Weil–Châtelet group


Weil–Châtelet group

In mathematics, particularly in arithmetic geometry, the Weil-Châtelet group of an abelian variety "A" defined over a field "K" is the abelian group of principal homogeneous spaces for "A", defined over "K". It is named for André Weil, who introduced the general group operation in it, and F. Châtelet. 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, it was proved that the group is trivial.

The Tate-Shafarevich group, named for John Tate and Igor 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. the p-adic fields 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 points 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 by Karl Rubin.

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{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 following exact 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 representations and to "p"-adic variations of motives in the context of Iwasawa theory.

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