title: Weil–Châtelet group
text: # Weil–Châtelet group
  
  In arithmetic geometry, the Weil–Châtelet group or WC-group of an algebraic group such as an abelian variety A defined over a field K is the abelian group of principal homogeneous spaces for A, defined over K. John Tate (1958) named it for François Châtelet (1946) who introduced it for elliptic curves, and André Weil (1955), who introduced it for more general groups. 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 $H^{1}(G_{K},A)$, where $G_{K}$ 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, Friedrich Karl Schmidt (1931) proved that the Weil–Châtelet group is trivial for elliptic curves, and Serge Lang (1956) proved that it is trivial for any connected algebraic group.
  
  