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The group of principal homogeneous spaces (cf. [[Principal homogeneous space|Principal homogeneous space]]) over an Abelian variety. It was shown by A. Weil [[#References|[1]]] 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|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 [[#References|[4]]], [[#References|[5]]]. 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 $VC ( A , k )$ [[#References|[6]]], [[#References|[10]]]. In the general case these numbers are different, but $rd ( D )$ is always a divisor of $1$ [[#References|[7]]]. The group $VC ( A , k )$ has been computed for local fields $k$ (see, for instance, [[#References|[6]]], [[#References|[8]]], [[#References|[9]]]).

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 [[#References|[5]]], [[#References|[8]]], [[#References|[11]]]. 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.

====References====
<table><tr><td valign="top">[1]</td> <td valign="top"> A. Weil, "On algebraic groups and homogeneous spaces" ''Amer. J. Math.'' , '''77''' (1955) pp. 493–512 {{MR|0074084}} {{ZBL|0065.14202}} </td></tr><tr><td valign="top">[2]</td> <td valign="top"> M.I. Bashmakov, "Cohomology of Abelian varieties over a number field" ''Russian Math. Surveys'' , '''27''' : 6 (1972) pp. 25–70 ''Uspekhi Mat. Nauk'' , '''27''' : 6 (1972) pp. 25–66</td></tr><tr><td valign="top">[3]</td> <td valign="top"> J. Cassels, "Diophantine equations with special reference to elliptic curves" ''J. London Math. Soc.'' , '''41''' (1966) pp. 193–291 {{MR|0199150}} {{ZBL|0138.27002}} </td></tr><tr><td valign="top">[4]</td> <td valign="top"> I.R. Shafarevich, "Birational equivalence of elliptic curves" ''Dokl. Akad. Nauk SSSR'' , '''114''' : 2 (1957) pp. 267–270 (In Russian)</td></tr><tr><td valign="top">[5]</td> <td valign="top"> I.R. Shafarevich, "Exponents of elliptic curves" ''Dokl. Akad. Nauk SSSR'' , '''114''' : 4 (1957) pp. 714–716 (In Russian)</td></tr><tr><td valign="top">[6]</td> <td valign="top"> I.R. Shafarevich, "Principal homogeneous spaces defined over a function field" ''Trudy Mat. Inst. Steklov.'' , '''64''' (1961) pp. 316–346 (In Russian)</td></tr><tr><td valign="top">[7]</td> <td valign="top"> S. Lang, J. Tate, "Principal homogeneous spaces over abelian varieties" ''Amer. J. Math.'' , '''80''' (1958) pp. 659–684 {{MR|0106226}} {{ZBL|0097.36203}} </td></tr><tr><td valign="top">[8]</td> <td valign="top"> A.P. Ogg, "Cohomology of Abelian varieties over function fields" ''Ann. of Math. (2)'' , '''76''' : 2 (1962) pp. 185–212</td></tr><tr><td valign="top">[9]</td> <td valign="top"> J.T. Tate, "WC-groups over $t$-adic fields" , ''Sem. Bourbaki'' , '''Exp. 156''' , Secr. Math. Univ. Paris (1957)</td></tr><tr><td valign="top">[10]</td> <td valign="top"> S. Lichtenbaum, "The period-index problem for elliptic curves" ''Amer. J. Math.'' , '''90''' : 4 (1968) pp. 1209–1223</td></tr><tr><td valign="top">[11]</td> <td valign="top"> M. Raynaud, "Caractéristique d'Euler–Poincaré d'un faisceau et cohomologie des variétés abéliennes (d'après Ogg–Shafarévitch et Grothendieck)" A. Grothendieck (ed.) J. Giraud (ed.) et al. (ed.) , ''Dix exposés sur la cohomologie des schémas'' , North-Holland &amp; Masson (1968) pp. 12–30</td></tr></table>



====Comments====
The Tate–Shafarevich group of certain elliptic curves over number fields has been recently computed ([[#References|[a1]]], [[#References|[a2]]], [[#References|[a5]]]). Also, some new results on the $D$-component of the Weil–Châtelet groups have been obtained [[#References|[a3]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top"> V.A. Kolyvagin, "Finiteness of $E ( Q )$ and $\square ( E / Q )$ for a class of Weil curves" ''Math. USSR Izv.'' , '''32''' (1989) pp. 523–541 ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''52''' (1988) pp. 522–540</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> V.A. Kolyvagin, "On the structure of Shafarevich–Tate groups" S. Block (ed.) et al. (ed.) , ''Algebraic geometry'' , ''Lect. notes in math.'' , '''1479''' , Springer (1991) pp. 94–121 {{MR|1181210}} {{ZBL|0753.14025}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> J. Milne, "Arithmetic duality theorems" , Acad. Press (1986)</td></tr><tr><td valign="top">[a4]</td> <td valign="top"> J.H. Silverman, "The arithmetic of elliptic curves" , Springer (1986) {{MR|0817210}} {{ZBL|0585.14026}} </td></tr><tr><td valign="top">[a5]</td> <td valign="top"> K. Rubin, "Tate–Shafarevich groups and $L$-functions of elliptic curves with complex multiplication" ''Invert. Math.'' , '''89''' (1987) pp. 527–560 {{MR|0903383}} {{ZBL|}} </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> V.A. Kolyvagin, "Euler systems" P. Cartier (ed.) et al. (ed.) , ''Grothendieck Festschrift'' , '''II''' , Birkhäuser (1990) pp. 435–484</td></tr><tr><td valign="top">[a7]</td> <td valign="top"> K. Rubin, "The work of Kolyvagin on the arithmetic of elliptic curves" W.P. Barth (ed.) et al. (ed.) , ''Arithmetic of Complex Manifolds'' , ''Lect. notes in math.'' , '''1399''' , Springer (1989) pp. 128–136</td></tr></table>