Three-dimensional purely quasi-monomial actions

0.36 MB·English

by  Akinari Hoshi

THREE-DIMENSIONAL PURELY QUASI-MONOMIAL ACTIONS
AKINARI HOSHI AND HIDETAKA KITAYAMA
Abstract. Let G be a finite subgroup of Autk(K(x1,...,xn)) where K/k is a finite field
extension and K(x1,...,xn) is the rational function field with n variables over K. The action
5 of G on K(x1,...,xn) is called quasi-monomial if it satisfies the following three conditions (i)
1 σ(K) K for any σ G; (ii) KG =k where KG is the fixed field under the action of G; (iii)
20 for an⊂y σ ∈ G and 1∈≤ j ≤ n, σ(xj) = cj(σ) ni=1xaiij where cj(σ) ∈ K× and [ai,j]1≤i,j≤n ∈
GL ( ). A quasi-monomialaction is called purely quasi-monomialif c (σ)=1 for any σ G,
n n Z Q j ∈
a any 1 j n. When k =K, a quasi-monomialaction is called monomial. The main problem
J is that≤, un≤der what situations, K(x1,...,xn)G is rational (= purely transcendental) over k.
5
For n = 1, the rationality problem was solved by Hoshi, Kang and Kitayama. For n = 2, the
1
problem was solved by Hajja when the action is monomial, by Voskresenskii when the action

] is faithful on K and purely quasi-monomial, which is equivalent to the rationality problem of
G
n-dimensionalalgebraick-toriwhichsplitoverK,andbyHoshi,KangandKitayamawhenthe
A
action is purely quasi-monomial. For n = 3, the problem was solved by Hajja, Kang, Hoshi
.
h and Rikuna when the action is purely monomial, by Hoshi, Kitayama and Yamasaki when the
t actionismonomialexceptforonecaseandbyKunyavskiiwhenthe actionisfaithfulonK and
a
m purely quasi-monomial. In this paper, we determine the rationality when n=3 andthe action
[ is purely quasi-monomial except for few cases. As an application, we will show the rationality

of some 5-dimensional purely monomial actions which are decomposable.
1
v
8
5
Contents
5
3
0 1. Introduction 2
.
1 2. Notation 8
0
3. Preliminaries 10
5
1 3.1. Reduction to lower degree 10
:
v 3.2. Explicit transcendental bases 10
i
X 3.3. Rationality of quadrics and conic bundles 12
r 3.4. Conjugacy classes move 12
a
4. Proof of Theorem 1.13 13
4.1. The 3rd crystal system (II) 13
4.2. The 4th crystal system (I), (II) 15
4.3. The 5th crystal system (I) 20
5. Proof of Theorem 1.14 21
5.1. The 7th crystal system (I), (II) 21
5.2. The 7th crystal system (III) 23
6. Proof of Proposition 1.15 27
7. Proof of Theorem 1.16 31
References 32
2010 Mathematics Subject Classification. Primary 12F20, 13A50, 14E08.
Key words and phrases. Rationality problem, monomial actions, Noether’s problem, algebraic tori.
This work was supported by JSPS KAKENHI Grant Numbers 24740014,25400027. Some part of this work
was done during the authors visited the National Center for Theoretic Sciences (Taipei Office), whose support
is gratefully acknowledged.
1
2 AKINARIHOSHIANDHIDETAKAKITAYAMA
1. Introduction
Let k be a field and K be a finitely generated field extension of k. K is called k-rational
(or rational over k) if K is purely transcendental over k, i.e. K is isomorphic to k(x ,...,x ),
1 n
the rational function field of n variables over k for some integer n. K is called stably k-
rational if K(y ,...,y ) is k-rational for some y ,...,y such that y ,...,y are algebraically
1 m 1 m 1 m
independent over K. When k is an infinite field, K is called retract k-rational if K is the
quotient field of some integral domain A where k A K satisfying the conditions that there
⊂ ⊂
exist apolynomial ringk[X ,...,X ], some non-zeroelement f k[X ,...,X ], andk-algebra
1 m 1 m
∈
morphisms ϕ : A k[X ,...,X ][1/f], ψ : k[X ,...,X ][1/f] A such that ψ ϕ = 1 .
1 m 1 m A
→ → ◦
K is called k-unirational if k K k(x ,...,x ) for some integer n. It is not difficult to see
1 n
⊂ ⊂
that “k-rational” “stably k-rational” “retract k-rational” “k-unirational”. The reader is
⇒ ⇒ ⇒
referred to the papers [MT86, CTS07, Swa83] for surveys of the various rationality problems,
e.g. Noether’s problem. We will restrict ourselves to consider the rationality problem of the
fixed field K(x ,...,x )G under the following special kind of actions:
1 n
Definition 1.1 ([HKK14, Definition 1.1]). Let G be a finite subgroup of Aut (K(x ,...,x )).
k 1 n
The action of G on K(x ,...,x ) is called quasi-monomial if it satisfies the following three
1 n
conditions:
(i) σ(K) K for any σ G;
⊂ ∈
(ii) KG = k where KG is the fixed field under the action of G;
(iii) for any σ G and 1 j n,
∈ ≤ ≤
n
σ(x ) = c (σ) xaij
j j i
i=1
Y
where c (σ) K× and [a ] GL ( ).
j i,j 1≤i,j≤n n
∈ ∈ Z
A quasi-monomial action is called purely quasi-monomial if c (σ) = 1 for any σ G, any
j
∈
1 j n in (iii). When k = K, a quasi-monomial action is just called monomial.
