi}(z_{0})=0$:

$\displaystyle 0\to C\oplus\langle z_{0}\rangle_{\mathbbm{Z}}\to
P\oplus\langle
z_{0}\rangle_{\mathbbm{Z}}\xrightarrow{\widetilde{\varphi}}I_{G}\to 0$

where $C\oplus\langle z_{0}\rangle_{\mathbbm{Z}}={\rm
Ker}(\widetilde{\varphi})$ with ${\rm rank}_{\mathbbm{Z}}(C\oplus\langle
z_{0}\rangle_{\mathbbm{Z}})=(2n+1)+1$.

Because ${\rm gcd}\\{2,n\\}=1$, there exist $u,v\in\mathbbm{Z}$ such that
$2u+nv=1$. Then we can get a $\mathbbm{Z}$-basis
$a_{2i-1}^{\prime}:=a_{2i-1}+vz_{0}$, $a_{2i}$ $(1\leq i\leq n)$,
$b_{l}^{\prime}:=b_{l}-uz_{0}$ $(1\leq l\leq 2)$ of $C\oplus\langle
z_{0}\rangle_{\mathbbm{Z}}\simeq\mathbbm{Z}[G/\langle
xy\rangle]\oplus\mathbbm{Z}[G/\langle
x^{2}y\rangle]\oplus\mathbbm{Z}[G/\langle x\rangle]$ with $\mathbbm{Z}$-rank
$(2n+1)+1=n+n+2$ on which $G$ acts by

$\displaystyle x$ $\displaystyle:a_{1}^{\prime}\mapsto
a_{3}^{\prime}\mapsto\cdots\mapsto a_{2n-1}^{\prime}\mapsto
a_{1}^{\prime},a_{2}\mapsto a_{4}\mapsto\cdots\mapsto a_{2n}\mapsto
a_{2},b_{1}^{\prime}\mapsto b_{1}^{\prime},b_{2}^{\prime}\mapsto
b_{2}^{\prime},$ $\displaystyle y$
$\displaystyle:a_{2i-1}^{\prime}\leftrightarrow a_{2n-(2i-1)}^{\prime}\ (1\leq
i\leq n),a_{2j}\leftrightarrow a_{2n-2j}\ (1\leq j\leq n-1),a_{2n}\mapsto
a_{2n},b_{1}^{\prime}\leftrightarrow b_{2}^{\prime}.$

Indeed, we can confirm that

$\displaystyle\ \ \ \begin{array}[]{ccc|cc}\lx@intercol\hfil\overbrace{\hskip
42.67912pt}^{n}\hfil\lx@intercol&\lx@intercol\hfil\,\overbrace{\hskip
34.1433pt}^{2}\hfil\lx@intercol\end{array}$ $\displaystyle\det$
$\displaystyle\left(\begin{array}[]{ccc|cc}1&&&0&v\\\
&\ddots&&\vdots&\vdots\\\ &&1&0&v\\\ \hline\cr 0&\cdots&0&1&-u\\\
1&\cdots&1&-1&-u\\\ \end{array}\right)=-2u-nv=-1.$

Then $C\oplus\langle z_{0}\rangle_{\mathbbm{Z}}\simeq\mathbbm{Z}[G/\langle
xy\rangle]\oplus\mathbbm{Z}[G/\langle
x^{2}y\rangle]\oplus\mathbbm{Z}[G/\langle x\rangle]$ is a permutation
$G$-lattice and hence $[C]=0$.∎

It follows from the exact sequence

$\displaystyle 0\to
J_{G}\to(P)^{\circ}\oplus\mathbbm{Z}\to(C)^{\circ}\oplus\mathbbm{Z}\to 0,$

Endo and Miyata [EM73, Proposition 1.10] and Lenstra [Len74, Proposition 1.5]
(see also Ono [Ono63, Proposition 1.2.2], Swan [Swa10, Lemma 3.1], Hoshi, Kang
and Kitayama [HKK14, Proof of Theorem 6.5]) that
$L(J_{G}\oplus(C)^{\circ}\oplus\mathbbm{Z})^{G}\simeq
L((P)^{\circ}\oplus\mathbbm{Z})^{G}$. Hence we get:

