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 $\mathrm {WC} (A/K)=H^{1}(G_{K},A)$ , where $G_{K}=\mathrm {Gal} (K^{alg}/K)$ 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

\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 $x 4 - 17 = 2 y 2$ 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 $3 x 3 + 4 y 3 + 5 z 3 = 0$ .[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 $Ш(A) \times Ш(Â) \to Q / Z$ , where $A$ is an abelian variety and $Â$ is its dual. Cassels introduced this for elliptic curves, when $A$ can be identified with $Â$ 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).

References

Cassels, John William Scott (1962), "Arithmetic on curves of genus 1. III. The Tate–Šafarevič and Selmer groups", Proceedings of the London Mathematical Society, Third Series, 12: 259–296, doi:10.1112/plms/s3-12.1.259, ISSN 0024-6115, MR 0163913
Cassels, John William Scott (1962b), "Arithmetic on curves of genus 1. IV. Proof of the Hauptvermutung", Journal für die reine und angewandte Mathematik, 211 (211): 95–112, doi:10.1515/crll.1962.211.95, ISSN 0075-4102, MR 0163915
Cassels, John William Scott (1991), Lectures on elliptic curves, London Mathematical Society Student Texts, vol. 24, Cambridge University Press, doi:10.1017/CBO9781139172530, ISBN 978-0-521-41517-0, MR 1144763
Hindry, Marc; Silverman, Joseph H. (2000), Diophantine geometry: an introduction, Graduate Texts in Mathematics, vol. 201, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98981-5
Greenberg, Ralph (1994), "Iwasawa Theory and p-adic Deformation of Motives", in Serre, Jean-Pierre; Jannsen, Uwe; Kleiman, Steven L. (eds.), Motives, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1637-0
Kolyvagin, V. A. (1988), "Finiteness of E(Q) and SH(E,Q) for a subclass of Weil curves", Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, 52 (3): 522–540, 670–671, ISSN 0373-2436, 954295
Lang, Serge; Tate, John (1958), "Principal homogeneous spaces over abelian varieties", American Journal of Mathematics, 80 (3): 659–684, doi:10.2307/2372778, ISSN 0002-9327, JSTOR 2372778, MR 0106226
Lind, Carl-Erik (1940). Untersuchungen über die rationalen Punkte der ebenen kubischen Kurven vom Geschlecht Eins (Thesis). Vol. 1940. University of Uppsala. 97 pp. MR 0022563.
Poonen, Bjorn; Stoll, Michael (1999), "The Cassels-Tate pairing on polarized abelian varieties", Annals of Mathematics, Second Series, 150 (3): 1109–1149, arXiv:math/9911267, doi:10.2307/121064, ISSN 0003-486X, JSTOR 121064, MR 1740984
Rubin, Karl (1987), "Tate–Shafarevich groups and L-functions of elliptic curves with complex multiplication", Inventiones Mathematicae, 89 (3): 527–559, Bibcode:1987InMat..89..527R, doi:10.1007/BF01388984, ISSN 0020-9910, MR 0903383
Selmer, Ernst S. (1951), "The Diophantine equation ax³+by³+cz³=0", Acta Mathematica, 85: 203–362, doi:10.1007/BF02395746, ISSN 0001-5962, MR 0041871
Shafarevich, I. R. (1959), "The group of principal homogeneous algebraic manifolds", Doklady Akademii Nauk SSSR (in Russian), 124: 42–43, ISSN 0002-3264, MR 0106227 English translation in his collected mathematical papers
Stein, William A. (2004), "Shafarevich–Tate groups of nonsquare order" (PDF), Modular curves and abelian varieties, Progr. Math., vol. 224, Basel, Boston, Berlin: Birkhäuser, pp. 277–289, MR 2058655
Swinnerton-Dyer, P. (1967), "The conjectures of Birch and Swinnerton-Dyer, and of Tate", in Springer, Tonny A. (ed.), Proceedings of a Conference on Local Fields (Driebergen, 1966), Berlin, New York: Springer-Verlag, pp. 132–157, MR 0230727
Tate, John (1958), WC-groups over p-adic fields, Séminaire Bourbaki; 10e année: 1957/1958, vol. 13, Paris: Secrétariat Mathématique, MR 0105420
Tate, John (1963), "Duality theorems in Galois cohomology over number fields", Proceedings of the International Congress of Mathematicians (Stockholm, 1962), Djursholm: Inst. Mittag-Leffler, pp. 288–295, MR 0175892, archived from the original on 2011-07-17
Weil, André (1955), "On algebraic groups and homogeneous spaces", American Journal of Mathematics, 77 (3): 493–512, doi:10.2307/2372637, ISSN 0002-9327, JSTOR 2372637, MR 0074084

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[FOOTNOTELangTate1958-1] Lang & Tate 1958.

[FOOTNOTEShafarevich1959-2] Shafarevich 1959.

[FOOTNOTELind1940-3] Lind 1940.

[FOOTNOTESelmer1951-4] Selmer 1951.

[FOOTNOTERubin1987-5] Rubin 1987.

[FOOTNOTEKolyvagin1988-6] Kolyvagin 1988.

[FOOTNOTECassels1962-7] Cassels 1962.

[FOOTNOTETate1963-8] Tate 1963.

[FOOTNOTESwinnerton-Dyer1967-9] Swinnerton-Dyer 1967.

[FOOTNOTEPoonenStoll1999-10] Poonen & Stoll 1999.

[FOOTNOTEStein2004-11] Stein 2004.