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Multiplicative Invariant Fields of Dimension $\le 6$

About this Title

Akinari Hoshi, Ming-chang Kang and Aiichi Yamasaki

Publication: Memoirs of the American Mathematical Society
Publication Year: 2023; Volume 283, Number 1403
ISBNs: 978-1-4704-6022-8 (print); 978-1-4704-7404-1 (online)
DOI: https://doi.org/10.1090/memo/1403
Published electronically: January 20, 2023
Keywords: Rationality problems, Noether’s problem, crystallographic groups, integral representations, unramified Brauer groups, algebraic tori

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Table of Contents

Chapters

  • 1. Introduction
  • 2. Preliminaries and the unramified Brauer groups
  • 3. CARAT ID of the $\mathbb {Z}$-classes in dimensions $5$ and $6$
  • 4. Proof of Theorem
  • 5. Classification of elementary abelian groups $(C_2)^k$ in $GL_n(\mathbb {Z})$ with $n\leq 7$
  • 6. The case $G=(C_2)^3$ with $H_u^2(G,M)\neq 0$
  • 7. The case $G=A_6$ with $H_u^2(G,M)\neq 0$ and Noether’s problem for $N\rtimes A_6$
  • 8. Some lattices of rank $2n+2, 4n$, and $p(p-1)$
  • 9. GAP computation: an algorithm to compute $H_u^2(G,M)$
  • 10. Tables: multiplicative invariant fields with non-trivial unramified Brauer groups

Abstract

The finite subgroups of $GL_4(\mathbb {Z})$ are classified up to conjugation in Brown, Büllow, Neubüser, Wondratscheck, and Zassenhaus (1978); in particular, there exist $710$ non-conjugate finite groups in $GL_4(\mathbb {Z})$. Each finite group $G$ of $GL_4(\mathbb {Z})$ acts naturally on $\mathbb {Z}^{\oplus 4}$; thus we get a faithful $G$-lattice $M$ with $\mathrm {rank}_\mathbb {Z} M=4$. In this way, there are exactly $710$ such lattices. Given a $G$-lattice $M$ with $\mathrm {rank}_\mathbb {Z} M=4$, the group $G$ acts on the rational function field $\mathbb {C}(M)≔\mathbb {C}(x_1,x_2,x_3,x_4)$ by multiplicative actions, i.e. purely monomial automorphisms over $\mathbb {C}$. We are concerned with the rationality problem of the fixed field $\mathbb {C}(M)^G$. A tool of our investigation is the unramified Brauer group of the field $\mathbb {C}(M)^G$ over $\mathbb {C}$. It is known that, if the unramified Brauer group, denoted by $\mathrm {Br}_u(\mathbb {C}(M)^G)$, is non-trivial, then the fixed field $\mathbb {C}(M)^G$ is not rational (= purely transcendental) over $\mathbb {C}$. A formula of the unramified Brauer group $\mathrm {Br}_u(\mathbb {C}(M)^G)$ for the multiplicative invariant field was found by Saltman in 1990. However, to calculate $\mathrm {Br}_u(\mathbb {C}(M)^G)$ for a specific multiplicatively invariant field requires additional efforts, even when the lattice $M$ is of rank equal to $4$. There is a direct decomposition $\mathrm {Br}_u(\mathbb {C}(M)^G)= B_0(G) \oplus H^2_u(G,M)$ where $H^2_u(G,M)$ is some subgroup of $H^2(G,M)$. The first summand $B_0(G)$, which is related to the faithful linear representations of $G$, has been investigated by many authors. But the second summand $H^2_u(G,M)$ doesn’t receive much attention except when the rank is $\le 3$. Theorem 1. Among the $710$ finite groups $G$, let $M$ be the associated faithful $G$-lattice with $\mathrm {rank}_\mathbb {Z} M=4$, there exist precisely $5$ lattices $M$ with $\mathrm {Br}_u(\mathbb {C}(M)^G)\neq 0$. In these situations, $B_0(G)=0$ and thus $\mathrm {Br}_u(\mathbb {C}(M)^G)\subset H^2(G,M)$. The $5$ groups are isomorphic to $D_4$, $Q_8$, $QD_8$, $SL_2(\mathbb {F}_3)$, $GL_2(\mathbb {F}_3)$ 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 $6079$ (resp. $85308$) finite subgroups $G$ in $GL_5(\mathbb {Z})$ (resp. $GL_6(\mathbb {Z})$). Let $M$ be the lattice with rank $5$ (resp. $6$) associated to each group $G$. Among these lattices precisely $46$ (resp. $1073$) of them satisfy the condition $\mathrm {Br}_u(\mathbb {C}(M)^G)\neq 0$. The GAP IDs (actually the CARAT IDs) of the corresponding groups $G$ may be determined explicitly. Motivated by these results, we construct $G$-lattices $M$ of rank $2n+2$, $4n$, $p(p-1)$ ($n$ is any positive integer and $p$ is any odd prime number) satisfying that $B_0(G)=0$ and $H^2_u(G,M)\neq 0$; and therefore $\mathbb {C}(M)^G$ are not rational over $\mathbb {C}$. For these $G$-lattices $M$, we prove that the flabby class $[M]^{fl}$ of $M$ is not invertible. We also construct an example of $(C_2)^3$-lattice (resp. $A_6$-lattice) $M$ of rank $7$ (resp. $9$) with $\mathrm {Br}_u(\mathbb {C}(M)^G)\neq 0$. As a consequence, we give a counter-example to Noether’s problem for $N\rtimes A_6$ over $\mathbb {C}$ where $N$ is some abelian group.

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