2017/10/05 by Hoshi, Akinari, Kang, Ming-chang, Yamasaki, Aiichi
#14E08 #14F22 #20C10 #20J06 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1710.01958
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 H_[nri(C(G),Q/Z)=0 for i ≥ 2. Hoshi, Kang and Kunyavskii [HKK] showed that, for a p-group of order p5 (p: an odd prime number), H_[nr2(C(G),Q/Z)≠ 0 if and only if G belongs to the isoclinism family Φ10. When p is an odd prime number, Peyre [Pe3] and Hoshi, Kang and Yamasaki [HKY1] exhibit some p-groups G which are of the form of a central extension of certain elementary abelian p-group by another one with H_[nr2(C(G),Q/Z)=0 and H_[nr3(C(G),Q/Z)≠ 0. However, it is difficult to tell whether H_[nr3(C(G),Q/Z) is non-trivial if G is an arbitrary finite group. In this paper, we are able to determine H_[nr3(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 H_[nr3(C(G),Q/Z)≠ 0 if and only if G belongs to 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 Φ7 and Φ10. Theorem 3. Let G be a group of order 55 or 75. Then H_[nr3(C(G),Q/Z)≠ 0 if and only if G belongs to 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 H_[nrd(C(G),Q/Z)=0 for any d≥ 2. Besides the degree three unramified cohomology groups, we compute also the stable cohomology groups.