Birational classification of fields of invariants for groups of order 128

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• 作者：Akinari Hoshi ^{hoshi@math.sc.niigata-u.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：12F12 ; 13A50 ; 14E08 ; 14F22 ; 16K50 ; 20C10
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：1 January 2016
• 年：2016
• 卷：445
• 期：Complete
• 页码：394-432
• 全文大小：690 K

文摘

Let G   be a finite group acting on the rational function field c10d3fcf92" title="Click to view the MathML source">C(x_g:g∈G)$\mathbb{C}(x_g:g\in G)$ by C$\mathbb{C}$-automorphisms h(x_g)=x_hg$h(x_g)=x_{hg}$ for any g,h∈G$g,h\in G$. Noether's problem asks whether the invariant field C(G)=k(x_g:g∈G)^G$\mathbb{C}(G)=k{(x_g:g\in G)}^G$ is rational (i.e. purely transcendental) over C$\mathbb{C}$. By Fischer's theorem, C(G)$\mathbb{C}(G)$ is rational over C$\mathbb{C}$ when G is a finite abelian group. Saltman and Bogomolov, respectively, showed that for any prime p there exist groups G   of order p⁹$p^9$ and of order p⁶$p^6$ such that C(G)$\mathbb{C}(G)$ is not rational over C$\mathbb{C}$ by showing the non-vanishing of the unramified Brauer group: Br_nr(C(G))≠0${\mathrm{Br}}_{\mathrm{nr}}(\mathbb{C}(G))\ne 0$, which is an avatar of the birational invariant H³(X,Z)_tors$H^3{(X,\mathbb{Z})}_{\mathrm{tors}}$ given by Artin and Mumford where X   is a smooth projective complex variety whose function field is C(G)$\mathbb{C}(G)$. For p=2$p=2$, Chu, Hu, Kang and Prokhorov proved that if G   is a 2-group of order ≤32, then C(G)$\mathbb{C}(G)$ is rational over C$\mathbb{C}$. Chu, Hu, Kang and Kunyavskii showed that if G   is of order 64, then C(G)$\mathbb{C}(G)$ is rational over C$\mathbb{C}$ except for the groups G   belonging to the two isoclinism families Φ₁₃${\mathrm{\Phi }}_{13}$ with Br_nr(C(G))=0${\mathrm{Br}}_{\mathrm{nr}}(\mathbb{C}(G))=0$ and Φ₁₆${\mathrm{\Phi }}_{16}$ with Br_nr(C(G))≃C₂${\mathrm{Br}}_{\mathrm{nr}}(\mathbb{C}(G))\simeq C_2$. Bogomolov and Böhning's theorem claims that if G₁$G_1$ and G₂$G_2$ belong to the same isoclinism family, then C(G₁)$\mathbb{C}(G_1)$ and C(G₂)$\mathbb{C}(G_2)$ are stably C$\mathbb{C}$-isomorphic. We investigate the birational classification of C(G)$\mathbb{C}(G)$ for groups G   of order 128 with Br_nr(C(G))≠0${\mathrm{Br}}_{\mathrm{nr}}(\mathbb{C}(G))\ne 0$. Moravec showed that there exist exactly 220 groups G   of order 128 with Br_nr(C(G))≠0${\mathrm{Br}}_{\mathrm{nr}}(\mathbb{C}(G))\ne 0$ forming 11 isoclinism families Φ_j${\mathrm{\Phi }}_j$. We show that if G₁$G_1$ and G₂$G_2$ belong to Φ₁₆${\mathrm{\Phi }}_{16}$, Φ₃₁${\mathrm{\Phi }}_{31}$, Φ₃₇${\mathrm{\Phi }}_{37}$, Φ₃₉${\mathrm{\Phi }}_{39}$, Φ₄₃${\mathrm{\Phi }}_{43}$, Φ₅₈${\mathrm{\Phi }}_{58}$, Φ₆₀${\mathrm{\Phi }}_{60}$ or Φ₈₀${\mathrm{\Phi }}_{80}$ (resp. Φ₁₀₆${\mathrm{\Phi }}_{106}$ or Φ₁₁₄${\mathrm{\Phi }}_{114}$), then C(G₁)$\mathbb{C}(G_1)$ and C(G₂)$\mathbb{C}(G_2)$ are stably C$\mathbb{C}$-isomorphic with Br_nr(C(G_i))≃C₂${\mathrm{Br}}_{\mathrm{nr}}(\mathbb{C}(G_i))\simeq C_2$. Explicit structures of non-rational fields C(G)$\mathbb{C}(G)$ are given for each cases including also the case Φ₃₀${\mathrm{\Phi }}_{30}$ with Br_nr(C(G))≃C₂×C₂${\mathrm{Br}}_{\mathrm{nr}}(\mathbb{C}(G))\simeq C_2\times C_2$.