Lattices over cyclic groups and Noether settings

For each positive integer m, let ωm be a primitive mth root of 1. Given a prime p, let a be an integer which is a primitive root of 1modp. Let Ip,m be the ideal in Z[ωm] generated by p and ωm−a. Let G be a cyclic group of order n such that the flasque class of any maximal projective ideal in ZG contains a maximal ideal U, of index p for some prime p such that n divides p−1, and such that, Ip,n is principal. It is not known whether the number of groups satisfying this property is finite or infinite. We show that if F is a field of characteristic zero containing all nth roots of 1, then for any ZG-lattice M the fixed subfield of F(M) is stably rational over F. If F also contains primitive mth roots of 1 for all natural numbers m, then for any finite G-module T the fixed subfield of the Noether setting of the group G′=T⋊G is stably rational over F.