\le 6维乘性不变域
Multiplicative Invariant Fields of Dimension \le 6
摘要：
$GL_4(\bm{Z})$的有限子群在\cite{BBNWZ}中被分类到共轭，特别地，在$GL_4(\bm{Z})$中存在$710$非共轭有限群. $GL_4(\bm{Z})$的每个有限群$G$自然地作用在$\bm{Z}^{\oplus 4}$上，从而我们得到一个忠实的$G$ -格$M$，其中有${\rm rank}_\bm{Z} M=4$ .这样，就有了$710$这样的格。给定一个具有${\rm rank}_\bm{Z} M=4$的$G$ -格$M$，群$G$通过乘法作用作用于有理函数域$\bm{C}(M):=\bm{C}(x_1,x_2,x_3,x_4)$，即$\bm{C}$上的纯单项自同构。本文研究了固定域$\bm{C}(M)^G$的合理性问题。我们的研究工具是场$\bm{C}(M)^G$在$\bm{C}$上的非分歧Brauer群。Saltman在1990年发现了乘法不变域的非分歧Brauer群${\rm Br}_u(\bm{C}(M)^G)$的一个公式。然而，计算特定乘法不变域的${\rm Br}_u(\bm{C}(M)^G)$需要额外的努力，即使当格$M$的秩等于$4$时。定理1.在$710$有限群$G$中，设$M$是与${\rm rank}_\bm{Z} M=4$相关联的忠实$G$ -格，则精确地存在与${\rm Br}_u(\bm{C}(M)^G)\neq 0$相关联的$5$格$M$.在这些情况下，$B_0(G)=0$和${\rm Br}_u(\bm{C}(M)^G)\subset H^2(G,M)$。五组$G$的{ \rm GAP ID}是{ \rm \cite{BBNWZ} }和{ \rm \cite{GAP} }中的{ \rm（4，12，4，12），（4，32，1，2），（4，32，3，2），（4，33，3，1），（4，33，6，1）}。定理2.在$GL_5(\bm{Z})$中存在$6079$有限子群$G$.令$M$是具有与每个组$G$相关联的秩$5$的格。在这些格中，恰好有$46$格满足条件${\rm Br}_u(\bm{C}(M)^G)\neq 0$。对于秩为$6$的格也得到了类似的结果。
英文摘要：
The finite subgroups of $GL_4(\bm{Z})$ are classified up to conjugation in \cite{BBNWZ}; in particular, there exist $710$ non-conjugate finite groups in $GL_4(\bm{Z})$. Each finite group $G$ of $GL_4(\bm{Z})$ acts naturally on $\bm{Z}^{\oplus 4}$; thus we get a faithful $G$-lattice $M$ with ${\rm rank}_\bm{Z} M=4$. In this way, there are exactly $710$ such lattices. Given a $G$-lattice $M$ with ${\rm rank}_\bm{Z} M=4$, the group $G$ acts on the rational function field $\bm{C}(M):=\bm{C}(x_1,x_2,x_3,x_4)$ by multiplicative actions, i.e. purely monomial automorphisms over $\bm{C}$. We are concerned with the rationality problem of the fixed field $\bm{C}(M)^G$. A tool of our investigation is the unramified Brauer group of the field $\bm{C}(M)^G$ over $\bm{C}$. A formula of the unramified Brauer group ${\rm Br}_u(\bm{C}(M)^G)$ for the multiplicative invariant field was found by Saltman in 1990. However, to calculate ${\rm Br}_u(\bm{C}(M)^G)$ for a specific multiplicatively invariant field requires additional efforts, even when the lattice $M$ is of rank equal to $4$. Theorem 1. Among the $710$ finite groups $G$, let $M$ be the associated faithful $G$-lattice with ${\rm rank}_\bm{Z} M=4$, there exist precisely $5$ lattices $M$ with ${\rm Br}_u(\bm{C}(M)^G)\neq 0$. In these situations, $B_0(G)=0$ and thus ${\rm Br}_u(\bm{C}(M)^G)\subset H^2(G,M)$. The {\rm GAP IDs} of the five groups $G$ are {\rm (4,12,4,12), (4,32,1,2), (4,32,3,2), (4,33,3,1), (4,33,6,1)} in {\rm \cite{BBNWZ}} and in {\rm \cite{GAP}}. Theorem 2. There exist $6079$ finite subgroups $G$ in $GL_5(\bm{Z})$. Let $M$ be the lattice with rank $5$ associated to each group $G$. Among these lattices precisely $46$ of them satisfy the condition ${\rm Br}_u(\bm{C}(M)^G)\neq 0$. A similar result for lattices of rank $6$ is found also.
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