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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w0975901.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w0975902.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w0975903.png" /> of principal homogeneous spaces over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w0975904.png" />, defined over k, has a group structure. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w0975905.png" /> is isomorphic to the first [[Galois cohomology|Galois cohomology]] group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w0975906.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w0975907.png" /> is always periodic; moreover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w0975908.png" />, it contains elements of arbitrary orders [[#References|[4]]], [[#References|[5]]]. According to Lang's theorem, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w0975909.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759010.png" /> is a finite field. The index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759011.png" />, equal to the smallest degree of an extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759012.png" /> for which there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759013.png" />-rational point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759014.png" />, is defined for any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759015.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759017.png" /> is an algebraic function field over an algebraically closed field of constants or a local field, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759018.png" /> becomes identical with the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759019.png" /> in the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759020.png" /> [[#References|[6]]], [[#References|[10]]]. In the general case these numbers are different, but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759021.png" /> is always a divisor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759022.png" /> [[#References|[7]]]. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759023.png" /> has been computed for local fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759024.png" /> (see, for instance, [[#References|[6]]], [[#References|[8]]], [[#References|[9]]]).
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759025.png" /> is a global field, the computation of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759026.png" /> is based on the reduction homomorphisms
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Out of 4 formulas, 4 were replaced by TEX code.-->
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759027.png" /></td> </tr></table>
+
{{TEX|semi-auto}}{{TEX|done}}
 +
{{TEX|auto}}
 +
{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759028.png" /> is an arbitrary valuation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759030.png" /> is the completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759031.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759032.png" />. The kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759033.png" /> of the homomorphism
+
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  $  A $
 +
over a field  $  k $,
 +
the set  $  { \mathop{\rm WC} } ( A, k) $
 +
of principal homogeneous spaces over  $  A $,
 +
defined over k, has a group structure. The group  $  { \mathop{\rm WC} } ( A, k) $
 +
is isomorphic to the first [[Galois cohomology|Galois cohomology]] group  $  H  ^ {1} ( k, A) $.  
 +
The group  $  { \mathop{\rm WC} } ( A, k) $
 +
is always periodic; moreover, if  $  k = \mathbf Q $,
 +
it contains elements of arbitrary orders [[#References|[4]]], [[#References|[5]]]. According to Lang's theorem,  $  { \mathop{\rm WC} } ( A, k) = 0 $
 +
if  $  k $
 +
is a finite field. The index  $  I = { \mathop{\rm ind} } _ {k} ( D) $,
 +
equal to the smallest degree of an extension  $  K/k $
 +
for which there exists a  $  K $-
 +
rational point  $  D $,
 +
is defined for any element  $  D \in { \mathop{\rm WC} } ( A, k) $.  
 +
If  $  { \mathop{\rm dim} }  A = 1 $
 +
and  $  k $
 +
is an algebraic function field over an algebraically closed field of constants or a local field,  $  I $
 +
becomes identical with the order of $  D $
 +
in the group  $  { \mathop{\rm WC} } ( A, k) $[[#References|[6]]], [[#References|[10]]]. In the general case these numbers are different, but  $  { \mathop{\rm ord} } ( D) $
 +
is always a divisor of  $  I $[[#References|[7]]]. The group  $  { \mathop{\rm WC} } ( A, k) $
 +
has been computed for local fields  $  k $(
 +
see, for instance, [[#References|[6]]], [[#References|[8]]], [[#References|[9]]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759034.png" /></td> </tr></table>
+
If  $  k $
 +
is a global field, the computation of the group  $  { \mathop{\rm WC} } ( A, k) $
 +
is based on the reduction homomorphisms
  
known as the Tate–Shafarevich group of the Abelian variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759035.png" />, has been computed only in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759036.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759037.png" /> has also been described in this case (up to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759038.png" />-component, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759039.png" /> is the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759040.png" />). The results of these calculations are used in the theory of elliptic surfaces. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759041.png" /> is an algebraic number field, the structure of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759042.png" /> has not been studied to any great extent.
+
$$
 +
\phi _ {v} :   \mathop{\rm WC} ( A, k )  \rightarrow  \mathop{\rm WC} ( A, k _ {v} ),
 +
$$
  
====References====
+
where $ v $
<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</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</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  (1957pp. 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</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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759043.png" />-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>
+
is an arbitrary valuation of  $ k $
 +
and $ k _ {v} $
 +
is the completion of $ k $
 +
with respect to  $ v $.  
 +
The kernel $ {\mathop{\amalg\kern-0.30em\amalg}} ( A) $
 +
of the homomorphism
 +
 
 +
$$
 +
\phi \sum \phi _ {v} \mathop{\rm WC} ( A, k\rightarrow \sum _ { v }
 +
  \mathop{\rm WC} ( A, k _ {v} ),
 +
$$
  
 +
known as the Tate–Shafarevich group of the Abelian variety  $  A $,
 +
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  $  \phi $
 +
has also been described in this case (up to the  $  p $-
 +
component, where  $  p $
 +
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  $  {\mathop{\amalg\kern-0.30em\amalg}} ( 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 $\mathfrak{p}$-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====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759044.png" />-component of the Weil–Châtelet groups have been obtained [[#References|[a3]]].
+
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 $  p $-
 +
component of the Weil–Châtelet groups have been obtained [[#References|[a3]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V.A. Kolyvagin,   "Finiteness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759046.png" /> 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</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)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> K. Rubin,   "Tate–Shafarevich groups and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097590/w09759047.png" />-functions of elliptic curves with complex multiplication" ''Invert. Math.'' , '''89''' (1987) pp. 527–560</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>
+
<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>

