# Tate–Shafarevich group
The **Tate–Shafarevich group** of an elliptic curve or abelian variety \( A \) over a global field \( K \), usually written \( \Sha(A/K) \) with the Cyrillic letter Sha (Ш), is the subgroup of the [Galois cohomology](https://www.edgepedia.org/galois-cohomology) group \( H^1(K, A) \) consisting of classes that restrict to zero in \( H^1(K_v, A) \) for every place \( v \) of \( K \)[1](https://sporadic.stanford.edu/reference/arithmetic_curves/sage/schemes/elliptic_curves/sha_tate.html) • [2](https://mathworld.wolfram.com/Tate-ShafarevichGroup.html). It measures the failure of the [Hasse principle](https://www.edgepedia.org/hasse-principle), the local-to-global principle, for the curve: its nonzero elements are represented by curves that have points over every completion of the base field but no rational point[3](https://androma.org/page/Arithmetic%20Geometry%20I%3A%20Elliptic%20Curves) • [4](https://planetmath.org/selmergroup).
| Key fact | Detail |
|---|---|
| Definition | \( \Sha(A/K) = \bigcap_v \ker\left[H^1(K, A) \to H^1(K_v, A)\right] \), the intersection over all places \( v \) of \( K \)[1](https://sporadic.stanford.edu/reference/arithmetic_curves/sage/schemes/elliptic_curves/sha_tate.html) • [5](https://wwwf.imperial.ac.uk/~anskor/abel%20var%20notes.pdf) |
| Geometric meaning | Isomorphism classes of torsors (principal homogeneous spaces) under \( A \) that are soluble over \( K_v \) for every \( v \) but not over \( K \)[3](https://androma.org/page/Arithmetic%20Geometry%20I%3A%20Elliptic%20Curves) |
| Structure | An abelian torsion group; finiteness is conjectured for every abelian variety over a global field but is not known in general[1](https://sporadic.stanford.edu/reference/arithmetic_curves/sage/schemes/elliptic_curves/sha_tate.html) • [5](https://wwwf.imperial.ac.uk/~anskor/abel%20var%20notes.pdf) |
| Relation to Selmer groups | For each \( n \) there is an exact sequence \( 0 \to A(K)/n \to \mathrm{Sel}_n(A) \to \Sha(A)[n] \to 0 \), and \( \mathrm{Sel}_n(A) \) is finite[5](https://wwwf.imperial.ac.uk/~anskor/abel%20var%20notes.pdf) |
| Known finiteness | A theorem of Kolyvagin and Gross–Zagier using Heegner points shows \( \Sha \) is finite when the L-series of \( E/\mathbb{Q} \) does not vanish at 1 or has a simple zero there[1](https://sporadic.stanford.edu/reference/arithmetic_curves/sage/schemes/elliptic_curves/sha_tate.html) |
| Canonical example | The Selmer curve \( 3x^3 + 4y^3 + 5z^3 = 0 \) has points everywhere locally and no rational point, giving a nonzero element of \( \Sha \) of its Jacobian[6](https://www.imo.universite-paris-saclay.fr/~harari/articles/submathz.pdf) |
## Definition and torsor interpretation
For an elliptic curve \( E \) over a global field \( K \), the group is the kernel of the product of global-to-local restriction maps
\[ \Sha(E/K) = \ker\left( H^1(K, E) \to \prod_v H^1(K_v, E) \right), \]
where \( v \) runs over all places of \( K \)[1](https://sporadic.stanford.edu/reference/arithmetic_curves/sage/schemes/elliptic_curves/sha_tate.html) • [3](https://androma.org/page/Arithmetic%20Geometry%20I%3A%20Elliptic%20Curves). The same construction applies to any abelian variety \( A \)[5](https://wwwf.imperial.ac.uk/~anskor/abel%20var%20notes.pdf), and more generally the kernel of the map \( H^1(k, G) \to \prod_v H^1(k_v, G) \) is written \( \mathrm{Shaf}(G) \) for an arbitrary algebraic group \( G \); for abelian varieties this kernel is the Tate–Shafarevich group[7](https://encyclopediaofmath.org/wiki/Galois_cohomology).
The cohomological definition has a direct geometric reading. The group \( H^1(K, E) \) classifies principal homogeneous spaces, or torsors, under \( E \): curves of genus 1 equipped with a simply transitive \( E \)-action, each having \( E \) as its Jacobian[8](https://www.jmilne.org/math/Books/EC2.pdf). A torsor \( X \) has trivial restriction to \( K_v \) exactly when \( X(K_v) \) is nonempty[5](https://wwwf.imperial.ac.uk/~anskor/abel%20var%20notes.pdf). So \( \Sha(E/K) \) consists of the isomorphism classes of homogeneous spaces under \( E \) that are locally soluble at every place[3](https://androma.org/page/Arithmetic%20Geometry%20I%3A%20Elliptic%20Curves). A nonzero element is thus a curve with points over \( \mathbb{R} \) and over every \( p \)-adic field \( \mathbb{Q}_p \), but with no rational point: a counterexample to the Hasse principle[8](https://www.jmilne.org/math/Books/EC2.pdf).
