ヴェイユ・シャトレ群

代数体 K 上定義されたアーベル多様体 A のテイト・シャファレヴィッチ群は、ヴェイユ・シャトレ群の元で K のすべての完備化で自明になるもの全体である。

エルンスト・セルマーにちなむ、A のアーベル多様体の同種 ${\displaystyle f\colon A\to B}$ についてのセルマー群も関係する群である。これはガロアコホモロジーを使って

${\displaystyle \mathrm {Sel} ^{(f)}(A/K)=\bigcap _{v}\mathrm {ker} (H^{1}(G_{K},\mathrm {ker} (f))\rightarrow H^{1}(G_{K_{v}},A_{v}[f])/\mathrm {im} (\kappa _{v}))}$

と定義できる。ここで A_v[ f ] は A_v の f ねじれで、${\displaystyle \kappa _{v}}$ は局所クンマー写像

${\displaystyle B_{v}(K_{v})/f(A_{v}(K_{v}))\rightarrow H^{1}(G_{K_{v}},A_{v}[f])}$

である。

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