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 $ 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