≤ ≤
When G Gal(K/k), i.e. G acts faithfully on K, and G acts on K(x ,...,x ) by purely
1 n
≃
quasi-monomial k-automorphisms, the rationality problem of K(x ,...,x )G coincides with
1 n
that of algebraic k-tori of dimension n which split over K (see [HKK14, Section 1]).
For quasi-monomial actions, the following results are already known (see a survey [Hos14]):
Theorem 1.2 (Voskresenskii [Vos67]). Let k be a field. All the two-dimensional algebraic k-
tori are k-rational. In particular, K(x ,x )G is k-rational if G acts on K faithfully and on
1 2
K(x ,x ) by purely quasi-monomial k-automorphisms.
1 2
Theorem 1.3 (Kunyavskii [Kun87], see also Kang [Kan12, Section 1]). Let k be a field.
All the three-dimensional algebraic k-tori are k-rational except for the 15 cases in the list of
[Kun87, Theorem 1]. In particular, K(x ,x ,x )G is k-rational if G acts on K faithfully and
1 2 3
on K(x ,x ,x ) by purely quasi-monomial k-automorphisms except for the 15 cases. For the
1 2 3
exceptional 15 cases, they are not k-rational; in fact, they are even not retract k-rational.
Theorem 1.4 (Hajja [Haj87]). Let k be a field and G be a finite group acting on k(x ,x ) by
1 2
monomial k-automorphisms. Then k(x ,x )G is k-rational.
1 2
Theorem 1.5 (Hajja, Kang [HK92, HK94], Hoshi, Rikuna [HR08]). Let k be a field and G be
a finite group acting on k(x ,x ,x ) by purely monomial k-automorphisms. Then k(x ,x ,x )G
1 2 3 1 2 3
is k-rational.
THREE-DIMENSIONAL PURELY QUASI-MONOMIAL ACTIONS 3
Let G be a finite group acting on K(x ,...,x ) by quasi-monomial k-automorphisms. There
1 n
exists a group homomorphism ρ : G GL ( ) defined by ρ (σ) = [a ] GL ( ) for
x n x i,j 1≤i,j≤n n
→ Z ∈ Z
any σ G where [a ] is given in (i) of Definition 1.1.
i,j 1≤i,j≤n
∈
Theorem 1.6 (Hoshi, Kang, Kitayama [HKK14, Proposition 1.12]). Let k be a field and G be
a finite group acting on K(x ,...,x ) by quasi-monomial k-automorphisms. Then there is a
1 n
normal subgroup N of G satisfying the following conditions:
(i) K(x ,...,x )N = KN(y ,...,y ) where each y is of the form axe1xe2 xen with a K×
1 n 1 n i 1 2 ··· n ∈
and e (we may take a = 1 if the action is a purely quasi-monomial action);
i
∈ Z
(ii) G/N acts on KN(y ,...,y ) by quasi-monomial k-automorphisms;
1 n
(iii) ρ : G/N GL ( ) is an injective group homomorphism.
y n
→ Z
By Theorem 1.6, we may assume that ρ : G GL ( ) is injective and thus G may
x n
→ Z
be regarded as a finite subgroup of GL ( ). The following theorem was already proved by
n
Z
Prokhorov [Pro10] when k = .
C
Theorem 1.7 (Hoshi, Kitayama, Yamasaki [HKY11, Yam12]). Let k be a field with char k = 2
6
and G be a finite subgroup of GL ( ) acting on K(x ,x ,x ) by monomial k-automorphisms.
3 1 2 3
Z
Then k(x ,x ,x )G is k-rational except for the 8 cases contained in [Yam12] and one additional
1 2 3
case. For the last exceptional case, k(x ,x ,x )G is also k-rational except for a minor situation.
1 2 3
In particular, if k is a quadratically closed field with char k = 2, then k(x ,x ,x )G is k-rational.
1 2 3
6
Indeed, for the exceptional 8 cases in Theorem 1.7, the necessary and sufficient condition for
the k-rationality of K(x ,x ,x )G was given in terms of k and c (σ). In particular, if it is not
1 2 3 j
k-rational, then it is not retract k-rational (see [Yam12]).
Let (a,b) (resp. [a,b) ) be the norm residue symbol of degree two over k when char k = 2
k k
6
(resp. char k = 2), see [Dra83, Chapter 11]. For dimension one (resp. two), the rationality
problem for quasi-monomial (resp. purely quasi-monomial) actions was solved by Hoshi, Kang
and Kitayama [HKK14].
Theorem 1.8 (Hoshi, Kang, Kitayama [HKK14, Proposition 1.13]). Let k be a field.
(1) Let G be a finite group acting on K(x) by purely quasi-monomial k-automorphisms. Then
K(x)G is k-rational.
(2) Let G be a finite group acting on K(x) by quasi-monomial k-automorphisms. Then K(x)G
is k-rational except for the following case: There is a normal subgroup N of G such that (i)
G/N = σ , (ii) K(x)N = k(α)(y) with α2 = a K×, σ(α) = α (if char k = 2), and
2
h i ≃ C ∈ − 6
α2 +α = a K, σ(α) = α+1 (if chark = 2), (iii) σ y = b/y for some b k×.
∈ · ∈
For the exceptional case, K(x)G = k(α)(y)G/N is k-rational if and only if the norm residue
2-symbol (a,b) = 0 (if char k = 2), and [a,b) = 0 (if char k = 2).
k k
6
Moreover, if K(x)G is not k-rational, then k is an infinite field, the Brauer group Br(k) is
non-trivial, and K(x)G is not k-unirational.
