- Weil–Châtelet group
In

mathematics , particularly inarithmetic geometry , the**Weil-Châtelet group**of anabelian variety "A" defined over a field "K" is theabelian group ofprincipal homogeneous space s for "A", defined over "K". It is named forAndré Weil , who introduced the general group operation in it, andF. Châtelet . It plays a basic role in thearithmetic of abelian varieties , in particular forelliptic curve s, because of its connection withinfinite descent .It can be defined directly from

Galois cohomology , as "H"^{1}("G"_{"K"},"A"), where "G"_{"K"}is theabsolute Galois group of "K". It is of particular interest forlocal field s andglobal field s, such asalgebraic number field s. For "K" afinite field , it was proved that the group is trivial.The

**Tate-Shafarevich group**, named forJohn Tate andIgor Shafarevich , 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" (i.e. thep-adic field s obtained from "K", as well as its real and complex completions). Thus, in terms of Galois cohomology, in can be written as:$igcap\_vmathrm\{ker\}(H^1(G\_K,A)\; ightarrow\; H^1(G\_\{K\_v\},A\_v)).$

It is often denoted Ш("A"/"K"), where Ш is the

Cyrillic letter "Sha ", for Shafarevich.Geometrically, the non-trivial elements of the Tate-Shafarevich group can be thought of as the homogeneous spaces of "A" that have "K"

_{v}-rational point s for every place "v" of "K", but no "K"-rational point.The Tate-Shafarevich group is conjectured to be finite; the first results on this were obtained byKarl Rubin .The

**Selmer group**, named afterErnst S. Selmer , of "A" with respect to anisogeny "f":"A"→"B" of abelian varieties is a related group which can be defined in terms of Galois cohomology as:$mathrm\{Se\}ell^\{(f)\}(A/K)=igcap\_vmathrm\{ker\}(H^1(G\_K,mathrm\{ker\}(f))\; ightarrow\; H^1(G\_K,A\_v\; [f]\; ))$

where "A"

_{v}["f"] denotes the "f"-torsion of "A"_{v}. Geometrically, the principal homogeneous spaces coming from elements of the Selmer group have "K"_{v}-rational points for all places "v" of "K". The Selmer group is finite. This has implications to the conjecture that the Tate-Shafarevich group is finite due to the followingexact sequence :0→"B"("K")/"f"("A"("K"))→Sel

^{(f)}("A"/"K")→Ш("A"/"K") ["f"] →0.Ralph Greenberg has generalized the notion of Selmer group to more general "p"-adic

Galois representation s and to "p"-adic variations of motives in the context ofIwasawa theory .**References***

*

*Wikimedia Foundation.
2010.*