Weil-Châtelet group

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 over a field , the set of principal homogeneous spaces over , defined over k, has a group structure. The group is isomorphic to the first Galois cohomology group . The group is always periodic; moreover, if , it contains elements of arbitrary orders [4], [5]. According to Lang's theorem, if is a finite field. The index , equal to the smallest degree of an extension for which there exists a -rational point , is defined for any element . If and is an algebraic function field over an algebraically closed field of constants or a local field, becomes identical with the order of in the group [6], [10]. In the general case these numbers are different, but is always a divisor of [7]. The group has been computed for local fields (see, for instance, [6], [8], [9]).

If is a global field, the computation of the group is based on the reduction homomorphisms

where is an arbitrary valuation of and is the completion of with respect to . The kernel of the homomorphism

known as the Tate–Shafarevich group of the Abelian variety , has been computed only in the case when is a field of algebraic functions of one variable over an algebraically closed field of constants [5], [8], [11]. The co-kernel of has also been described in this case (up to the -component, where is the characteristic of ). The results of these calculations are used in the theory of elliptic surfaces. If is an algebraic number field, the structure of the group 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 -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 -component of the Weil–Châtelet groups have been obtained [a3].

References

[a1]	V.A. Kolyvagin, "Finiteness of and 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 -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