Multiplicative Invariant Fields of Dimension ≤ 6

The finite subgroups of GL(4)(Z) are classified up to conjugation in Brown, Bu spacing diaeresis llow, Neubu spacing diaeresis ser, Wondratscheck, and Zassenhaus (1978); in particular, there exist 710 non-conjugate finite groups in GL(4)(Z). Each finite group G of GL4(Z) acts naturally on Z(circle plus 4); thus we get a faithful G-lattice M with rank(Z)M = 4. In this way, there are exactly 710 such lattices. Given a G-lattice M with rank(Z)M = 4, the group G acts on the rational function field C(M) := C(x(1), x(2), x(3), x(4)) by multi-plicative actions, i.e. purely monomial automorphisms over C. We are concerned with the rationality problem of the fixed field C(M)(G). A tool of our investigation is the unramified Brauer group of the field C(M)G over C. It is known that, if the unramified Brauer group, denoted by Br-u(C(M)(G)), is non-trivial, then the fixed field C(M)(G) is not rational (= purely transcendental) over C. A formula of the un-ramified Brauer group Br-u(C(M)(G)) for the multiplicative invariant field was found by Saltman in 1990. However, to calculate Bru(C(M)G) for a specific multiplica-tively invariant field requires additional efforts, even when the lattice M is of rank equal to 4. There is a direct decomposition Br-u(C(M)(G)) = B-0(G) (R) H-u(2)(G, M) where H-u(2)(G, M) is some subgroup of H2(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-u(2)(G, M) doesn't receive much attention ex-cept when the rank is < 3. Theorem 1. Among the 710 finite groups G, let M be the associated faithful G-lattice with rank(Z)M = 4, there exist precisely 5 lattices M with Br-u(C(M)(G)) not equal 0. In these situations, B-0(G) = 0 and thus Br-u(C(M)(G)) subset of H-2(G, M). The 5 groups are isomorphic to D-4, Q(8), QD(8), SL2(F-3), GL(2)(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, Bu spacing diaeresis llow, Neubu spacing diaeresis ser, Wondratscheck, and Zassenhaus (1978) and in The GAP Group (2008). Theorem 2. There exist 6079 (resp. 85308) finite subgroups G in GL(5)(Z) (resp. GL(6)(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 Br-u(C(M)(G)) not equal 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 inte-ger and p is any odd prime number) satisfying that B-0(G) = 0 and H-u(2)(G, M) not equal 0; and therefore C(M)G are not rational over 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. A6-lattice) M of rank 7 (resp. 9) with Bru(C(M)(G)) not equal 0. As a consequence, we give a counter-example to No ether's problem for N X A(6) over C where N is some abelian group.