######  Corollary 4.1.

Let $G\simeq D_{n}=\langle x,y\mid x^{n}=y^{2}=1,y^{-1}xy=x^{-1}\rangle\leq
S_{2n}$ $(n\geq 3:$ odd$)$ and $H=\\{1\\}$. Then
$R^{(1)}_{K/k}(\mathbbm{G}_{m})\times R_{k_{1}/k}(\mathbbm{G}_{m,k_{1}})\times
R_{k_{2}/k}(\mathbbm{G}_{m,k_{2}})\times R_{k_{3}/k}(\mathbbm{G}_{m,k_{3}})$
of dimension $(2n-1)+n+n+2=4n+1$ and $R_{L/k}(\mathbbm{G}_{m,L})\times
R_{L/k}(\mathbbm{G}_{m,L})\times\mathbbm{G}_{m,k}$ of dimension $2n+2n+1=4n+1$
are birationally $k$-equivalent where $R_{k_{i}/k}(\mathbbm{G}_{m,k_{i}})$ and
$R_{L/k}(\mathbbm{G}_{m,L})$ are $k$-rational with $k_{1}=L^{\langle
xy\rangle}$, $k_{2}=L^{\langle x^{2}y\rangle}$, $k_{3}=L^{\langle x\rangle}$,
$\langle xy\rangle\simeq\langle x^{2}y\rangle\simeq C_{2}$, $\langle
x\rangle\simeq C_{n}$ and $[k_{1}:k]=[k_{2}:k]=n$, $[k_{3}:k]=2$.

######  Remark 4.2.

We see that the $G$-lattice $C$ is not permutation but stably permutation as
in Corollary 3.2.

Refinement of the proof of Theorem 2.5 (Endo [End11, Theorem 3.1]): stably
rational cases with $G\simeq D_{n}$ $(n\geq 3:$ odd$)$ and $H\simeq C_{2}$.

Let $K/k$ be a finite non-Galois separable field extension, $L/k$ be the
Galois closure of $K/k$ and $G={\rm Gal}(L/k)$. By Lemma 3.1, we may assume
that $G\simeq D_{n}=\langle x,y\mid x^{n}=y^{2}=1,y^{-1}xy=x^{-1}\rangle\leq
S_{n}$ $(n\geq 3:$ odd$)$ and $H=\langle xy\rangle$ where $x=(1\ 2\ \cdots\
n)$, $y=(1\ n)(2\ n-1)\cdots(\tfrac{n-1}{2}\ \tfrac{n+3}{2})$, $xy=(2\ n)(3\
n-1)\cdots(\tfrac{n+1}{2}\ \tfrac{n+1}{2}+1)$.

We take the exact sequence

$\displaystyle 0\to
I_{G/H}\to\mathbbm{Z}[G/H]\xrightarrow{\varepsilon}\mathbbm{Z}\to 0$

where $\varepsilon$ is the augmentation map. We will construct an exact
sequence of $G$-lattices

$\displaystyle 0\to C\to P\to I_{G/H}\to 0$

with $P$ permutation and $C$ stably permutation, i.e. $[C]=0$. Then we get a
flabby resolution

$\displaystyle 0\to J_{G/H}\to(P)^{\circ}\to(C)^{\circ}\to 0$

of $J_{G/H}$. This implies that $[J_{G/H}]^{fl}=[(C)^{\circ}]=0$ and hence
$R^{(1)}_{K/k}(\mathbbm{G}_{m})$ is stably $k$-rational (by Theorem 2.1).