Let (resp. , , ) be the symmetric group (resp. the alternating group, the dihedral
n n n n
S A D C
group, the cyclic group) of degree n of order n! (resp. n!/2, 2n, n).
Theorem 1.9 (Hoshi, Kang, Kitayama [HKK14, Theorem 1.14]). Let k be a field and G be a
finite group acting on K(x,y) by purely quasi-monomial k-automorphisms. Define N = σ
{ ∈
G : σ(x) = x, σ(y) = y , H = σ G : σ(α) = α for all α K . Then K(x,y)G is k-rational
} { ∈ ∈ }
except possibly for the following situation: (1) char k = 2 and (2) (G/N,HN/N) ( , ) or
4 2
6 ≃ C C
( , ).
4 2
D C
More precisely, in the exceptional situation we may choose u,v k(x,y) satisfying that
∈
k(x,y)HN/N = k(u,v) (and therefore K(x,y)HN/N = K(u,v)) such that
4 AKINARIHOSHIANDHIDETAKAKITAYAMA
(i) when (G/N,HN/N) ( , ), KN = k(√a) for some a k k2, G/N = σ , then
4 2 4
≃ C C ∈ \ h i ≃ C
σ acts on KN(u,v) by σ : √a √a, u 1, v 1; or
7→ − 7→ u 7→ −v
(ii) when (G/N,HN/N) ( , ), KN = k(√a,√b) is a biquadratic extension of k with
4 2
≃ D C
a,b k k2, G/N = σ,τ , then σ and τ act on KN(u,v) by σ : √a √a, √b √b,
4
∈ \ h i ≃ D 7→ − 7→
u 1, v 1, τ : √a √a, √b √b, u u, v v.
7→ u 7→ −v 7→ 7→ − 7→ 7→ −
For Case (i), K(x,y)G is k-rational if and only if the norm residue 2-symbol (a, 1) = 0.
k
−
For Case (ii), K(x,y)G is k-rational if and only if (a, b) = 0.
k
−
Moreover, if K(x,y)G is not k-rational, then k is an infinite field, the Brauer group Br(k) is
non-trivial, and K(x,y)G is not k-unirational.
The following definition gives an equivalent definition of quasi-monomial actions. This def-
inition follows the approach of Saltman’s definition of twisted multiplicative actions [Sal90a,
Sal90b], [Kan09, Definition 2.2].
Definition 1.10. Let G be a finite group. A G-lattice M is a finitely generated [G]-module
Z
which is -free as an abelian group, i.e. M = x with a [G]-module structure.
Z 1≤i≤nZ · i Z
Let K/k be a field extension such that G acts on K with KG = k. Consider a short exact
L
sequence of [G]-modules α : 1 K× M M 0 where M is a G-lattice and K× is
α
Z → → → →
regarded as a [G]-module through the G-action on K. The [G]-module structure (written
Z Z
multiplicatively) of M may be described as follows: For each x M (where 1 j n), take
α j
∈ ≤ ≤
a pre-image u of x . As an abelian group, M is the direct product of K× and u ,...,u .
j j α 1 n
h i
If σ G and σ x = a x M, we find that σ u = c (σ) uaij M for a
∈ · j 1≤i≤n ij i ∈ · j j · 1≤i≤n i ∈ α
unique c (σ) K× determined by the group extension α.
j
∈ P Q
Using the same idea, once a group extension α : 1 K× M M 0 is given,
α
→ → → →
we may define a quasi-monomial action of G on the rational function field K(x ,...,x ) as
1 n
follows: If σ x = a x M, then define σ x = c (σ) xaij K(x ,...,x )
· j 1≤i≤n ij i ∈ · j j 1≤i≤n i ∈ 1 n
and σ α = σ(α) for α K where σ(α) is the image of α under σ via the prescribed action of
· P ∈ Q
G on K. This quasi-monomial action is well-defined (see [Sal90a, page 538] for details). The
field K(x ,...,x ) with such a G-action will be denoted by K (M) to emphasize the role of
1 n α
the extension α; its fixed field is denoted as K (M)G. We will say that G acts on K (M) by
α α
quasi-monomial k-automorphisms.
If k = K, then k (M)G is nothing but the fixed field associated to the monomial action.
α
Iftheextensionαsplits, thenwemaytakeu ,...,u M satisfyingthatσ u = uaij.
1 n ∈ α · j 1≤i≤n i
Hence the associated quasi-monomial action of G on K(x ,...,x ) becomes a purely quasi-
1 n
Q
monomial action. In this case, we will write K (M) and K (M)G as K(M) and K(M)G
α α
respectively (the subscript α is omitted because the extension α plays no important role). We
will say that G acts on K(M) by purely quasi-monomial k-automorphisms. Again k(M)G is
the fixed field associated to the purely monomial action.
As an application of Theorem 1.9, we have the following theorems:
Theorem 1.11 (Hoshi, Kang, Kitayama [HKK14, Theorem 1.16]). Let k be a field, G be
a finite group and M be a G-lattice with rank M = 4 such that G acts on k(M) by purely
Z
monomial k-automorphisms. If M is decomposable, i.e. M = M M as [G]-modules where
1 2
⊕ Z
1 rank M 3, then k(M)G is k-rational.
1
≤ Z ≤
Theorem 1.12 (Hoshi, Kang, Kitayama [HKK14, Theorem 6.2]). Let k be a field, G be a finite
group and M be a G-lattice such that G acts on k(M) by purely monomial k-automorphisms.
