View source for Weil-Châtelet group

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]]]).

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

<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>

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

<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>

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.

====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</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  (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</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>



====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]]].

====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>