Weil–Châtelet Group
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In
arithmetic geometry In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties. ...
, the Weil–Châtelet group or WC-group of an
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...
such as an
abelian variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group ...
''A'' defined over a field ''K'' is the
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
of
principal homogeneous space In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non-e ...
s for ''A'', defined over ''K''. named it for who introduced it for
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...
s, and , who introduced it for more general groups. It plays a basic role in the
arithmetic of abelian varieties In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back to the studies of Pierre de Fermat on what are now recognized as elliptic curves; and has b ...
, in particular for elliptic curves, because of its connection with
infinite descent In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold f ...
. It can be defined directly from
Galois cohomology In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated with a field extension ''L''/''K'' acts in a na ...
, as H^1(G_K,A), where G_K is the
absolute Galois group In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' ...
of ''K''. It is of particular interest for
local field In mathematics, a field ''K'' is called a non-Archimedean local field if it is complete with respect to a metric induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. In general, a local field is a locally compact t ...
s and
global field In mathematics, a global field is one of two types of fields (the other one is local fields) that are characterized using valuations. There are two kinds of global fields: *Algebraic number field: A finite extension of \mathbb *Global functio ...
s, such as
algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
s. For ''K'' a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
, proved that the Weil–Châtelet group is trivial for elliptic curves, and proved that it is trivial for any connected algebraic group.


See also

The
Tate–Shafarevich group In arithmetic geometry, the Tate–Shafarevich group of an abelian variety (or more generally a group scheme) defined over a number field consists of the elements of the Weil–Châtelet group \mathrm(A/K) = H^1(G_K, A), where G_K = \mathrm(K ...
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^(A/K)=\bigcap_v\mathrm(H^1(G_K,\mathrm(f))\rightarrow H^1(G_,A_v /\mathrm(\kappa_v)) where ''A''v 'f''denotes the ''f''- torsion of ''A''v and \kappa_v is the local Kummer map : B_v(K_v)/f(A_v(K_v))\rightarrow H^1(G_,A_v .


References

* * * * * * * * * * English translation in his collected mathematical papers. * * {{DEFAULTSORT:Weil-Chatelet group Number theory