Assume that (i) M = M M as [G]-modules where rank M = 3 and rank M = 2, (ii)
1 2 1 2
⊕ Z Z Z
either M or M is a faithful G-lattice. Then k(M)G is k-rational except the following situation:
1 2
char k = 2, G = σ,τ and M = x , M = y such that σ : x x ,
6 h i ≃ D4 1 1≤i≤3Z i 2 1≤j≤2Z j 1 ↔ 2
L L
THREE-DIMENSIONAL PURELY QUASI-MONOMIAL ACTIONS 5
x x x x , y y y , τ : x x , x x x x , y y where
3 1 2 3 1 2 1 1 3 2 1 2 3 1 2
7→ − − − 7→ 7→ − ↔ 7→ − − − ↔
the [G]-module structure of M is written additively. For the exceptional case, k(M)G is not
Z
retract k-rational.
The aim of this paper is to investigate the rationality problem of K(x ,x ,x )G for purely
1 2 3
quasi-monomial k-automorphisms. The followings are main results of this paper.
Note that (i) by Theorem 1.6, we may assume N = σ G : σ(x ) = x (i = 1,2,3) = 1
i i
{ ∈ }
and hence G may be regarded as a finite subgroup of GL ( ); (ii) when H = 1 the answer to
3
Z
the rationality problem in dimension 3 was given by Kunyavskii (see Theorem 1.3).
There exist 73 finite subgroups G (1 i 7) contained in GL ( ) up to conjugation
i,j,k 3
≤ ≤ Z
which are classified into 7 crystal systems (see Section 2 for details).
Theorem 1.13 (The groups G do not belong to the 7th crystal system in dimension 3). Let k
be a field with char k = 2 and G be a finite subgroup of GL ( ) acting on K(x ,x ,x ) by purely
3 1 2 3
6 Z
quasi-monomial k-automorphisms. Assume that G does not belong to the 7th crystal system in
dimension 3 and H = σ G σ(α) = α for any α K = 1.
{ ∈ | ∈ } 6
(1) If G does not belong to the 4th crystal system in dimension 3, i.e. G is either not a 2-group
or a 2-group of exponent 2, then K(x ,x ,x )G is k-rational;
1 2 3
(2) When G belongs to the 4th crystal system in dimension 3, i.e. G is a 2-group of exponent
4, G is -conjugate to one of the following 8 groups: σ , σ , I ,
4A 4 4A 3 4 2
Q h± i ≃ C h − i ≃ C × C
σ , λ , σ ,λ , I where
4A 1 4 4A 1 3 4 2
h± ± i ≃ D h − i ≃ D ×C
0 1 0 1 0 0
− −
σ = 1 0 0 , λ = 0 1 0
4A 1
   
0 0 1 0 0 1
−
   
and I is the 3 3 identity matrix. Then K(x ,x ,x )G is k-rational except possibly for the
3 1 2 3
×
following cases with H = σ2 or σ2 , I :
h 4Ai h 4A − 3i
(i) The case where H = σ2 .
h 4Ai ≃ C2
(i-1) When G is -conjugate to σ , (resp. σ , I ) we may take K = k(√a) (resp.
4A 4A 3
Q h± i h − i
Kh−I3i = k(√a)) on which G acts by σ : √a √a, and K(x ,x ,x )G is k-rational if
4A 1 2 3
± 7→ −
and only if (a, 1) = 0.
k
−
(i-2) When G is -conjugate to σ , λ (resp. σ ,λ , I ) we may take K =
4A 1 4A 1 3
Q h± ± i h − i
k(√a,√b) (resp. Kh−I3i = k(√a,√b)) on which G acts by σ : √a √a,√b √b,
4A
± 7→ − 7→
λ : √a √a,√b √b, and K(x ,x ,x )G is k-rational if and only if (a, b) = 0.
1 1 2 3 k
± 7→ 7→ − −
(ii) The case where H = σ2 , I .
h 4A − 3i ≃ C2 ×C2
(ii-1) When G is -conjugate to σ , I , we may take K = k(√a) on which G acts by
4A 3
Q h − i
σ : √a √a, and K(x ,x ,x )G is k-rational if and only if (a, 1) = 0.
4A 1 2 3 k
7→ − −
(ii-2) When G is -conjugate to σ ,λ , I , we may take K = k(√a,√b) on which G acts
4A 1 3
Q h − i
by σ : √a √a,√b √b, λ : √a √a,√b √b, and K(x ,x ,x )G is k-rational if
4A 1 1 2 3
7→ − 7→ 7→ 7→ −
and only if (a, b) = 0.
k
−
Moreover, if K(x ,x ,x )G is not k-rational, then k is an infinite field, the Brauer group
1 2 3
Br(k) is non-trivial, and K(x ,x ,x )G is not k-unirational.
1 2 3
In particular, the k-rationality of K(x ,x ,x )G does not depend on the -conjugacy class
1 2 3
Q
of G and the sign of σ and λ .
4A 1
± ±
There exist 15 finite subgroups G (1 j 5,1 k 3) of GL ( ) which belong to the
7,j,k 3
≤ ≤ ≤ ≤ Z
7th crystal system (see Section 2).
Theorem 1.14 (The groups G belong to the 7th crystal system in dimension 3). Let k be a
field with char k = 2 and G be a finite subgroup of GL ( ) acting on K(x ,x ,x ) by purely
3 1 2 3
6 Z
quasi-monomial k-automorphisms. Assume that G = G belongs to the 7th crystal system in
7,j,k
6 AKINARIHOSHIANDHIDETAKAKITAYAMA
dimension 3 and H = σ G σ(α) = α for any α K = 1.
