Tate–Shafarevich group

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

${\displaystyle \bigcap _{v}\mathrm {ker} \left(H^{1}\left(G_{K},A\right)\rightarrow H^{1}\left(G_{K_{v}},A_{v}\right)\right).}$

This group was introduced by Serge Lang and John Tate^[1] and Igor Shafarevich.^[2] Cassels introduced the notation Ш(A/K), where Ш is the Cyrillic letter "Sha", for Shafarevich, replacing the older notation TS or TŠ.

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 A that have K_v-rational points for every place v of K, but no K-rational point. Thus, the group measures the extent to which the Hasse principle fails to hold for rational equations with coefficients in the field K. Carl-Erik Lind gave an example of such a homogeneous space, by showing that the genus 1 curve x⁴ − 17 = 2y² has solutions over the reals and over all p-adic fields, but has no rational points.^[3] Ernst S. Selmer gave many more examples, such as 3x³ + 4y³ + 5z³ = 0.^[4]

The special case of the Tate–Shafarevich group for the finite group scheme consisting of points of some given finite order n 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.^[5] 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).^[6]

Cassels–Tate pairing

The Cassels–Tate pairing is a bilinear pairing Ш(A) × Ш(Â) → Q/Z, where A is an abelian variety and  is its dual. Cassels introduced this for elliptic curves, when A can be identified with  and the pairing is an alternating form.^[7] 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.^[8] A choice of polarization on A gives a map from A to Â, which induces a bilinear pairing on Ш(A) with values in Q/Z, 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 Ш is finite then it is a square. For more general abelian varieties it was sometimes incorrectly believed for many years that the order of Ш is a square whenever it is finite; this mistake originated in a paper by Swinnerton-Dyer,^[9] who misquoted one of the results of Tate.^[8] 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,^[10] and Stein gave some examples where the power of an odd prime dividing the order is odd.^[11] If the abelian variety has a principal polarization then the form on Ш is skew symmetric which implies that the order of Ш 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 Ш is a square (if it is finite). On the other hand building on the results just presented Konstantinous showed that for any squarefree number n there is an abelian variety A defined over Q and an integer m with |Ш| = n ⋅ m².^[12] In particular Ш is finite in Konstantinous' examples and these examples confirm a conjecture of Stein. Thus modulo squares any integer can be the order of Ш.

See also

• Birch and Swinnerton-Dyer conjecture

Citations

References