Tate–Shafarevich group

In <a href="/facts/Arithmetic_geometry/Bs1mfNu4">arithmetic geometry</a>, the Tate–Shafarevich group Ш(A/K) of an <a href="/facts/Abelian_variety/dGSDVruG">abelian variety</a> A (or more generally a <a href="/facts/Group_scheme/bJW14FE2">group scheme</a>) defined over a <a href="/facts/Number_field/e0K0TXSh">number field</a> K consists of the elements of the <a href="/facts/Weil%25E2%2580%2593Ch%25C3%25A2telet_group/RbpHyuQU">Weil–Châtelet group</a> 
 
 
 
 
 W
 C
 
 (
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 H
 
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 {\displaystyle \mathrm {WC} (A/K)=H^{1}(G_{K},A)}
 
, where 
 
 
 
 
 G
 
 K
 
 
 =
 
 G
 a
 l
 
 (
 
 K
 
 a
 l
 g
 
 
 
 /
 
 K
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 {\displaystyle G_{K}=\mathrm {Gal} (K^{alg}/K)}
 
 is the <a href="/facts/Absolute_Galois_group/fvcDwzWq">absolute Galois group</a> of K, that become trivial in all of the completions of K (i.e., the real and complex completions as well as the <a href="/facts/P-adic_field/SDgDbkLF">p-adic fields</a> obtained from K by completing with respect to all its Archimedean and non Archimedean <a href="/facts/Valuation_(algebra)/FXadrz5c">valuations</a> v). Thus, in terms of <a href="/facts/Galois_cohomology/nzC1EVYT">Galois cohomology</a>, Ш(A/K) can be defined as

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    {\displaystyle \bigcap _{v}\mathrm {ker} \left(H^{1}\left(G_{K},A\right)\rightarrow H^{1}\left(G_{K_{v}},A_{v}\right)\right).}

This group was introduced by <a href="/facts/Serge_Lang/I7MW2rUu">Serge Lang</a> and <a href="/facts/John_Tate_(mathematician)/nBntLCG0">John Tate</a> and <a href="/facts/Igor_Shafarevich/n6UINBX1">Igor Shafarevich</a>. <a href="/facts/J._W._S._Cassels/AC33eVoA">Cassels</a> introduced the notation Ш(A/K), where Ш is the <a href="/facts/Cyrillic/0btN3oWC">Cyrillic</a> letter "<a href="/facts/Sha_(Cyrillic)/D6hcCyhN">Sha</a>", for Shafarevich, replacing the older notation TS or TŠ.

Tate–Shafarevich group open-in-new

Tate–Shafarevich group