{ ∈ | ∈ } 6
(1) If G = G (1 j 5), G (1 j 5) or G (j = 1,4), then K(x ,x ,x )G is
7,j,1 7,j,2 7,j,3 1 2 3
≤ ≤ ≤ ≤
k-rational;
(2) When G = G (j = 2,3,5), G = τ ,λ ,σ , I , G = τ ,λ ,σ , β
7,j,3 7,2,3 3 3 3B 3 4 2 7,3,3 3 3 3B 3
h − i ≃ A ×C h − i
, G = τ ,λ ,σ ,β , I where
4 7,5,3 3 3 3B 3 3 4 2
≃ S h − i ≃ S ×C
0 1 1 0 1 1 0 0 1 1 0 1
− − −
τ = 1 0 1 , λ = 0 1 0 , σ = 1 0 0 , β = 0 1 1
3 3 3B 3
 −   −     − 
0 0 1 1 1 0 0 1 0 0 0 1
− − −
       
and I is the 3 3 identity matrix. Then K(x ,x ,x )G is k-rational except possibly for the
3 1 2 3
×
following cases with H = τ ,λ or τ ,λ ,σ :
3 3 3 3 3B
h i h i
(i) The case where H = τ ,λ . We have
3 3 2 2
h i ≃ C ×C
K(x ,x ,x )G7,2,3 = K(u ,u ,u )hσ3B,−I3i,
1 2 3 1 2 3
K(x ,x ,x )G7,3,3 = K(u ,u ,u )hσ3B,−β3i,
1 2 3 1 2 3
K(x ,x ,x )G7,5,3 = K(u ,u ,u )hσ3B,β3,−I3i
1 2 3 1 2 3
where K(u ,u ,u ) = K(x ,x ,x )H and
1 2 3 1 2 3
σ : u u , u u , u u ,
3B 1 2 2 3 3 1
7→ 7→ 7→
u +u +u u +u u u u +u
1 2 3 1 2 3 1 2 3
β : u − , u − , u − ,
3 1 2 3
− 7→ u u 7→ u u 7→ u u
2 3 1 2 1 3
β : u u , u u , u u ,
3 1 1 2 3 3 2
7→ 7→ 7→
u +u +u u u +u u +u u
1 2 3 1 2 3 1 2 3
I : u − , u − , u − .
3 1 2 3
− 7→ u u 7→ u u 7→ u u
2 3 1 3 1 2
(ii) The case where H = τ ,λ ,σ . We have
3 3 3B 4
h i ≃ A
K(x ,x ,x )G7,2,3 = K(s ,s ,s )h−I3i,
1 2 3 1 2 3
K(x ,x ,x )G7,3,3 = K(s ,s ,s )h−β3i,
1 2 3 1 2 3
K(x ,x ,x )G7,5,3 = K(s ,s ,s )hβ3,−I3i
1 2 3 1 2 3
where K(s ,s ,s ) = K(x ,x ,x )H and
1 2 3 1 2 3
1+3s2 1 6s2 9s4 +2s +10s2s +4s4s s2 3s2s2
β : s s , s 1, s − − 1 − 1 2 1 2 1 2 − 2 − 1 2,
3 1 1 2 3
− 7→ 7→ s 7→ s s
2 2 3
β : s s , s s , s s ,
3 1 1 2 2 3 3
7→ − 7→ 7→
1+3s2 1 6s2 9s4 +2s +10s2s +4s4s s2 3s2s2
I : s s , s 1, s − − 1 − 1 2 1 2 1 2 − 2 − 1 2.
3 1 1 2 3
− 7→ − 7→ s 7→ s s
2 2 3
We do not know the rationality of K(x ,x ,x )G in Theorem 1.14 (2) (i) with H .
1 2 3 2 2
≃ C ×C
For the case (2) (ii) of Theorem 1.14 with H , we will give the following partial result. It
4
≃ A
turns out that the fixed field K(x ,x ,x )G has a conic bundle structure.
1 2 3
Proposition 1.15. Let k be a field with char k = 2 and G = G (j = 2,3,5). Assume that
7,j,3
6
H = σ G σ(α) = α for any α K = τ ,λ ,σ .
3 3 3B 4
{ ∈ | ∈ } h i ≃ A
(1) G = G = H, I . There exist w ,w ,w K(x ,x ,x )H such that K(x ,x ,x )H =
7,2,3 3 1 2 3 1 2 3 1 2 3
h − i ∈
K(w ,w ,w ) and
1 2 3
I : w w , w w ,
3 1 1 2 2
− 7→ 7→ −
w (1+w +w2)2 (2+3w +4w2 +2w3)w2 +(w +2)w4
w 1 1 1 − 1 1 1 2 1 2.
3
7→ w
3
THREE-DIMENSIONAL PURELY QUASI-MONOMIAL ACTIONS 7
In particular, we have K(x ,x ,x )G = k(X,Y,W ,W ) where K = k(√b) and
1 2 3 1 2
X2 bY2 = W (1+W +W2)2 b(2+3W +4W2 +2W3)W2 +b2(W +2)W4.
− 1 1 1 − 1 1 1 2 1 2
(2) G = G = H, β . There exist u ,u ,u K(x ,x ,x )H such that K(x ,x ,x )H =
7,3,3 3 1 2 3 1 2 3 1 2 3
h − i ∈
K(u ,u ,u ) and
1 2 3
2(5 u2)(u2 u2 +1)
β : u u , u u , u − 2 1 − 2 .