The group is known to be an abelian torsion group[1](https://sporadic.stanford.edu/reference/arithmetic_curves/sage/schemes/elliptic_curves/sha_tate.html). Its \( n \)-torsion \( \Sha(A)[n] \) is finite for every \( n \), because it is a quotient of the finite \( n \)-[Selmer group](https://www.edgepedia.org/selmer-group)[5](https://wwwf.imperial.ac.uk/~anskor/abel%20var%20notes.pdf).
## Relation to Selmer groups and descent
The \( \varphi \)-Selmer group of an elliptic curve, for an isogeny or for multiplication by \( n \), is defined as the kernel of \( H^1(K, E[\varphi]) \to \prod_p H^1(K_p, E) \), and it contains the image of \( E(K) \)[4](https://planetmath.org/selmergroup). For an abelian variety \( A \) and an integer \( n \), the two constructions fit into the exact sequence
\[ 0 \to A(K)/n \to \mathrm{Sel}_n(A) \to \Sha(A)[n] \to 0. \]
The Selmer group is finite and computable in principle, so a descent by \( n \)-isogeny bounds the rank of \( A(K) \) from above; the error term in that bound is exactly \( \Sha(A)[n] \)[5](https://wwwf.imperial.ac.uk/~anskor/abel%20var%20notes.pdf). Computing the Cassels–Tate pairing on a Selmer group can shrink this bound: Fisher and Newton extended Cassels's method for computing the pairing on the 2-Selmer group to 3-Selmer groups, making the method practical in sufficiently small examples and improving the rank upper bounds obtained by 3-descent[9](https://arxiv.org/abs/1306.1410). A related method for 3-isogeny Selmer groups reduces the global part of the computation to solving norm equations \( N_{L/K}(\xi) = a \) in degree-\( p \) extensions[10](https://arxiv.org/abs/1711.02432).
The group also enters the problem of computing ranks directly. No algorithm is currently known for computing the rank of an abelian variety over a number field, even in theory; there is a candidate algorithm, but it terminates only if the \( p \)-primary part of \( \Sha(A) \) is finite for some prime \( p \)[11](https://math.univ-lyon1.fr/~roblot/ihp/weakmw.pdf).
## The Cassels–Tate pairing
There is an antisymmetric bilinear pairing[23](https://www.maths.tcd.ie/EMIS/journals/Annals/150_3/poonen.pdf)
\[ \langle \cdot, \cdot \rangle : \Sha(A) \times \Sha(A^t) \to \mathbb{Q}/\mathbb{Z}, \]
where \( A^t \) is the dual abelian variety[5](https://wwwf.imperial.ac.uk/~anskor/abel%20var%20notes.pdf). It is built from local invariants: a class in \( \Sha(A) \) is represented by a torsor \( X \) with a \( K_v \)-point \( P_v \) for each completion, and the pairing sums local pairings over all places, with the product formula ensuring the result is well defined[5](https://wwwf.imperial.ac.uk/~anskor/abel%20var%20notes.pdf) • [9](https://arxiv.org/abs/1306.1410).
The pairing was introduced in [J. W. S. Cassels](https://www.edgepedia.org/j-w-s-cassels)'s series *Arithmetic on Curves of Genus 1*; part III, on the Tate–Šafarevič and Selmer groups, appeared in the *Proceedings of the London Mathematical Society* in 1962, occupying pages 259 to 296 of volume s3-12[12](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s3-12.1.259). Cassels, of Trinity College, Cambridge, showed in the first paper of the series (1959) how to compute the pairing when \( p = 3 \) for curves of the form \( x^3 + y^3 = k \)[10](https://arxiv.org/abs/1711.02432). The Cassels–Tate pairing need not be alternating when \( A \) is principally polarized so that \( A = A^t \); consequently, if the \( p \)-primary part of \( \Sha(A) \) is finite, its order is a square for odd \( p \) and a square or twice a square for \( p = 2 \)[5](https://wwwf.imperial.ac.uk/~anskor/abel%20var%20notes.pdf) • [24](https://eref.uni-bayreuth.de/id/eprint/7145/).
