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]

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) × 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.^[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 ⋅ m².^[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.

See also

• Birch and Swinnerton-Dyer conjecture

Citations

1. Lang & Tate 1958.
2. Shafarevich 1959.
3. Lind 1940.
4. Selmer 1951.
5. Rubin 1987.
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).
9. Cassels 1962.
10. Tate 1963.
11. Swinnerton-Dyer 1967.
12. Poonen & Stoll 1999.
13. Stein 2004.
14. Konstantinous 2024.

References

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