| Abstract |
"The finite subgroups of GL4(Z) are classified up to conjugation in Brown, Bullow, Neubuser, Wondratscheck, and Zassenhaus (1978); in particular, there exist 710 non-conjugate finite groups in GL4(Z). Each finite group G of GL4(Z) acts naturally on Z [plus inside a circle] 4; thus we get a faithful G-lattice M with rankZM [equals] 4. In this way, there are exactly 710 such lattices. Given a G-lattice M with rankZM [equals] 4, the group G acts on the rational function field C(M) :[equals] C(x1, x2, x3, x4) by multiplicative 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 Bru(C(M)G), is non-trivial, then the fixed field C(M)G is not rational ([equals] purely transcendental) over C. A formula of the unramified Brauer group Bru(C(M)G) for the multiplicative invariant field was found by Saltman in 1990. However, to calculate Bru(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 Bru(C(M)G) [equals] B0(G) [plus sign inside a circle] H[superscript]2 [subscript]u(G,M) where H[superscript]2 [subscript]u(G,M) is some subgroup of H2(G,M). The first summand B0(G), which is related to the faithful linear representations of G, has been investigated by many authors. But the second summand H[superscript]2 [subscript]u(G,M) doesn't receive much attention except when the rank is [less than or equal to] 3. Theorem 1. Among the 710 finite groups G, let M be the associated faithful G-lattice with rankZM [equals] 4, there exist precisely 5 lattices M with Bru(C(M)G) [x inside a square] [equals] 0. In these situations, B0(G) [equals] 0 and thus Bru(C(M)G) [is a proper subset of] H2(G,M). The 5 groups are isomorphic to D4, Q8, QD8, SL2(F3), GL2(F3) 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, Bullow, Neubuser, Wondratscheck, and Zassenhaus (1978) and in The GAP Group (2008). Theorem 2. There exist 6079 (resp. 85308) finite subgroups G in GL5(Z) (resp. GL6(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 Bru(C(M)G) [x inside a square] [equals] 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[plus sign]2, 4n, p(p[minus sign]1) (n is any positive integer and p is any odd prime number) satisfying that B0(G) [equals] 0 and H[superscript]2 [subscript]u(G,M) [x inside a square] [equals] 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 (C2)3-lattice (resp. A6-lattice) M of rank 7 (resp. 9) with Bru(C(M)G) [x inside a square] [equals] 0. As a consequence, we give a counter-example to Noether's problem for N [x inside a square] A6 over C where N is some abelian group"-- Provided by publisher. |