3 1 1 2 2 3
− 7→ 7→ − 7→ u
3
We have K(x ,x ,x )G = k(X,Y,U ,U ) where X2 dY2 = 2(5 dU2)(U2 dU2 + 1) and
1 2 3 1 2 − − 2 1 − 2
K = k(√d). Moreover, if √5 k, then the following conditions are equivalent:
∈
(i) K(x ,x ,x )G is k-rational;
1 2 3
(ii) K(x ,x ,x )G is k-unirational;
1 2 3
(iii) X2 dY2 2v2 2v2 +2dv2 has a non-trivial k-zero with K = k(√d).
− − 0 − 1 2
In particular, if √5,√2 k or √5,√ 1 k, then K(x ,x ,x )G is k-rational.
1 2 3
∈ − ∈
(3) G = G = H,β , I . There exist p ,p ,p K(x ,x ,x )H such that K(x ,x ,x )H
7,5,3 3 3 1 2 3 1 2 3 1 2 3
h − i ∈
= K(p ,p ,p ) and
1 2 3
β : p p , p p , p p ,
3 1 1 2 2 3 3
7→ − 7→ − 7→ −
1 5p2 7p2 (p2 p2)(3p2 +17p2)+9(p2 p2)3
I : p p , p p , p − − 1 − 2 − 1 − 2 1 2 1 − 2 .
3 1 1 2 2 3
− 7→ 7→ − 7→ p
3
In particular, we have K(x ,x ,x )G = k(X,Y,P ,P ) where K = k(√a,√b) and
1 2 3 1 2
X2 bY2 = 1 5P2 7bP2 a(P2 bP2)(3P2 +17bP2)+9a2(P2 bP2)3.
− −a − 1 − 2 − 1 − 2 1 2 1 − 2
(4) For G = G = H, I , we assume that √ 3 k. Then there exist t ,t ,t
7,2,3 3 1 2 3
h − i − ∈ ∈
K(x ,x ,x )H such that K(x ,x ,x )H = K(t ,t ,t ) and
1 2 3 1 2 3 1 2 3
(t2 +4)(t2 t2 +1)
I : t t , t t , t 1 1 − 2 .
3 1 1 2 2 3
− 7→ − 7→ 7→ t
3
In particular, if √ 3,√ 1 k, then K(x ,x ,x )G is k-rational.
1 2 3
− − ∈
(5) If char k = 3, then K(x ,x ,x )G is k-rational for G = G (j = 2,3,5).
1 2 3 7,j,3
As an application of Theorems 1.13 and 1.14 and Proposition 1.15, we get the following
theorem which complements to Theorem 1.12:
Theorem 1.16. Let k be a field with char k = 2, G be a finite group and M be a G-lattice such
6
that G acts on k(M) by purely monomial k-automorphisms. Assume that (i) M = M M
1 2
⊕
as [G]-modules where rank M = 3 and rank M = 2, (ii) both M and M are not faithful
1 2 1 2
Z Z Z
G-lattices. Let N = σ G σ(α) = α for any α k(M ) (i = 1,2). Then k(M)G is k-
i i
{ ∈ | ∈ }
rational except the following situation: (G/N ,N N /N ,G/N ,N N /N ) (G , , , ),
1 1 2 1 2 1 2 2 7,2,3 4 4 2
≃ A C C
(G , , , ) and (G , , , ). Moreover, if char k = 3, then k(M)G is k-rational.
7,3,3 4 4 2 7,5,3 4 4 2
A C C A D C
This paper is organized as follows. In Section 2, we recall the classification of subgroups of
GL ( ) up to conjugation. Section 3 contains some rationality results which will be used in the
3
Z
paper. A technique of conjugacy classes move, which is described in Subsection 3.4, is useful
and will be used in the proof of Theorems 1.13 and 1.14 to the cases of 3rd crystal system
(II), the 4th crystal system (II) and 7th crystal system (II) in Subsections 4.1, 4.2 and 5.1
respectively. Section 4 contains the proof of Theorem 1.13. The proof of Theorem 1.14 is given
in Section 5. In Section 6, the proof of Proposition 1.15 will be given. We will prove Theorem
1.16 in Section 7.
Acknowledgment. The authors thank Ming-chang Kang for many helpful comments and
valuable suggestions.
8 AKINARIHOSHIANDHIDETAKAKITAYAMA
2. Notation
Let (resp. , , ) be the symmetric group (resp. the alternating group, the dihedral
n n n n
S A D C
group, the cyclic group) of degree n of order n! (resp. n!/2, 2n, n). Let I be the 3 3 identity
3
×
matrix. We define the following matrices:
0 1 0 0 0 1 0 1 0 0 1 0
− −
σ = 1 1 0 , σ = 1 0 0 , σ = 1 0 0 , σ = 0 1 1 ,
3A 3B 4A 4B
 −       − 
0 0 1 0 1 0 0 0 1 1 1 0
−
       
1 0 0 1 0 0 0 1 0
− − −
τ = 0 1 0 , λ = 0 1 0 , β = 1 0 0 ,
1 1 1
 −     − 
0 0 1 0 0 1 0 0 1
−
     
0 1 0 0 0 1 1 0 0
τ = 1 0 0 , λ = 1 1 1 , β = 0 1 0 ,
2 2 2
   − − −   
1 1 1 1 0 0 1 1 1
− − − − − −
     
0 1 1 0 1 1 1 0 1
− − −
τ = 1 0 1 , λ = 0 1 0 , β = 0 1 1 ,
3 3 3
 −   −   − 
0 0 1 1 1 0 0 0 1
− − −
     
0 1 0 1 0 0
−
α = 1 0 0 , I = 0 1 0 .
