# User:Maximilian Janisch/latexlist/Algebraic Groups/Weil-Châtelet group

The group of principal homogeneous spaces (cf. Principal homogeneous space) over an Abelian variety. It was shown by A. Weil  and, in one special case, by F. Châtelet, that for an arbitrary Abelian variety $4$ over a field $k$, the set $VC ( A , k )$ of principal homogeneous spaces over $4$, defined over k, has a group structure. The group $VC ( A , k )$ is isomorphic to the first Galois cohomology group $H ^ { 1 } ( k , A )$. The group $VC ( A , k )$ is always periodic; moreover, if $k = Q$, it contains elements of arbitrary orders , . According to Lang's theorem, $WC ( A , k ) = 0$ if $k$ is a finite field. The index $I = \operatorname { ind } _ { k } ( D )$, equal to the smallest degree of an extension $K / k$ for which there exists a $K$-rational point $\Omega$, is defined for any element $D \in W C ( A , k )$. If $\operatorname { dim } A = 1$ and $k$ is an algebraic function field over an algebraically closed field of constants or a local field, $1$ becomes identical with the order of $\Omega$ in the group $VC ( A , k )$ , . In the general case these numbers are different, but $rd ( D )$ is always a divisor of $1$ . The group $VC ( A , k )$ has been computed for local fields $k$ (see, for instance, , , ).

If $k$ is a global field, the computation of the group $VC ( A , k )$ is based on the reduction homomorphisms

\begin{equation} \phi _ { v } : \operatorname { WC } ( A , k ) \rightarrow WC ( A , k _ { v } ) \end{equation}

where $v$ is an arbitrary valuation of $k$ and $k _ { j }$ is the completion of $k$ with respect to $v$. The kernel $\square ( A )$ of the homomorphism

\begin{equation} \phi = \sum \phi _ { v } : WC ( A , k ) \rightarrow \sum _ { v } WC ( A , k _ { v } ) \end{equation}

known as the Tate–Shafarevich group of the Abelian variety $4$, has been computed only in the case when $k$ is a field of algebraic functions of one variable over an algebraically closed field of constants , , . The co-kernel of $( 1 )$ has also been described in this case (up to the $D$-component, where $D$ is the characteristic of $k$). The results of these calculations are used in the theory of elliptic surfaces. If $k$ is an algebraic number field, the structure of the group $\square ( A )$ has not been studied to any great extent.

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Maximilian Janisch/latexlist/Algebraic Groups/Weil-Châtelet group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Maximilian_Janisch/latexlist/Algebraic_Groups/Weil-Ch%C3%A2telet_group&oldid=44071