Weil–Châtelet group

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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 H¹(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

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

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.

[edit] References

• Hindry, Marc & Silverman, Joseph H. (2000), Diophantine geometry: an introduction, vol. 201, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98981-5 
• Greenberg, Ralph (1994), “Iwasawa Theory and p-adic Deformation of Motives”, in Serre, Jean-Pierre; Jannsen, Uwe & Kleiman, Steven L., Motives, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1637-0