3
  −  − 
0 0 1 0 0 1
−
   
There exist exactly 73 finite subgroups contained in GL ( ) up to conjugation which are clas-
3
Z
sified into 7 crystal systems (see [BBNWZ78, Table 1]).
The 1st crystal system ( -reducible):
Z
G = I , G = I .
1,1,1 3 1,2,1 3 2
{ } h− i ≃ C
The 2nd crystal system ( -reducible):
Z
G = λ , G = α ,
2,1,1 1 2 2,1,2 2
h i ≃ C h− i ≃ C
G = λ , G = α ,
2,2,1 1 2 2,2,2 2
h− i ≃ C h i ≃ C
G = λ , I , G = α, I .
2,3,1 1 3 2 2 2,3,2 3 2 2
h − i ≃ C ×C h− − i ≃ C ×C
The 3rd crystal system (I) ( -reducible):
Z
G = τ ,λ , G = τ , α ,
3,1,1 1 1 2 2 3,1,2 1 2 2
h i ≃ C ×C h − i ≃ C ×C
G = τ , λ , G = τ ,α ,
3,2,1 1 1 2 2 3,2,2 1 2 2
h − i ≃ C ×C h i ≃ C ×C
G = α,β ,
3,2,3 1 2 2
h− i ≃ C ×C
G = τ ,λ , I , G = τ , α, I .
3,3,1 1 1 3 2 2 2 3,3,2 1 3 2 2 2
h − i ≃ C ×C ×C h − − i ≃ C ×C ×C
The 3rd crystal system (II) ( -irreducible, but -reducible):
Z Q
G = τ ,λ , ∗G = τ ,λ ,
3,1,3 2 2 2 2 3,1,4 3 3 2 2
h i ≃ C ×C h i ≃ C ×C
G = τ , λ , G = τ , λ ,
3,2,4 2 2 2 2 3,2,5 3 3 2 2
h − i ≃ C ×C h − i ≃ C ×C
∗G = τ ,λ , I , ∗G = τ ,λ , I .
3,3,3 2 2 3 2 2 2 3,3,4 3 3 3 2 2 2
h − i ≃ C ×C ×C h − i ≃ C ×C ×C
The 4th crystal system (I) ( -reducible):
Z
G = σ , G = σ ,
4,1,1 4A 4 4,2,1 4A 4
h i ≃ C h− i ≃ C
G = σ , I , G = σ ,λ ,
4,3,1 4A 3 4 2 4,4,1 4A 1 4
h − i ≃ C ×C h i ≃ D
THREE-DIMENSIONAL PURELY QUASI-MONOMIAL ACTIONS 9
G = σ , λ , G = σ ,λ ,
4,5,1 4A 1 4 4,6,1 4A 1 4
h − i ≃ D h− i ≃ D
G = σ , λ , G = σ ,λ , I .
4,6,2 4A 1 4 4,7,1 4A 1 3 4 2
h− − i ≃ D h − i ≃ D ×C
The 4th crystal system (II) ( -irreducible, but -reducible):
Z Q
G = σ , G = σ ,
4,1,2 4B 4 4,2,2 4B 4
h i ≃ C h− i ≃ C
∗G = σ , I , ∗G = σ ,λ ,
4,3,2 4B 3 4 2 4,4,2 4B 3 4
h − i ≃ C ×C h i ≃ D
G = σ , λ , G = σ , λ ,
4,5,2 4B 3 4 4,6,3 4B 3 4
h − i ≃ D h− − i ≃ D
∗G = σ ,λ , ∗G = σ ,λ , I .
4,6,4 4B 3 4 4,7,2 4B 3 3 4 2
h− i ≃ D h − i ≃ D ×C
The 5th crystal system (I) ( -irreducible, but -reducible):
Z Q
G = σ , G = σ , I ,
5,1,1 3B 3 5,2,1 3B 3 6
h i ≃ C h − i ≃ C
G = σ , α , G = σ ,α ,
5,3,1 3B 3 5,4,1 3B 3
h − i ≃ S h i ≃ S
G = σ , α, I .
5,5,1 3B 3 6
h − − i ≃ D
The 5th crystal system (II) ( -reducible):
Z
G = σ , G = σ , I ,
5,1,2 3A 3 5,2,2 3A 3 6
h i ≃ C h − i ≃ C
G = σ , α , G = σ , β ,
5,3,2 3A 3 5,3,3 3A 1 3
h − i ≃ S h − i ≃ S
G = σ ,β , G = σ ,α ,
5,4,2 3A 1 3 5,4,3 3A 3
h i ≃ S h i ≃ S
G = σ , α, I , G = σ , β , I .
5,5,2 3A 3 6 5,5,3 3A 1 3 6
h − − i ≃ D h − − i ≃ D
The 6th crystal system ( -reducible):
Z
G = σ ,τ , G = σ , τ ,
6,1,1 3A 1 6 6,2,1 3A 1 6
h i ≃ C h − i ≃ C
G = σ ,τ , I , G = σ ,τ , β ,
6,3,1 3A 1 3 6 2 6,4,1 3A 1 1 6
h − i ≃ C ×C h − i ≃ D
G = σ ,τ ,β , G = σ , τ ,β ,
6,5,1 3A 1 1 6 6,6,1 3A 1 1 6
h i ≃ D h − i ≃ D
G = σ , τ , β , G = σ ,τ , β , I .