Let $e_{1},\ldots,e_{n}$ be the standard $\mathbbm{Z}$-basis of
$\mathbbm{Z}[G/H]$ via the left regular representation of $G/H$:

$\displaystyle x$ $\displaystyle:e_{1}\mapsto\cdots\mapsto e_{n}\mapsto
e_{1},$ $\displaystyle y$ $\displaystyle:e_{i}\leftrightarrow e_{n+1-i}\
(1\leq i\leq\tfrac{n-1}{2}),e_{\tfrac{n+1}{2}}\mapsto e_{\tfrac{n+1}{2}}.$

Define $f_{i}:=e_{i}-e_{i+1}$ $(1\leq i\leq n-1)$ and $f_{n}:=e_{n}-e_{1}$.
Then $f_{1},\ldots,f_{n-1}$ becomes a $\mathbbm{Z}$-basis of $I_{G/H}$ with
$\sum_{i=1}^{n}f_{i}=0$, i.e. $f_{n}=-\sum_{i=1}^{n-1}f_{i}$:

$\displaystyle x$ $\displaystyle:f_{1}\mapsto\cdots\mapsto
f_{n-1}\mapsto-\sum_{i=1}^{n-1}f_{i},$ $\displaystyle y$
$\displaystyle:f_{i}\leftrightarrow-f_{n+1-i}\ (1\leq i\leq\tfrac{n+1}{2}).$

We consider a permutation $G$-lattice $P\simeq\mathbbm{Z}[G]$ with
$\mathbbm{Z}$-basis $\alpha_{1},\ldots,\alpha_{n}$,
$\beta_{1},\ldots,\beta_{n}$ on which $G$ acts by

$\displaystyle x$
$\displaystyle:\alpha_{1}\mapsto\cdots\mapsto\alpha_{n}\mapsto\alpha_{1},\beta_{1}\mapsto\cdots\mapsto\beta_{n}\mapsto\beta_{1},$
$\displaystyle y$ $\displaystyle:\alpha_{i}\leftrightarrow\beta_{n+1-i}\
(1\leq i\leq n).$

We define a surjective $G$-homomorphism

$\displaystyle\varphi:P\rightarrow I_{G/H},\ $ $\displaystyle\alpha_{i}\mapsto
f_{i},\ \beta_{i}\mapsto-f_{i}\ (1\leq i\leq n).$

Then we get an exact sequence of $G$-lattices

$\displaystyle 0\to C\to P\xrightarrow{\varphi}I_{G/H}\to 0$

where $C={\rm Ker}(\varphi)$ with ${\rm rank}_{\mathbbm{Z}}C=n+1$. We find a
$\mathbbm{Z}$-basis $a_{i}=\alpha_{i}+\beta_{i}$ $(1\leq i\leq n)$,
$b_{1}=\sum_{i=1}^{n}\alpha_{i}$ of $C$ and the action of $G$ on $C=\langle
a_{1},\ldots,a_{n},b_{1}\rangle_{\mathbbm{Z}}$ is given by

$\displaystyle x$ $\displaystyle:a_{1}\mapsto\cdots\mapsto a_{n}\mapsto
a_{1},b_{1}\mapsto b_{1},$ $\displaystyle y$
$\displaystyle:a_{i}\leftrightarrow a_{n+1-i}\ (1\leq
i\leq\tfrac{n-1}{2}),a_{\tfrac{n+1}{2}}\mapsto
a_{\tfrac{n+1}{2}},b_{1}\mapsto\sum_{i=1}^{n}a_{i}-b_{1}.$

As in the proof of Theorem 1.1, we consider the trivial $G$-lattice $\langle
z_{0}\rangle_{\mathbbm{Z}}\simeq\mathbbm{Z}$ and extend the map $\varphi$ from
$P$ to $P\oplus\langle z_{0}\rangle_{\mathbbm{Z}}$ by
$\widetilde{\varphi}(z_{0})=0$:

$\displaystyle 0\to C\oplus\langle z_{0}\rangle_{\mathbbm{Z}}\to
P\oplus\langle
z_{0}\rangle_{\mathbbm{Z}}\xrightarrow{\widetilde{\varphi}}I_{G/H}\to 0$

where $C\oplus\langle z_{0}\rangle_{\mathbbm{Z}}={\rm
Ker}(\widetilde{\varphi})$ with ${\rm rank}_{\mathbbm{Z}}(C\oplus\langle
z_{0}\rangle_{\mathbbm{Z}})=(n+1)+1$.

