Multiplicative Invariant Fields of Dimension ≤6
Authors: Hoshi Akinari , Kang Ming-chang , Yamasaki Aiichi 
Published: January 2023

The finite subgroups of

G

L
4

(

Z

)

GL_4(\mathbb {Z})

are classified up to conjugation in Brown, Büllow, Neubüser, Wondratscheck, and Zassenhaus (1978); in particular, there exist

710
710

non-conjugate finite groups in

G

L
4

(

Z

)

GL_4(\mathbb {Z})

. Each finite group

G
G

of

G

L
4

(

Z

)

GL_4(\mathbb {Z})

acts naturally on

Z

⊕
4

\mathbb {Z}^{\oplus 4}

; thus we get a faithful

G
G

-lattice

M
M

with

r
a
n
k

Z

M
=
4

\mathrm {rank}_\mathbb {Z} M=4

. In this way, there are exactly

710
710

such lattices. Given a

G
G

-lattice

M
M

with

r
a
n
k

Z

M
=
4

\mathrm {rank}_\mathbb {Z} M=4

, the group

G
G

acts on the rational function field

C

(
M
)
≔

C

(

x
1

,

x
2

,

x
3

,

x
4

)

\mathbb {C}(M)≔\mathbb {C}(x_1,x_2,x_3,x_4)

by multiplicative actions, i.e. purely monomial automorphisms over

C

\mathbb {C}

. We are concerned with the rationality problem of the fixed field

C

(
M

)
G

\mathbb {C}(M)^G

. A tool of our investigation is the unramified Brauer group of the field

C

(
M

)
G

\mathbb {C}(M)^G

over

C

\mathbb {C}

. It is known that, if the unramified Brauer group, denoted by

B
r

u

(

C

(
M

)
G

)

\mathrm {Br}_u(\mathbb {C}(M)^G)

, is non-trivial, then the fixed field

C

(
M

)
G

\mathbb {C}(M)^G

is not rational (= purely transcendental) over

C

\mathbb {C}

. A formula of the unramified Brauer group

B
r

u

(

C

(
M

)
G

)

\mathrm {Br}_u(\mathbb {C}(M)^G)

for the multiplicative invariant field was found by Saltman in 1990. However, to calculate

B
r

u

(

C

(
M

)
G

)

\mathrm {Br}_u(\mathbb {C}(M)^G)

for a specific multiplicatively invariant field requires additional efforts, even when the lattice

M
M

is of rank equal to

4
4

. There is a direct decomposition
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper B normal r Subscript u Baseline left-parenthesis double-struck upper C left-parenthesis upper M right-parenthesis Superscript upper G Baseline right-parenthesis equals upper B 0 left-parenthesis upper G right-parenthesis circled-plus upper H Subscript u Superscript 2 Baseline left-parenthesis upper G comma upper M right-parenthesis">

B
r

u

(

C

(
M

)
G

)
=

B
0

(
G
)
⊕

H
u
2

(
G
,
M
)

\mathrm {Br}_u(\mathbb {C}(M)^G)= B_0(G) \oplus H^2_u(G,M)

where

H
u
2

(
G
,
M
)

H^2_u(G,M)

is some subgroup of

H
2

(
G
,
M
)

H^2(G,M)

. The first summand

B
0

(
G
)

B_0(G)

, which is related to the faithful linear representations of

G
G

, has been investigated by many authors. But the second summand

H
u
2

(
G
,
M
)

H^2_u(G,M)

doesn’t receive much attention except when the rank is

≤
3

\le 3

. Theorem 1. Among the

710
710

finite groups

G
G

, let

M
M

be the associated faithful

G
G

-lattice with

r
a
n
k

Z

M
=
4

\mathrm {rank}_\mathbb {Z} M=4

, there exist precisely

5
5

lattices

M
M

with

B
r

u

(

C

(
M

)
G

)
≠
0

\mathrm {Br}_u(\mathbb {C}(M)^G)\neq 0

. In these situations,

B
0

(
G
)
=
0

B_0(G)=0

and thus
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper B normal r Subscript u Baseline left-parenthesis double-struck upper C left-parenthesis upper M right-parenthesis Superscript upper G Baseline right-parenthesis subset-of upper H squared left-parenthesis upper G comma upper M right-parenthesis">

B
r

u

(

C

(
M

)
G

)
⊂

H
2

(
G
,
M
)

\mathrm {Br}_u(\mathbb {C}(M)^G)\subset H^2(G,M)

. The

5
5

groups are isomorphic to

D
4

D_4

,

Q
8

Q_8

,

Q

D
8

QD_8

,

S

L
2

(

F

3

)

SL_2(\mathbb {F}_3)

,

G

L
2

(

F

3

)

GL_2(\mathbb {F}_3)

whose GAP IDs are (4,12,4,12), (4,32,1,2), (4,32,3,2), (4,33,3,1), (4,33,6,1) respectively in Brown, Büllow, Neubüser, Wondratscheck, and Zassenhaus (1978) and in The GAP Group (2008). Theorem 2. There exist

6079
6079

(resp.

85308
85308

) finite subgroups

G
G

in

G

L
5

(

Z

)

GL_5(\mathbb {Z})

(resp.

G

L
6

(

Z

)

GL_6(\mathbb {Z})

). Let

M
M

be the lattice with rank

5
5

(resp.

6
6

) associated to each group

G
G

. Among these lattices precisely

46
46

(resp.

1073
1073

) of them satisfy the condition

B
r

u

(

C

(
M

)
G

)
≠
0

\mathrm {Br}_u(\mathbb {C}(M)^G)\neq 0

. The GAP IDs (actually the CARAT IDs) of the corresponding groups

G
G

may be determined explicitly. Motivated by these results, we construct

G
G

-lattices

M
M

of rank

2
n
+
2

2n+2

,

4
n

4n

,

p
(
p
−
1
)

p(p-1)

(

n
n

is any positive integer and

p
p

is any odd prime number) satisfying that

B
0

(
G
)
=
0

B_0(G)=0

and

H
u
2

(
G
,
M
)
≠
0

H^2_u(G,M)\neq 0

; and therefore

C

(
M

)
G

\mathbb {C}(M)^G

are not rational over

C

\mathbb {C}

. For these

G
G

-lattices

M
M

, we prove that the flabby class

[
M

]

f
l

[M]^{fl}

of

M
M

is not invertible. We also construct an example of

(

C
2

)
3

(C_2)^3

-lattice (resp.

A
6

A_6

-lattice)

M
M

of rank

7
7

(resp.

9
9

) with

B
r

u

(

C

(
M

)
G

)
≠
0

\mathrm {Br}_u(\mathbb {C}(M)^G)\neq 0

. As a consequence, we give a counter-example to Noether’s problem for

N
⋊

A
6

N\rtimes A_6

over

C

\mathbb {C}

where

N
N

is some abelian group.