6,6,2 3A 1 1 6 6,7,1 3A 1 1 3 6 2
h − − i ≃ D h − − i ≃ D ×C
The 7th crystal system (I) ( -irreducible):
Q
G = τ ,λ ,σ , G = τ ,λ ,σ , I ,
7,1,1 1 1 3B 4 7,2,1 1 1 3B 3 4 2
h i ≃ A h − i ≃ A ×C
G = τ ,λ ,σ , β , G = τ ,λ ,σ ,β ,
7,3,1 1 1 3B 1 4 7,4,1 1 1 3B 1 4
h − i ≃ S h i ≃ S
G = τ ,λ ,σ ,β , I .
7,5,1 1 1 3B 1 3 4 2
h − i ≃ S ×C
The 7th crystal system (II) ( -irreducible):
Q
G = τ ,λ ,σ , ∗G = τ ,λ ,σ , I ,
7,1,2 2 2 3B 4 7,2,2 2 2 3B 3 4 2
h i ≃ A h − i ≃ A ×C
∗G = τ ,λ ,σ , β , G = τ ,λ ,σ ,β ,
7,3,2 2 2 3B 2 4 7,4,2 2 2 3B 2 4
h − i ≃ S h i ≃ S
∗G = τ ,λ ,σ ,β , I .
7,5,2 2 2 3B 2 3 4 2
h − i ≃ S ×C
The 7th crystal system (III) ( -irreducible):
Q
∗G = τ ,λ ,σ , ∗G = τ ,λ ,σ , I ,
7,1,3 3 3 3B 4 7,2,3 3 3 3B 3 4 2
h i ≃ A h − i ≃ A ×C
∗G = τ ,λ ,σ , β , ∗G = τ ,λ ,σ ,β ,
7,3,3 3 3 3B 3 4 7,4,3 3 3 3B 3 4
h − i ≃ S h i ≃ S
∗G = τ ,λ ,σ ,β , I .
7,5,3 3 3 3B 3 3 4 2
h − i ≃ S ×C
The 15 groups ∗G with marked above are in the Kunyavskii’s list of [Kun87, Theorem
i,j,k
∗
1], i.e. the only 15 groups in GL ( ) up to conjugation for which K(x ,x ,x )Gi,j,k are not
3 1 2 3
Z
retract k-rational under the faithful action of G on K (see Theorem 1.3).
i,j,k
10 AKINARIHOSHIANDHIDETAKAKITAYAMA
3. Preliminaries
3.1. Reduction to lower degree.
Theorem 3.1 (Miyata [Miy71, Lemma, page 70], Ahmad, Hajja, Kang [AHK00, Theorem
3.1]). Let L be a field, L(x) be the rational function field of one variable over L and G be a
finite group acting on L(x). Suppose that, for any σ G, σ(L) L and σ(x) = a x+b where
σ σ
∈ ⊂
a , b L and a = 0. Then L(x)G = LG(f) for some polynomial f L[x]G.
σ σ σ
∈ 6 ∈
Corollary 3.2. Let k be a field with char k = 2 and G be a finite subgroup of GL ( ) acting on
n
6 Z
K(x ,...,x ) by purely quasi-monomial k-automorphisms. If G is -reducible of (n 1,1)-type
1 n
Z −
and K(x ,...,x )G is k-rational, then K(x ,...,x )G is k-rational.
1 n−1 1 n
Proof. Put x′ = (x 1)/(x +1) and apply Theorem 3.1. (cid:3)
n n − n
The following lemmas are a restatement of Hilbert 90 (see also [Miy71, Lemma, page 70]).
Theorem 3.3 (Endo, Miyata [EM73, Proposition 1.1]). Let L/k be a finite Galois extension
of fields with Galois group G which acts on L(x ,...,x ) by k-automorphisms. Suppose that
1 n
for any σ G,
∈ n
σ(αx ) = σ(α) a (σ)x , α,a (σ) L.
i ij j ij
∈
j=1
X
Then L(x ,...,x )G is k-rational.
1 n
Theorem 3.4 (Hajja, Kang [HK95, Theorem 1]). Let L be a field and G be a finite group
acting on L(x ,...,x ). Suppose that
1 n
(i) σ(L) L for any σ G;
⊂ ∈
(ii) the restriction of the actions of G to L is faithful;
(iii) for any σ G,
∈
σ(x ) x
1 1
. .
 ..  = A(σ) .. +B(σ)
σ(x ) x
n n
   
   
where A(σ) GL (L) and B(σ) is an n 1 matrix over L. Then there exist z ,...,z
n 1 n
∈ × ∈
L(x ,...,x ) such that L(x ,...,x ) = L(z ,...,z ) with σ(z ) = z for any σ G and
1 n 1 n 1 n i i
∈
1 i n.
≤ ≤
3.2. Explicit transcendental bases.
Lemma 3.5 (Hashimoto, Hoshi, Rikuna [HHR08, page1176], Hoshi, Kang, Kitayama [HKK14,
Lemma 3.3], see also [HKY11, Lemma 3.4]). Let k be a field and τ act on k(x ,x ) by k-
1 2
automorphisms
1 1
τ : x , x .
1 2
7→ x 7→ x
1 2
Then k(x ,x )hτi = k(t ,t ) where
1 2 1 2
x1x2−1 if char k = 2,
x x +1
1 2 x1−x2 6
t = , t =
1 x1 +x2 2 ( xx12((xx2221++11)) if char k = 2.
Lemma 3.6 (Hajja, Kang [HK94, Lemma 2.7], [Kan04, Theorem 2.4]). Let k be a field and
I act on k(x ,x ) by
2 1 2
−
a b
I : x , x , a k×
2 1 2
− 7→ x 7→ x ∈
1 2