High Quality Content by WIKIPEDIA articles! In mathematics, particularly in arithmetic geometry, the Weil-Chatelet 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 Andre Weil, who introduced the general group operation in it, and Francois Chatelet. 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(GK,A), where GK is the absolute Galois group of K. It is of particular interest 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, of an abelian variety A defined over a number field K consists of the elements of the Weil-Chatelet group 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). Thus, in terms of Galois cohomology.

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