Latest revision as of 16:59, 1 July 2020


The group of principal homogeneous spaces (cf. Principal homogeneous space) over an Abelian variety. It was shown by A. Weil [1] and, in one special case, by F. Châtelet, that for an arbitrary Abelian variety $ A $ over a field $ k $, the set $ { \mathop{\rm WC} } ( A, k) $ of principal homogeneous spaces over $ A $, defined over k, has a group structure. The group $ { \mathop{\rm WC} } ( A, k) $ is isomorphic to the first Galois cohomology group $ H ^ {1} ( k, A) $. The group $ { \mathop{\rm WC} } ( A, k) $ is always periodic; moreover, if $ k = \mathbf Q $, it contains elements of arbitrary orders [4], [5]. According to Lang's theorem, $ { \mathop{\rm WC} } ( A, k) = 0 $ if $ k $ is a finite field. The index $ I = { \mathop{\rm ind} } _ {k} ( D) $, equal to the smallest degree of an extension $ K/k $ for which there exists a $ K $- rational point $ D $, is defined for any element $ D \in { \mathop{\rm WC} } ( A, k) $. If $ { \mathop{\rm dim} } A = 1 $ and $ k $ is an algebraic function field over an algebraically closed field of constants or a local field, $ I $ becomes identical with the order of $ D $ in the group $ { \mathop{\rm WC} } ( A, k) $[6], [10]. In the general case these numbers are different, but $ { \mathop{\rm ord} } ( D) $ is always a divisor of $ I $[7]. The group $ { \mathop{\rm WC} } ( A, k) $ has been computed for local fields $ k $( see, for instance, [6], [8], [9]).

If $ k $ is a global field, the computation of the group $ { \mathop{\rm WC} } ( A, k) $ is based on the reduction homomorphisms

$$ \phi _ {v} : \mathop{\rm WC} ( A, k ) \rightarrow \mathop{\rm WC} ( A, k _ {v} ), $$

where $ v $ is an arbitrary valuation of $ k $ and $ k _ {v} $ is the completion of $ k $ with respect to $ v $. The kernel $ {\mathop{\amalg\kern-0.30em\amalg}} ( A) $ of the homomorphism

$$ \phi = \sum \phi _ {v} : \mathop{\rm WC} ( A, k) \rightarrow \sum _ { v } \mathop{\rm WC} ( A, k _ {v} ), $$

known as the Tate–Shafarevich group of the Abelian variety $ A $, 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 [5], [8], [11]. The co-kernel of $ \phi $ has also been described in this case (up to the $ p $- component, where $ p $ 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 $ {\mathop{\amalg\kern-0.30em\amalg}} ( A) $ has not been studied to any great extent.

References

[1] A. Weil, "On algebraic groups and homogeneous spaces" Amer. J. Math. , 77 (1955) pp. 493–512 MR0074084 Zbl 0065.14202
[2] 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
[3] J. Cassels, "Diophantine equations with special reference to elliptic curves" J. London Math. Soc. , 41 (1966) pp. 193–291 MR0199150 Zbl 0138.27002
[4] I.R. Shafarevich, "Birational equivalence of elliptic curves" Dokl. Akad. Nauk SSSR , 114 : 2 (1957) pp. 267–270 (In Russian)
[5] I.R. Shafarevich, "Exponents of elliptic curves" Dokl. Akad. Nauk SSSR , 114 : 4 (1957) pp. 714–716 (In Russian)
[6] I.R. Shafarevich, "Principal homogeneous spaces defined over a function field" Trudy Mat. Inst. Steklov. , 64 (1961) pp. 316–346 (In Russian)
[7] S. Lang, J. Tate, "Principal homogeneous spaces over abelian varieties" Amer. J. Math. , 80 (1958) pp. 659–684 MR0106226 Zbl 0097.36203
[8] A.P. Ogg, "Cohomology of Abelian varieties over function fields" Ann. of Math. (2) , 76 : 2 (1962) pp. 185–212
[9] J.T. Tate, "WC-groups over $\mathfrak{p}$-adic fields" , Sem. Bourbaki , Exp. 156 , Secr. Math. Univ. Paris (1957)
[10] S. Lichtenbaum, "The period-index problem for elliptic curves" Amer. J. Math. , 90 : 4 (1968) pp. 1209–1223
[11] 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 & Masson (1968) pp. 12–30

Comments

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

References

[a1] 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
[a2] 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 MR1181210 Zbl 0753.14025
[a3] J. Milne, "Arithmetic duality theorems" , Acad. Press (1986)
[a4] J.H. Silverman, "The arithmetic of elliptic curves" , Springer (1986) MR0817210 Zbl 0585.14026
[a5] K. Rubin, "Tate–Shafarevich groups and $L$-functions of elliptic curves with complex multiplication" Invert. Math. , 89 (1987) pp. 527–560 MR0903383
[a6] V.A. Kolyvagin, "Euler systems" P. Cartier (ed.) et al. (ed.) , Grothendieck Festschrift , II , Birkhäuser (1990) pp. 435–484
[a7] 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
How to Cite This Entry:
Weil-Châtelet group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weil-Ch%C3%A2telet_group&oldid=23131
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article