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Weil–Châtelet group

{{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 K is the abelian group of principal homogeneous spaces for A, defined over K. {{harvs|txt|last=Tate|first=John|author-link=John Tate (mathematician)|year=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|first=André|year=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 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, {{harvs|txt|last=Schmidt | first=Friedrich Karl |authorlink=Friedrich Karl Schmidt|year=1931}} proved that the Weil–Châtelet group is trivial for elliptic curves, and {{harvs|txt|last=Lang | first=Serge | author-link=Serge Lang |year=1956}} proved that it is trivial for any connected 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\colon A\to B of abelian varieties is a related group which can be defined in terms of Galois cohomology as

:\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 Av[f] denotes the f-torsion of Av and \kappa_v is the local Kummer map

: B_v(K_v)/f(A_v(K_v))\rightarrow H^1(G_{K_v},A_v[f]).

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 |mr=0163913 | year=1962 | journal=Proceedings of the London Mathematical Society |series=Third Series | issn=0024-6115 | volume=12 | pages=259–296}}
  • {{Citation | last1=Cassels | first1=John William Scott | title=Lectures on elliptic curves | url=https://books.google.com/books?id=zgqUAuEJNJ4C | publisher=Cambridge University Press | series=London Mathematical Society Student Texts | isbn=978-0-521-41517-0 |mr=1144763 | year=1991 | volume=24 | doi=10.1017/CBO9781139172530}}
  • {{Citation | last1=Châtelet | first1=François | title=Méthode galoisienne et courbes de genre un |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}}
  • {{springer|title=Weil-Châtelet group|id=p/w097590}}
  • {{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Algebraic groups over finite fields | jstor=2372673 |mr=0086367 | year=1956 | journal=American Journal of Mathematics | issn=0002-9327 | volume=78 | issue=3 | pages=555–563 | doi=10.2307/2372673}}
  • {{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | last2=Tate | first2=John | author2-link=John Tate (mathematician) | title=Principal homogeneous spaces over abelian varieties | jstor=2372778 |mr=0106226 | year=1958 | journal=American Journal of Mathematics | issn=0002-9327 | volume=80 | issue=3 | pages=659–684 | doi=10.2307/2372778}}
  • {{Citation | last=Schmidt | first=Friedrich Karl |authorlink=Friedrich Karl Schmidt| title=Analytische Zahlentheorie in Körpern der Charakteristik p | doi=10.1007/BF01174341 | year=1931 | journal=Mathematische Zeitschrift | issn=0025-5874 | volume=33 | pages=1–32}}
  • {{Citation | last=Shafarevich | first=Igor R. | authorlink=Igor Shafarevich| title=The group of principal homogeneous algebraic manifolds | language=Russian |mr=0106227| year=1959 | journal=Doklady Akademii Nauk SSSR | issn=0002-3264 | volume=124 | pages=42–43}} English translation in his collected mathematical papers.
  • {{Citation | last1=Tate | first1=John | author1-link=John Tate (mathematician) | 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 |mr=0105420 | year=1958 | volume=13}}
  • {{Citation | last1=Weil | first1=André | author1-link=André Weil | title=On algebraic groups and homogeneous spaces | jstor=2372637 |mr=0074084 | year=1955 | journal=American Journal of Mathematics | issn=0002-9327 | volume=77 | issue=3 | pages=493–512 | doi=10.2307/2372637}}

{{DEFAULTSORT:Weil-Chatelet group}}

Category:Number theory