Because ${\rm gcd}\\{2,n\\}=1$, there exist $u,v\in\mathbbm{Z}$ such that
$2u+nv=1$. Then we can get a $\mathbbm{Z}$-basis
$a_{i}^{\prime}:=a_{i}+vz_{0}$ $(1\leq i\leq n)$,
$b_{l}^{\prime}:=b_{l}-uz_{0}$ $(1\leq l\leq 2)$ of $C\oplus\langle
z_{0}\rangle_{\mathbbm{Z}}\simeq\mathbbm{Z}[G/\langle
xy\rangle]\oplus\mathbbm{Z}[G/\langle x\rangle]$ with $\mathbbm{Z}$-rank
$(n+1)+1=n+2$ on which $G$ acts by

$\displaystyle x$ $\displaystyle:a_{1}^{\prime}\mapsto\cdots\mapsto
a_{n}^{\prime}\mapsto a_{1}^{\prime},b_{1}^{\prime}\mapsto
b_{1}^{\prime},b_{2}^{\prime}\mapsto b_{2}^{\prime},$ $\displaystyle y$
$\displaystyle:a_{i}^{\prime}\leftrightarrow a_{n+1-i}^{\prime}\ (1\leq
i\leq\tfrac{n-1}{2}),a_{\tfrac{n+1}{2}}^{\prime}\mapsto
a_{\tfrac{n+1}{2}}^{\prime},b_{1}^{\prime}\leftrightarrow b_{2}^{\prime}.$

Indeed, we can confirm that

$\displaystyle\ \ \ \begin{array}[]{ccc|cc}\lx@intercol\hfil\overbrace{\hskip
42.67912pt}^{n}\hfil\lx@intercol&\lx@intercol\hfil\,\overbrace{\hskip
34.1433pt}^{2}\hfil\lx@intercol\end{array}$ $\displaystyle\det$
$\displaystyle\left(\begin{array}[]{ccc|cc}1&&&0&v\\\
&\ddots&&\vdots&\vdots\\\ &&1&0&v\\\ \hline\cr 0&\cdots&0&1&-u\\\
1&\cdots&1&-1&-u\\\ \end{array}\right)=-2u-nv=-1.$

Then we find that $C\oplus\langle
z_{0}\rangle_{\mathbbm{Z}}\simeq\mathbbm{Z}[G/\langle
xy\rangle]\oplus\mathbbm{Z}[G/\langle x\rangle]$ is a permutation $G$-lattice
and hence $[C]=0$.∎

It follows from the exact sequence

$\displaystyle 0\to
J_{G/H}\to(P)^{\circ}\oplus\mathbbm{Z}\to(C)^{\circ}\oplus\mathbbm{Z}\to 0,$

Endo and Miyata [EM73, Proposition 1.10] and Lenstra [Len74, Proposition 1.5]
(see also Ono [Ono63, Proposition 1.2.2], Swan [Swa10, Lemma 3.1], Hoshi, Kang
and Kitayama [HKK14, Proof of Theorem 6.5]) that
$L(J_{G/H}\oplus(C)^{\circ}\oplus\mathbbm{Z})^{G}\simeq
L((P)^{\circ}\oplus\mathbbm{Z})^{G}$. Hence we get:

######  Corollary 4.3.

Let $G\simeq D_{n}=\langle x,y\mid x^{n}=y^{2}=1,y^{-1}xy=x^{-1}\rangle\leq
S_{n}$ $(n\geq 3:$ odd$)$ and $H=\langle xy\rangle\simeq C_{2}$. Then
$R^{(1)}_{K/k}(\mathbbm{G}_{m})\times R_{k_{1}/k}(\mathbbm{G}_{m,k_{1}})\times
R_{k_{2}/k}(\mathbbm{G}_{m,k_{2}})$ of dimension $(n-1)+n+2=2n+1$ and
$R_{L/k}(\mathbbm{G}_{m,L})\times\mathbbm{G}_{m,k}$ of dimension $2n+1$ are
birationally $k$-equivalent where $R_{k_{i}/k}(\mathbbm{G}_{m,k_{i}})$ and
$R_{L/k}(\mathbbm{G}_{m,L})$ are $k$-rational with $k_{1}=L^{\langle
xy\rangle}$, $k_{2}=L^{\langle x\rangle}$, $\langle xy\rangle\simeq C_{2}$,
$\langle x\rangle\simeq C_{n}$ and $[k_{1}:k]=n$, $[k_{2}:k]=2$.

