Tate–Shafarevich group
In arithmetic geometry, the Tate–Shafarevich group of an abelian variety (or more generally a group scheme) defined over a number field consists of the elements of the Weil–Châtelet group , where is the absolute Galois group of , that become trivial in all of the completions of (i.e., the real and complex completions as well as the -adic fields obtained from by completing with respect to all its Archimedean and non Archimedean valuations ). Thus, in terms of Galois cohomology, can be defined as This group was introduced by Serge Lang and John Tate and Igor Shafarevich. Cassels introduced the notation , where is the Cyrillic letter "Sha", for Shafarevich, replacing the older notation or . Elements of the Tate–Shafarevich group Geometrically, the non-trivial elements of the Tate–Shafarevich group can be thought of as the homogeneous spaces of that have -rational points for every place of , but no -rational point. Thus, the group measures the extent to which the Hasse principle fails to hold for rational equations with coefficients in the field . Carl-Erik Lind gave an example of such a homogeneous space, by showing that the genus 1 curve has solutions over the reals and over all -adic