## Finiteness
A conjecture of Shafarevich and Tate asserts that \( \Sha(A) \) is finite for any abelian variety over any global field[5](https://wwwf.imperial.ac.uk/~anskor/abel%20var%20notes.pdf). The conjecture is open in general. For elliptic curves over \( \mathbb{Q} \) of rank at least 2, the finiteness of the Tate–Shafarevich group is an open problem for every individual curve[13](https://mathoverflow.net/questions/255498/is-the-tate-shafarevich-group-of-a-rational-elliptic-curve-finite).
Several classes of results are known. The theorem of Kolyvagin and Gross–Zagier using Heegner points gives finiteness whenever the L-series of \( E/\mathbb{Q} \) does not vanish at 1 or has a simple zero there[1](https://sporadic.stanford.edu/reference/arithmetic_curves/sage/schemes/elliptic_curves/sha_tate.html). For linear algebraic groups the analogous kernel \( \mathrm{Shaf}(G) \) was proved finite by A. Borel[7](https://encyclopediaofmath.org/wiki/Galois_cohomology). In the function field case, the [Birch and Swinnerton-Dyer conjecture](https://www.edgepedia.org/birch-and-swinnerton-dyer-conjecture) for an elliptic curve over a function field is known to be equivalent to the finiteness of its Tate–Shafarevich group[14](https://www.claymath.org/library/monographs/MPPc.pdf), and finiteness has been proved for a class of elliptic curves over function fields of analytic rank greater than 1[15](https://arxiv.org/abs/2207.11904).
Finiteness also matters in practice: some computational results, such as work of Swinnerton-Dyer on ranks, are conditional on the unproven finiteness of \( \Sha \) for elliptic curves over quadratic fields[16](https://www.claymath.org/wp-content/uploads/2022/03/cmip07c.pdf).
## Examples
The classical example is Selmer's cubic. The plane curve
\[ 3x^3 + 4y^3 + 5z^3 = 0 \]
in \( \mathbb{P}^2_{\mathbb{Q}} \) is a genus one curve with points over \( \mathbb{R} \) and over every \( \mathbb{Q}_p \), but no rational point[6](https://www.imo.universite-paris-saclay.fr/~harari/articles/submathz.pdf). Its class in \( \Sha \) of its Jacobian is nonzero and has order 3; explicitly, it is an element of \( \Sha(E/\mathbb{Q}) \) for \( E : x^3 + y^3 + 60z^3 = 0 \) (Selmer, 1961)[17](https://mathoverflow.net/questions/447387/order-4-element-of-tate-shafarevich-group). The curve acquires a rational point over a cyclic extension of \( \mathbb{Q} \) of degree 6, so its class has trivial restriction to \( \Sha \) over that extension[6](https://www.imo.universite-paris-saclay.fr/~harari/articles/submathz.pdf).
Milne's notes give an infinite family: for primes \( p \equiv 1 \pmod 8 \) with 2 not a fourth power modulo \( p \), the elliptic curve \( Y^2 Z = X^3 + pXZ^2 \) has rank 0 and \( \Sha(E/\mathbb{Q})[2] \cong (\mathbb{Z}/2\mathbb{Z})^2 \), so its 2-torsion is nontrivial and the Hasse principle fails for curves in the associated family[8](https://www.jmilne.org/math/Books/EC2.pdf).
## Distribution and heuristics
Because \( \Sha \) is hard to compute, its expected behavior in families is studied through probabilistic models. Delaunay proposed Cohen–Lenstra-style heuristics for the distribution of Tate–Shafarevich groups of elliptic curves over \( \mathbb{Q} \)[18](https://wiki.sagemath.org/MSRItemp/CohenLenstra). Bhargava, Kane, Lenstra, Poonen, and Rains conjectured a distribution on the exact sequence
\[ 0 \to E(k) \otimes \mathbb{Q}_p/\mathbb{Z}_p \to \mathrm{Sel}_{p^\infty} E \to \Sha[p^\infty] \to 0, \]
and showed that this single conjecture would explain many known theorems and conjectures on ranks, Selmer groups, and Tate–Shafarevich groups, while agreeing with Delaunay's predictions[19](https://arxiv.org/abs/1304.3971). In the Poonen–Rains model, the \( p \)-Selmer group is an intersection of maximal isotropic subspaces in a quadratic space over \( \mathbb{F}_p \); modeling one subspace as random explains known distribution results and predicts that the average rank of elliptic curves over a fixed number field is at most \( 1/2 \)[20](https://arxiv.org/abs/1009.0287). A special case of the Bhargava–Kane–Lenstra–Poonen–Rains heuristics is the minimalist conjecture, that 50 percent of elliptic curves have rank 0 and 50 percent have rank 1[21](https://mathematics.stanford.edu/events/student-algebraic-geometry/geometric-distribution-ranks-and-selmer-groups-over-function). A related heuristic based on skew-symmetric matrices, developed by Park, Poonen, Voight, and Wood, suggests that only finitely many elliptic curves have rank greater than 21[22](https://people.math.harvard.edu/~mmwood/Publications/).