######  Remark 4.4.

We see that the $G$-lattice $C$ is not permutation but stably permutation as
in Corollary 3.2.

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# Autoregressive Transformers for Disruption Prediction in Nuclear Fusion
Plasmas

Lucas Spangher
Plasma Science and Fusion Center
Massachusetts Institute of Technology
<EMAIL_ADDRESS>
William F. Arnold
Kim Jaechul School of AI
KAIST
<EMAIL_ADDRESS>
Alexander Spangher
Information Sciences Institute
University of Southern California Andrew Maris
Plasma Science and Fusion Center
Massachusetts Institute of Technology
Cristina Rea
Plasma Science and Fusion Center
Massachusetts Institute of Technology
<EMAIL_ADDRESS>
###### Abstract

The physical sciences require models tailored to specific nuances of different
dynamics. In this work, we study outcome predictions in nuclear fusion
tokamaks, where a major challenge are disruptions, or the loss of plasma
stability with damaging implications for the tokamak. Although disruptions are
difficult to model using physical simulations, machine learning (ML) models
have shown promise in predicting these phenomena. Here, we first study several
variations on masked autoregressive transformers, achieving an average of 5%
increase in Area Under the Receiving Operating Characteristic metric above
existing methods. We then compare transformer models to limited context neural
networks in order to shed light on the “memory” of plasma effected by tokamaks
controls. With these model comparisons, we argue for the persistence of a
memory throughout the plasma in the context of tokamaks that our model
exploits.

##  1 Introduction

Nuclear fusion has been long sought as a promising method of generating
carbon-free energy without major land-use requirements or high-level
radioactive waste. One major approach to obtaining fusion in laboratory
plasmas is through magnetic confinement, which leverages the tendency of
plasmas to remain confined perpendicular to the magnetic field lines. The
fusion approach with the highest technological readiness level Wurzel and Hsu
(2021) is the tokamak, a toroidal chamber with external magnetic coils and a
large current that runs toroidally through the plasma with the goal of
generating net positive energy (see Figure 1.)

A major problem facing fusion tokamaks are disruptions, or loss of plasma
stability with damaging implications for the tokamak. Disruptions are
difficult to model using physical simulations due in part to the heterogeneity
of causes that can lead to instability, the multi-scale nature of plasma
dynamics, and to a wealth of unobserved factors that affect known physical
dynamics. Thus, there is an emerging literature on methods other than “first
principle models” for disruption prediction.

The goal in this prediction problem is to provide sufficient time to engage
control actions or emergency mitigation systems, which curb the deleterious
consequences of disruptions. Numerous ML models have been proposed to predict
the likelihood that a plasma disruption will occur Murari and Contributors
(2013); Kates-Harbeck et al. (2019). Prior work has investigated the use of
random forest models to monitor plasma states Rea et al.  atlargest tokamak in
the US, DIII-D, followed by current state of the art predictors, such as the
Hybrid Deep Learner (HDL) proposed in 2021, which utilize classical neural
techniques such as a convolutional LSTM network Zhu et al. (2020).

However, none of these models have the desired accuracy for commercial scale
tokamaks where disruption prediction accuracy must be nearly flawless (>95%
true positive rate) Maris et al. (2023). Also, we hypothesize, none are tuned
to the specific needs of the problem. It is unclear how long the “memory” of
the plasma is – while some argue that as little as 200 milliseconds of history
is needed to capture the evolution of the plasma temperature and density Zhu
et al. (2020); Rea et al. , others argue that certain instabilities or tokamak
control errors that occur near the start of a seconds-long shot may impact the
end behavior of the shotsMurari and Contributors (2013).

In this work, we apply variations on the Transformer architecture Vaswani et
al. (2017); Bengio et al. (2009), which has become popular in recent years to
model long term, event-based time-series. Our two contributions are:

  1. 1.

We create a state-of-the-art model on a data from three major tokamaks. We
find an average of .05 improvement in Area under the Receiver Operating
Characteristic Curve (AUC) over the HDL baseline (Table 1.) We find various
pretraining steps help augment performance.

