Tate–Shafarevich group

In arithmetic geometry, the Tate–Shafarevich group Template:Math of an abelian variety Template:Math (or more generally a group scheme) defined over a number field Template:Math consists of the elements of the Weil–Châtelet group $\mathrm{W}\mathrm{C}(A/K)=H^1(G_K,A)$, where $G_K=\mathrm{G}\mathrm{a}\mathrm{l}(K^{alg}/K)$ is the absolute Galois group of Template:Math, that become trivial in all of the completions of Template:Math (i.e., the real and complex completions as well as the [[p-adic field|Template:Math-adic fields]] obtained from Template:Math by completing with respect to all its Archimedean and non Archimedean valuations Template:Math). Thus, in terms of Galois cohomology, Template:Math can be defined as

$\bigcap _v\mathrm{k}\mathrm{e}\mathrm{r}(H^1(G_K,A)\to H^1(G_{K_v},A_v)).$

This group was introduced by Serge Lang and John TateTemplate:Sfn and Igor Shafarevich.Template:Sfn Cassels introduced the notation Template:Math, where Template:Math is the Cyrillic letter "Sha",Template:Sfn for Shafarevich, replacing the older notation Template:Math or Template:Math.Template:Sfn

Elements of the Tate–Shafarevich group

Geometrically, the non-trivial elements of the Tate–Shafarevich group can be thought of as the homogeneous spaces of Template:Math that have Template:Math-rational points for every place Template:Math of Template:Math, but no Template:Math-rational point. Thus, the group measures the extent to which the Hasse principle fails to hold for rational equations with coefficients in the field Template:Math. Carl-Erik Lind gave an example of such a homogeneous space, by showing that the genus 1 curve Template:Math has solutions over the reals and over all Template:Math-adic fields, but has no rational points.Template:Sfn Ernst S. Selmer gave many more examples, such as Template:Math.Template:Sfn

The special case of the Tate–Shafarevich group for the finite group scheme consisting of points of some given finite order Template:Math of an abelian variety is closely related to the Selmer group.

Tate–Shafarevich conjecture

The Tate–Shafarevich conjecture states that the Tate–Shafarevich group is finite. Karl Rubin proved this for some elliptic curves of rank at most 1 with complex multiplication.Template:Sfn Victor A. Kolyvagin extended this to modular elliptic curves over the rationals of analytic rank at most 1 (The modularity theorem later showed that the modularity assumption always holds).Template:Sfn Theorem 1.1 of Nikolaev treats simple abelian varieties over number fields.

It is known that the Tate–Shafarevich group is a torsion group, thus the conjecture is equivalent to stating that the group is finitely generated.

Cassels–Tate pairing

The Cassels–Tate pairing is a bilinear pairing Template:Math, where Template:Math is an abelian variety and Template:Math is its dual. Cassels introduced this for elliptic curves, when Template:Math can be identified with Template:Math and the pairing is an alternating form.Template:Sfn The kernel of this form is the subgroup of divisible elements, which is trivial if the Tate–Shafarevich conjecture is true. Tate extended the pairing to general abelian varieties, as a variation of Tate duality.Template:Sfn A choice of polarization on A gives a map from Template:Math to Template:Math, which induces a bilinear pairing on Template:Math with values in Template:Math, but unlike the case of elliptic curves this need not be alternating or even skew symmetric.

For an elliptic curve, Cassels showed that the pairing is alternating, and a consequence is that if the order of Template:Math is finite then it is a square. For more general abelian varieties it was sometimes incorrectly believed for many years that the order of Template:Math is a square whenever it is finite; this mistake originated in a paper by Swinnerton-Dyer,Template:Sfn who misquoted one of the results of Tate.Template:Sfn Poonen and Stoll gave some examples where the order is twice a square, such as the Jacobian of a certain genus 2 curve over the rationals whose Tate–Shafarevich group has order 2,Template:Sfn and Stein gave some examples where the power of an odd prime dividing the order is odd.Template:Sfn If the abelian variety has a principal polarization then the form on Template:Math is skew symmetric which implies that the order of Template:Math is a square or twice a square (if it is finite), and if in addition the principal polarization comes from a rational divisor (as is the case for elliptic curves) then the form is alternating and the order of Template:Math is a square (if it is finite). On the other hand building on the results just presented Konstantinou showed that for any squarefree number Template:Math there is an abelian variety Template:Math defined over Template:Math and an integer Template:Math with Template:Math.Template:Sfn In particular Template:Math is finite in Konstantinou's examples and these examples confirm a conjecture of Stein. Thus modulo squares any integer can be the order of Template:Math.

See also

• Birch and Swinnerton-Dyer conjecture

References

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