Weil–Châtelet 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. John Tate (1958) named it for François Châtelet (1946) who introduced it for elliptic curves, and André 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 ${\displaystyle H^{1}(G_{K},A)}$, where ${\displaystyle G_{K}}$ 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, Friedrich Karl Schmidt (1931) proved that the Weil–Châtelet group is trivial for elliptic curves, and Serge Lang (1956) proved that it is trivial for any connected algebraic group.

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 ${\displaystyle f\colon A\to B}$ of abelian varieties is a related group which can be defined in terms of Galois cohomology as

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

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

${\displaystyle B_{v}(K_{v})/f(A_{v}(K_{v}))\rightarrow H^{1}(G_{K_{v}},A_{v}[f])}$.

• Cassels, John William Scott (1962), "Arithmetic on curves of genus 1. III. The Tate–Šafarevič and Selmer groups", Proceedings of the London Mathematical Society, Third Series, 12: 259–296, doi:10.1112/plms/s3-12.1.259, ISSN 0024-6115, MR 0163913
• Cassels, John William Scott (1991), Lectures on elliptic curves, London Mathematical Society Student Texts, vol. 24, Cambridge University Press, doi:10.1017/CBO9781139172530, ISBN 978-0-521-41517-0, MR 1144763
• Châtelet, François (1946), "Méthode galoisienne et courbes de genre un", Annales de l'Université de Lyon Sect. A. (3), 9: 40–49, MR 0020575
• Hindry, Marc; Silverman, Joseph H. (2000), Diophantine geometry: an introduction, Graduate Texts in Mathematics, vol. 201, 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. (eds.), Motives, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1637-0
• "Weil-Châtelet group", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Lang, Serge (1956), "Algebraic groups over finite fields", American Journal of Mathematics, 78 (3): 555–563, doi:10.2307/2372673, ISSN 0002-9327, JSTOR 2372673, MR 0086367
• Lang, Serge; Tate, John (1958), "Principal homogeneous spaces over abelian varieties", American Journal of Mathematics, 80 (3): 659–684, doi:10.2307/2372778, ISSN 0002-9327, JSTOR 2372778, MR 0106226
• Schmidt, Friedrich Karl (1931), "Analytische Zahlentheorie in Körpern der Charakteristik p", Mathematische Zeitschrift, 33: 1–32, doi:10.1007/BF01174341, ISSN 0025-5874
• Shafarevich, Igor R. (1959), "The group of principal homogeneous algebraic manifolds", Doklady Akademii Nauk SSSR (in Russian), 124: 42–43, ISSN 0002-3264, MR 0106227 English translation in his collected mathematical papers.
• Tate, John (1958), WC-groups over p-adic fields, Séminaire Bourbaki; 10e année: 1957/1958, vol. 13, Paris: Secrétariat Mathématique, MR 0105420
• Weil, André (1955), "On algebraic groups and homogeneous spaces", American Journal of Mathematics, 77 (3): 493–512, doi:10.2307/2372637, ISSN 0002-9327, JSTOR 2372637, MR 0074084