  2. 2.

We investigate whether long term memory is helpful in modeling the fusion
reaction, giving us scientific insight into the dynamics of the process.

##  2 Dataset Provenance and Preparation

Our dataset is composed of trials from three different tokamaks, $r_{1}$,
$r_{2}$ and $r_{3}$.111$r_{1}$ is MIT’s Alcator C-Mod, $r_{2}$ is General
Atomic’s DIII-D, and $r_{3}$ is the Chinese Academy of Science’s EAST tokamak.
Each $r_{i}$ differs considerably222For a longer discussion on the physical
differences between the three, see Granetz et al. (2019). Each tokamak is run
continuously for a “shot” sequence lasting seconds to minutes, generating
gigabytes of multi-modal data, of which we focus on global state variables
(the rest include 1D states ("profiles"), 2D cross-sections, and camera
photos); we focus on shots’ global state variables, listed in Table 3. Our
objective is to understand how well this model would deploy to a new tokamak,
which is similar to $r_{1}$. Thus, we restrict our test data to data generated
by $r_{1}$. We build off previous work from our lab for the training set
composition, which contains different proportions of $r_{1}$, $r_{2}$, and
$r_{3}$ data [citation withheld].

Each shot is of length $n$, and we discard shots that are shorter than 125ms.
Each tokamak’s diagnostics store metrics from each shot at a different
sampling rate. We normalize across tokamaks by discretizing at 5ms and
interpolating via forward-fill. Roughly 10% of shots end in disruptions. We
truncate our time series to $X=1,...,n-\nu$, where $n$ is the full length and
$\nu=40ms$, the minimum amount of time for a disruption mitigation system to
activate. For a description of the features chosen, please see Table 2 in the
Appendix.

##  3 Model Details

Figure 1: Sequence to Label classification task using GPT2-like transformers.

###  3.1 Modeling Task

We seek to perform label prediction (Figure 1), where $X$ is described above,
and $y$ is 1 if the shot ended in disruption and 0 otherwise. We follow Zhu et
al. (2020) in several augmentations. First, we generate additional pairs
$(\overline{X},\overline{y})$ from each $(X,y)$ by randomly sampled starting
and ending windows of the shot. Next, in order enforce greater reasoning about
the start of instabilities, we set a $\tau_{r_{i}}$ variable per
tokamak333$\tau(r_{i})$ is the average length of time before a disruption
occurs which is tokamak-dependent, see Table 3: if
$(\overline{X},\overline{y})$ of length $n$ ends before $n-\tau_{r_{i}}$, it
is labelled 0.

###  3.2 Model Architecture

We adapt an autoregressive transformer based on GPT444We utilize Huggingface’s
implementation:
https://github.com/huggingface/transformers/blob/main/src/transformers/models/gpt2/modeling_gpt2.py
Radford et al. (2018). GPT-based models, which are typically used to model
long sequences of discrete word-tokens, have an embedding layer that is used
to project discrete tokens into a continuous embedding space. The dimensions
of our data are already continuous variables, so we treat them as embeddings
directly. These vectors are normalized, based on training-set mean and
variance.

We perform a hyperparameter sweep using a bayesian hyperop strategy Wu et al.
(2019) over number of layers, dropout proportions, learning rate schedulers,
optimizer $\beta$ parameters, and other hyperparameters of interest. All
training was done on a cluster of 48 A100 Nvidia GPUs and tested on Area Under
Receiver Operator Characteristc (AUC).

###  3.3 Model Augmentations: State Pretraining and Data-cutoff Currica
Induction

State Pretraining: We pretrain the model first on a next state prediction
(NSP) task. We model NSP with a regression head on the model and Mean Average
Error (MAE) loss. After convergence, which is observed to be 50k steps, we
replace the regression head with a classification head and perform our
sequence classification task.

