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]

It is known that the Tate–Shafarevich group is a torsion group,^[7]^[8] thus the conjecture is equivalent to stating that the group is finitely generated.

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.^[9] 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.^[10] 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,^[11] who misquoted one of the results of Tate.^[10] 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,^[12] and Stein gave some examples where the power of an odd prime dividing the order is odd.^[13] 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 $| Ш | = n \cdot m 2$ .^[14] 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 $Ш$ .

Citations

^ Lang & Tate 1958.
^ Shafarevich 1959.
^ Lind 1940.
^ Selmer 1951.
^ Rubin 1987.
^ Kolyvagin 1988.
^ Kolyvagin, V. A. (1991), "On the structure of shafarevich-tate groups", Algebraic Geometry, Lecture Notes in Mathematics, vol. 1479, Springer Berlin Heidelberg, pp. 94–121, doi:10.1007/bfb0086267, ISBN 978-3-540-54456-2
^ Poonen, Bjorn (2024-09-01). "THE SELMER GROUP, THE SHAFAREVICH-TATE GROUP, AND THE WEAK MORDELL-WEIL THEOREM" (PDF).
^ Cassels 1962.
^ ^a ^b Tate 1963.
^ Swinnerton-Dyer 1967.
^ Poonen & Stoll 1999.
^ Stein 2004.
^ Konstantinous 2024.

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
Konstantinous, Alexandros (2024-04-25). "A note on the order of the Tate-Shafarevich group modulo squares". arXiv:2404.16785 [math.NT].

[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.

[7] Kolyvagin, V. A. (1991), "On the structure of shafarevich-tate groups", Algebraic Geometry, Lecture Notes in Mathematics, vol. 1479, Springer Berlin Heidelberg, pp. 94–121, doi:10.1007/bfb0086267, ISBN 978-3-540-54456-2

[8] Poonen, Bjorn (2024-09-01). "THE SELMER GROUP, THE SHAFAREVICH-TATE GROUP, AND THE WEAK MORDELL-WEIL THEOREM" (PDF).

[FOOTNOTECassels1962-9] Cassels 1962.

[FOOTNOTETate1963-10] Tate 1963.

[FOOTNOTESwinnerton-Dyer1967-11] Swinnerton-Dyer 1967.

[FOOTNOTEPoonenStoll1999-12] Poonen & Stoll 1999.

[FOOTNOTEStein2004-13] Stein 2004.

[FOOTNOTEKonstantinous2024-14] Konstantinous 2024.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]