Weil–Châtelet group

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Not to be confused with Weil 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. Tate (1958) named it for François Châtelet (1946) who introduced it for elliptic curves, and 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 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, Schmidt (1931) proved that the Weil–Châtelet group is trivial for elliptic curves, and Lang (1956) proved that it is trivial for any algebraic group.

See also

The Tate–Shafarevich group of an abelian variety A defined over a number field K consists of the elements of the Weil–Châtelet group that become trivial in all of the completions of K.

The Selmer group, named after Ernst S. Selmer, of A with respect to an isogeny f:A→B of abelian varieties is a related group which can be defined in terms of Galois cohomology as

{\mathrm {Sel}}^{{(f)}}(A/K)=\bigcap _{v}{\mathrm {ker}}(H^{1}(G_{K},{\mathrm {ker}}(f))\rightarrow H^{1}(G_{{K_{v}}},A_{v}[f])/{\mathrm {im}}(\kappa _{v}))

where Av[f] denotes the f-torsion of Av and \kappa _{v} is the local Kummer map B_{v}(K_{v})/f(A_{v}(K_{v}))\rightarrow H^{1}(G_{{K_{v}}},A_{v}[f]).

References

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