Data-cutoff Curriculum Learning: We notice the model often misclassifies
samples with rapidly evolving end dynamics. We theorize that exposing the
model to the full data, without the 40ms cutoff, might improve its predictions
for the final 40ms. Using curriculum learning Bengio et al. (2009), we
sequentially train the model on easier tasks: predicting datasets with cutoffs
from 0ms to 30ms, in 10ms increments, for 50k steps each. We then test the
model on the main task. For example, a 3 step curriculum trains the model on
data with 20ms cutoff, then 30ms, then 40. We experiment with varying
curriculum lengths from zero to five steps.

##  4 Results and Discussion

###  4.1 Performance metrics

Compared to the baseline HDL predictor, our vanilla model achieves favorable
AUC across all cases, and our augmented model improves further on the vanilla
case (Table 1.) Wall-clock time on forward inference ranges from 1-3ms, which
is suitable for real-time deployment.

| Training Set Composition | AUCs
---|---|---
| Existing: $r^{2},r^{3}$ | “New”: $r^{1}$ | HDL | GPT | Aug-GPT
Case | Non-disruptions (ND) | Disruptions (D) | ND | D |  |  |
1 | None | None | All | 20 | .640 | .781 | .790
2 | None | None | All | All | .799 | .836 | .843
3 | None | All | All | All | .808 | .831 | .841
4 | All | All | None | None | .588 | .622 | .621
5 | All | All | All | 20 | .703 | .699 | .704
6 | All | All | All | All | .794 | .803 | .814
Mean |  |  |  |  | .72 | .76 | .77
Table 1: A table containing different training cases and output metrics
between our transformer and HDL, the baseline. Ours improves prior SOTA by 5%
AUC on average. Please note that “GPT” refers to the vanilla GPT, and “Aug-
GPT” refers to the augmented GPT.

###  4.2 Curriculum Training and State Pretraining

Curriculum Learning: We observe, interestingly, that the model achieves the
highest accuracy, recall, and precision ($\sim 10\%$ gain in each) following a
curriculum in which the models start training 10ms from shots’ ends and
progress successively – i.e. four curriculum steps. The result matches
physical intuition: at the very end of disruptive shots, the plasma has
already touched the tokamak’s walls, and thus the physics of the fluid have
changed.

State pretraining: we observe that model performance consistently improves. We
also observe that the transformer model seems to be a very good state
predictor, which may be a side benefit of the model for other uses.

###  4.3 Long Term Memory

We argue that plasma time-series exhibit long-term memory through three ways:

  1. 1.

A comparison of model performance finds the vanilla GPT significantly
outperforming the baseline HDL, which explicitly encodes limited memory.

  2. 2.

Ablations on data context show that model performance decreases as available
context decreases from 480ms to 80ms, indicating that the model may be using
data context from throughout the shot (Figure 3.)

  3. 3.

Global attention maps, which explicitly encode regions of the data to which
the model pays attention, does not tend towards zero even when the whole shot
is observed (Figure 3)

###  4.4 Model interpretation.

Generally we may interpret characteristics of the data that the transformer
picks out. First, we find the attention spikes maps to loop voltage spikes, of
which higher voltage generally correlates to increased system pressure.
Second, we find that shaded regions of the attention map correspond to an
uptick in $\beta_{p}$ and a drop in $Li$. These characteristics are consistent
with those of a cooling plasma prone for edge instabilities Zohm (1996). Both
of these observations give confidence that the transformer is picking up on
known physical events.

Figure 2: Exploration of transformer behavior. Here, we ablate on limiting
data context to test whether performance is directly effected by physical
memory. Figure 3: Here we show a further exploration of transformer behavior.
In the top plot, we show normalized input features, corresponding to the
transformer’s attention map below. To read the attention map, an unrolling
sequence starts at the top left and then progresses by traversing down.

##  5 Conclusion and Future Work

Transformers are promising models for the physical sciences. We present a case
where of transformers, as well as augmentations, achieve better performance
than baseline. We demonstrate that the model picks up on some physics, and
benefits from explicit memory of the system.

