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A duality theorem for Tate–Shafarevich groups of curves over algebraically closed fields
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg ( IF 0.4 ) Pub Date : 2018-10-01 , DOI: 10.1007/s12188-018-0196-7
Timo Keller

In this note, we prove a duality theorem for the Tate–Shafarevich group of a finite discrete Galois module over the function field K of a curve over an algebraically closed field: there is a perfect duality of finite groups for F a finite étale Galois module on K of order invertible in K and with $$F' = {{\mathrm{Hom}}}(F,\mathbf{Q}/\mathbf {Z}(1))$$F′=Hom(F,Q/Z(1)). Furthermore, we prove that $$\mathrm {H}^1(K,G) = 0$$H1(K,G)=0 for G a simply connected, quasisplit semisimple group over K not of type $$E_8$$E8.

中文翻译:

代数闭域上 Tate-Shafarevich 曲线群的对偶定理

在本笔记中,我们证明了在代数闭域上曲线的函数域 K 上有限离散伽罗瓦模的 Tate-Shafarevich 群的对偶定理:对于 F 有限元伽罗瓦模存在有限群的完美对偶性在 K 的阶次可逆的 K 和 $$F' = {{\mathrm{Hom}}}(F,\mathbf{Q}/\mathbf {Z}(1))$$F'=Hom(F, Q/Z(1))。此外,我们证明 $$\mathrm {H}^1(K,G) = 0$$H1(K,G)=0 对于 G 是一个在 K 上的单连通、准分裂半单群,不是 $$E_8$$ 类型E8.
更新日期:2018-10-01
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