Weil-Châtelet group
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In mathematics, the Weil-Châtelet group of an abelian variety A defined over a field K is the abelian group of principal homogeneous spaces for A, defined over K. It is named for André Weil, who introduced the general group operation in it, and F. Châtelet. It plays a basic role in the arithmetic of abelian varieties, in particular for elliptic curves, because of its connection with infinite descent.
It can be defined directly from Galois cohomology, as H1(A). It is of interest mainly for local fields and global fields, such as algebraic number fields. For K a finite field, it was proved that the group is trivial.
The Tate-Shafarevich group, named for John Tate and Igor Shafarevich, is defined for a number field K as the elements of the Weil-Châtelet group, in this context also called the Selmer group, named after Ernst S. Selmer, that become trivial in all of the completions of K (i.e. the p-adic fields obtained from K, as well as its real and complex completions). The Tate-Shafarevich group was long conjectured to be finite; the first results on this were obtained by Karl Rubin.