One shortcoming of the model is that it is optimized for discrete time series,
like sentences, where words are some fundamental unit of the process. Ours,
meanwhile, is a time series characterizing a dynamic process, where sampling
rate is relatively arbitrary and may change, not only between tokamaks, but
between diagnostics within a single tokamak. While impressive that the
transformer can learn attention on discrete spikes in $v_{loop}$, we
ultimately desire a model that may capture continuous patterns and subtle
variations. Thus in future work, we hope to evaluate the performance of an
Autoformer Wu et al. (2022), which is a transformer whose discrete attention
blocks are replaced with autocorrelational FFT decompositions, making it more
suited for continuous sequences.

## Acknowledgments and Disclosure of Funding

We thank PSFC for providing ample compute resources and Matteo Bonotto,
Tommasso Gallingani, Daniele Bigone, and Francesco Cannarile for consistent
technical collaboration, advice, ideas and feedback. This work was supported
by Eni S.p.A. through the MIT Energy Initiative and by Commonwealth Fusion
Systems under SPARC RPP021 funding. We are grateful to Robert Granetz for
advice on physical interpretation of the model.

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##  Appendix A Data Feature Decsription

Feature | Definition | Relevant instab.
---|---|---
Locked mode indicator | Locked mode mag. field normalized to toroidal field | MHD
Rotating mode indicator | Std. dev. of Mirnov array normalized by toroidal field | MHD
$\beta_{p}$ | Plasma pressure normalized by poloidal magnetic field | MHD
$\ell_{i}$ | Normalized plasma internal inductance | MHD
$q_{95}$ | Safety factor at 95th flux surface | MHD
$n/n_{G}$ | Electron density normalized by Greenwald density | Dens. limit
$\Delta z_{\textrm{center}}$ | Vertical position error of plasma current centroid | Vert. Stab.
$\Delta z_{\textrm{lower}}$ | Gap between plasma and lower divertor | Shaping
$\kappa$ | Plasma elongation | Shaping
$P_{\textrm{rad}}/P_{\textrm{input}}$ | Radiated power normalized by input power | Impurities
$I_{\textrm{p,error}}/I_{\textrm{p,prog}}$ | Plasma current error normalized by programmed current | Impurities
$V_{\textrm{loop}}$ | Toroidal “loop” voltage | Impurities
Table 2: The input features of the model, their definitions, and a
categorization of the type of instability the signal indicates.

##  Appendix B Further Experimental Details

tokamak | $\tau$ | Number of Shots | Average Shot Length | Initial Sampling Rate
---|---|---|---|---
C-Mod | 50 ms | 4000 | .52s | .005ms
DIII-D | 150 ms | 8000 | 3.7s | .01ms
EAST | 400 ms | 11000 | 5.3s | .025ms
Table 3: Metrics on a dataset composed of multiple tokamaks.

##  Appendix C Supplemental Background on Fusion Devices and Plasma Dynamics

The two major approaches to generation are inertial confinement, which uses
lasers or motion to heat up plasmaEdwards et al. [2013], and magnetic
confinement, which leverages extreme magnetic fields to pressurize gas into
plasma, and then pressurize plasma into “burning plasma”, i.e. ones that
undergo nuclear fusion Creely et al. [2020].

Tokamaks exert magnetic fields of roughly 5 Tesla in a toroidal vacuum chamber
filled with a gas of hydrogren, deuterium and tritium isoptopes555Deuterium
and tritium isotopes are hydrogen atoms that have one and two extra neutrons,
respectively. Under extreme conditions, this gas enters a plasma state, which
is characterized by the highly charged flow of subatomic particles (ions and
electrons.) Tokamaks achieve some degree of magnetic-field symmetry, so
tokamak plasmas are stable under certain conditions. However,
magnetohydrodynamic (MHD) instabilities may still arise. For example, a
majority of electrons may slow in toroidal rotations relative to poloidal
rotations, creating areas where the induced magnetic force of the plasma
pressures certain points of the toroid and not others. Left untreated,
instabilities like this may grow out of control and lead to a loss of plasma
confinement, causing an end to the reaction and a fast quench of the
temperature and current onto the walls of the tokamak. Given that the
generated plasma has a core temperature on par with the center of the sun,
this may be a serious problem, resulting in damage to the walls and downtime,
making plasma disruptions a fundamental hurdle facing tokamak
commercializability Spangher et al. [2019].