## References
1. [Tate-Shafarevich group, Sage 9.3 Reference Manual: Elliptic curves](https://sporadic.stanford.edu/reference/arithmetic_curves/sage/schemes/elliptic_curves/sha_tate.html)
2. [Tate-Shafarevich Group, Wolfram MathWorld](https://mathworld.wolfram.com/Tate-ShafarevichGroup.html)
3. [Arithmetic Geometry I: Elliptic Curves, Androma](https://androma.org/page/Arithmetic%20Geometry%20I%3A%20Elliptic%20Curves)
4. [Selmer group, PlanetMath](https://planetmath.org/selmergroup)
5. [Abelian varieties over local and global fields, lecture notes by A. Skorobogatov, Imperial College London](https://wwwf.imperial.ac.uk/~anskor/abel%20var%20notes.pdf)
6. [D. Harari, On Tate–Shafarevich groups of one-dimensional motives, Université Paris-Saclay](https://www.imo.universite-paris-saclay.fr/~harari/articles/submathz.pdf)
7. [Galois cohomology, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Galois_cohomology)
8. [J. Milne, Elliptic Curves](https://www.jmilne.org/math/Books/EC2.pdf)
9. [T. Fisher, R. Newton, Computing the Cassels-Tate pairing on the 3-Selmer group of an elliptic curve, arXiv:1306.1410](https://arxiv.org/abs/1306.1410)
10. [T. Fisher et al., Computing the Cassels-Tate pairing on 3-isogeny Selmer groups via cubic norm equations, arXiv:1711.02432](https://arxiv.org/abs/1711.02432)
11. [B. Poonen, The Selmer group, the Shafarevich-Tate group, and the weak Mordell-Weil theorem, Arizona Winter School lectures](https://math.univ-lyon1.fr/~roblot/ihp/weakmw.pdf)
12. [J. W. S. Cassels (1962), Arithmetic on Curves of Genus 1, III. The Tate-Šafarevič and Selmer Groups, Proc. London Math. Soc. s3-12: 259-296](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s3-12.1.259)
13. [Is the Tate-Shafarevich group of a rational elliptic curve finite?, MathOverflow](https://mathoverflow.net/questions/255498/is-the-tate-shafarevich-group-of-a-rational-elliptic-curve-finite)
14. [The Birch and Swinnerton-Dyer Conjecture, Clay Mathematics Institute monograph](https://www.claymath.org/library/monographs/MPPc.pdf)
15. [P. Hamacher, Z. Yang, X. Zhao, Finiteness of the Tate-Shafarevich Group for some Elliptic Curves of Analytic Rank > 1, arXiv:2207.11904](https://arxiv.org/abs/2207.11904)
16. [Clay Mathematics Proceedings volume on elliptic curves and ranks](https://www.claymath.org/wp-content/uploads/2022/03/cmip07c.pdf)
17. [Order 4 element of Tate-Shafarevich group, MathOverflow](https://mathoverflow.net/questions/447387/order-4-element-of-tate-shafarevich-group)
18. [Cohen-Lenstra heuristics reading group, SageMath Wiki](https://wiki.sagemath.org/MSRItemp/CohenLenstra)
19. [M. Bhargava, D. Kane, H. W. Lenstra Jr., B. Poonen, E. Rains, Modeling the distribution of ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves, arXiv:1304.3971](https://arxiv.org/abs/1304.3971)
20. [B. Poonen, E. Rains, Random maximal isotropic subspaces and Selmer groups, arXiv:1009.0287](https://arxiv.org/abs/1009.0287)
21. [A. Landesman, The geometric distribution of ranks and Selmer groups over function fields, Stanford Student Algebraic Geometry Seminar](https://mathematics.stanford.edu/events/student-algebraic-geometry/geometric-distribution-ranks-and-selmer-groups-over-function)
22. [Publications of Melanie Matchett Wood](https://people.math.harvard.edu/~mmwood/Publications/)
23. [maths.tcd.ie](https://www.maths.tcd.ie/EMIS/journals/Annals/150_3/poonen.pdf)
24. [The Cassels-Tate pairing on polarized abelian varieties - ERef Bayreuth](https://eref.uni-bayreuth.de/id/eprint/7145/)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Number theory › Arithmetic geometry › Galois representations and Galois cohomology*
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