Weil–Châtelet group

Template:Distinguish 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. Template:Harvs named it for Template:Harvs who introduced it for elliptic curves, and Template:Harvs, 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, Template:Harvs proved that the Weil–Châtelet group is trivial for elliptic curves, and Template:Harvs proved that it is trivial for any connected 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 $f:A\to B$ of abelian varieties is a related group which can be defined in terms of Galois cohomology as

${\mathrm{S}\mathrm{e}\mathrm{l}}^{(f)}(A/K)=\bigcap _v\mathrm{k}\mathrm{e}\mathrm{r}(H^1(G_K,\mathrm{k}\mathrm{e}\mathrm{r}(f))\to H^1(G_{K_v},A_v[f])/\mathrm{i}\mathrm{m}({\kappa }_v))$

where A_v[f] denotes the f-torsion of A_v and ${\kappa }_v$ is the local Kummer map

$B_v(K_v)/f(A_v(K_v))\to H^1(G_{K_v},A_v[f])$.

References

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