Degree three unramified cohomology groups and Noether's problem for groups of order 243

Abstract

Let k be a field and G be a finite group acting on the rational function field k(xg:g∈G) by k-automorphisms defined as h(xg)=xhg for any g,h∈G. We denote the fixed field k(xg:g∈G)G by k(G). Noether's problem asks whether k(G) is rational (= purely transcendental) over k. It is well-known that if C(G) is stably rational over C, then all the unramified cohomology groups Hnri(C(G),Q/Z)=0 for i≥2. Hoshi, Kang and Kunyavskii [32] showed that, for a p-group of order p5 (p: an odd prime number), Hnr2(C(G),Q/Z)≠0 if and only if G belongs to the isoclinism family Φ10. When p is an odd prime number, Peyre [55] and Hoshi, Kang and Yamasaki [33] exhibit some p-groups G which are of the form of a central extension of certain elementary abelian p-group by another one with Hnr2(C(G),Q/Z)=0 and Hnr3(C(G),Q/Z)≠0. However, it is difficult to tell whether Hnr3(C(G),Q/Z) is non-trivial if G is an arbitrary finite group. In this paper, we are able to determine Hnr3(C(G),Q/Z) where G is any group of order p5 with p=3,5,7. Theorem 1. Let G be a group of order 35. Then Hnr3(C(G),Q/Z)≠0 if and only if G belongs to the isoclinism family Φ7. Theorem 2. If G is a group of order 35, then the fixed field C(G) is rational if and only if G does not belong to the isoclinism families Φ7 and Φ10. Theorem 3. Let G be a group of order 55 or 75. Then Hnr3(C(G),Q/Z)≠0 if and only if G belongs to the isoclinism families Φ6, Φ7 or Φ10. Theorem 4. If G is the alternating group An, the Mathieu group M11, M12, the Janko group J1 or the group PSL2(Fq), SL2(Fq), PGL2(Fq) (where q is a prime power), then Hnrd(C(G),Q/Z)=0 for any d≥2. Besides the degree three unramified cohomology groups, we compute also the stable cohomology groups.