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2013-02-26T16:47:10Z

97.76.167.210: Minor grammar fix (then vs than)

{{distinguish|Weil 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 (mathematics)|field]] ''K'' is the [[abelian group]] of [[principal homogeneous space]]s for ''A'', defined over ''K''. {{harvtxt|Tate|1958}} named it for {{harvs|txt|authorlink=François Châtelet (mathematician)|first=François |last=Châtelet|year=1946}} who introduced it for elliptic curves, and {{harvs|txt|authorlink=André Weil|last=Weil|year=1955}}, who introduced it for more general groups. It plays a basic role in the [[arithmetic of abelian varieties]], in particular for [[elliptic curve]]s, 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 field]]s and [[global field]]s, such as [[algebraic number field]]s. For ''K'' a [[finite field]], {{harvtxt|Schmidt|1931}} proved that the Weil–Châtelet group is trivial for elliptic curves, and {{harvtxt|Lang|1956}} proved that it is trivial for any 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''→''B'' of abelian varieties is a related group which can be defined in terms of Galois cohomology as

:<math>\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))</math>

where ''A''v[''f''] denotes the ''f''-[[torsion (algebra)|torsion]] of ''A''v and <math>\kappa_v</math> is the local Kummer map
<math>B_v(K_v)/f(A_v(K_v))\rightarrow H^1(G_{K_v},A_v[f])</math>.

== References ==

*{{Citation | last1=Cassels | first1=John William Scott | title=Arithmetic on curves of genus 1. III. The Tate–Šafarevič and Selmer groups | doi=10.1112/plms/s3-12.1.259 | id={{MR|0163913}} | year=1962 | journal=Proceedings of the London Mathematical Society. Third Series | issn=0024-6115 | volume=12 | pages=259–296}}
*{{Citation | last1=Cassels | first1=John William Scott | title=Lectures on elliptic curves | url=http://books.google.com/books?id=zgqUAuEJNJ4C | publisher=[[Cambridge University Press]] | series=London Mathematical Society Student Texts | isbn=978-0-521-41517-0 | id={{MR|1144763}} | year=1991 | volume=24}}
*{{Citation | last1=Châtelet | first1=François | title=Méthode galoisienne et courbes de genre un | id={{MR|0020575}} | year=1946 | journal=Annales de L'Université de Lyon Sect. A. (3) | volume=9 | pages=40–49}}
* {{Citation | last2=Silverman | first2=Joseph H. | author2-link=Joseph H. Silverman | last1=Hindry | first1=Marc | author1-link=Marc Hindry | title=Diophantine geometry: an introduction | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-98981-5 | year=2000 | volume=201}}
* {{Citation | last1=Greenberg | first1=Ralph | author1-link=Ralph Greenberg | editor1-last=Serre | editor1-first=Jean-Pierre | editor1-link=Jean-Pierre Serre | editor2-last=Jannsen | editor2-first=Uwe | editor3-last=Kleiman | editor3-first=Steven L. | title=Motives | publisher=[[American Mathematical Society]] | location=Providence, R.I. | isbn=978-0-8218-1637-0 | year=1994 | chapter=Iwasawa Theory and p-adic Deformation of Motives}}
*{{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Algebraic groups over finite fields | url=http://www.jstor.org/stable/2372673 | id={{MR|0086367}} | year=1956 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=78 | pages=555–563}}
*{{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | last2=Tate | first2=John | author2-link=John Tate | title=Principal homogeneous spaces over abelian varieties | url=http://www.jstor.org/stable/2372778 | id={{MR|0106226}} | year=1958 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=80 | pages=659–684}}
*{{Citation | last1=Schmidt | first1=Friedrich Karl | title=Analytische Zahlentheorie in Körpern der Charakteristik p | url=http://dx.doi.org/10.1007/BF01174341 | publisher=Springer Berlin / Heidelberg | doi=10.1007/BF01174341 | year=1931 | journal=[[Mathematische Zeitschrift]] | issn=0025-5874 | volume=33 | pages=1–32}}
*{{Citation | last1=Shafarevich | first1=I. R. | title=The group of principal homogeneous algebraic manifolds | language=Russian | id={{MR|0106227}} English translation in his collected mathematical papers | year=1959 | journal=Doklady Akademii Nauk SSSR | issn=0002-3264 | volume=124 | pages=42–43}}
*{{Citation | last1=Tate | first1=John | author1-link=John Tate | title=WC-groups over p-adic fields | url=http://www.numdam.org/item?id=SB_1956-1958__4__265_0 | publisher=Secrétariat Mathématique | location=Paris | series=Séminaire Bourbaki; 10e année: 1957/1958 | id={{MR|0105420}} | year=1958 | volume=13}}
*{{Citation | last1=Weil | first1=André | author1-link=André Weil | title=On algebraic groups and homogeneous spaces | url=http://www.jstor.org/stable/2372637 | id={{MR|0074084}} | year=1955 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=77 | pages=493–512}}

{{DEFAULTSORT:Weil-Chatelet group}}
[[Category:Number theory]]

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