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| Sha (Tate–Shafarevich group) | |
|---|---|
| Name | Sha (Tate–Shafarevich group) |
| Occupation | Mathematical object |
Sha (Tate–Shafarevich group) is an arithmetic invariant associated to an abelian variety over a global field that measures the failure of the Hasse principle for principal homogeneous spaces. It plays a central role in the arithmetic of elliptic curves, abelian varieties, and the formulation of major conjectures such as the Birch and Swinnerton-Dyer conjecture and the Bloch–Kato conjecture.
The group is defined for an abelian variety A over a global field K such as Galois-related number fields or function fields over Weil's foundations and is denoted by an often-cited Tate symbol. For an abelian variety A and its dual A^∨, one considers principal homogeneous spaces (torsors) under A that are locally trivial at every completion K_v for places v of K, using completions like those in Tate's local duality framework. Those torsors form a subgroup of the Galois cohomology group H^1(K,A), and Sha is the kernel of the product map to the direct sum of local cohomology groups H^1(K_v,A) over places tied to Chebotarev density phenomena and Artin-type reciprocity laws. Basic properties include that Sha is a torsion abelian group conjecturally finite, related by duality to the corresponding group for A^∨ via a Cassels–Tate pairing originating from work of Cassels and Deligne's cohomological methods, and that its behavior interacts with arithmetic of Wiles-type modularity results and Faltings finiteness theorems.
Cohomologically, Sha sits inside the exact sequence in Galois cohomology derived from the long exact sequence of nonabelian and abelian cohomology applied to the structure of A, with references to Cartan-style spectral sequences and Serre's Galois cohomology treatments. It is the subgroup of H^1(K,A) annihilated by localisation maps to H^1(K_v,A) for all v, connecting to local duality theorems of Tate, Poitou–Tate duality named after Poitou and Tate, and to the étale cohomology frameworks developed by Grothendieck and Deligne. These interpretations link Sha to Selmer groups arising in descent theory as in work of Atkin, Stark, and Shanks, with Selmer groups fitting into exact sequences that compare global and local points via the Kummer sequence influenced by Kummer's ideas.
A central open problem asserts that Sha is finite for an elliptic curve or more generally for an abelian variety over a number field, a prediction embedded in the Birch and Swinnerton-Dyer conjecture that relates the order of Sha to the leading coefficient of the L-function L(A,s) at s=1. The conjectural formula combines contributions from the regulator (linking to Néron and Tate heights), the Tamagawa numbers (drawing on Néron and Chevalley), the period lattice (following Mordell and Mordell-type results), and the cardinaility of Sha. Progress toward this conjecture uses modularity theorems of Wiles, Taylor, and Conrad et al., Gross–Zagier formulae from Gross and Zagier, and Iwasawa theory developed by Ribet, Iwasawa, and Mazur, which connect special values of L-functions to Selmer groups and hence to Sha.
For elliptic curves over Q, instances where Sha has been computed or bounded use descent techniques pioneered by Cremona, visibility methods related to Mordell and Birch investigations, and explicit computations by Bhargava and collaborators in average-rank studies. Notable examples include curves with trivial Sha, curves where Sha contains nontrivial 2-torsion discovered via 2-descent inspired by Schoof and Silverman methods, and analytic verifications of finiteness in settings handled by Rubin and Wiles under hypotheses like the vanishing of Shafarevich–Tate for CM curves via Heegner constructions of Birch and Gross–Zagier results. Over function fields, finiteness and explicit structure are better understood through work of Grothendieck and Artin using geometric tools like the Néron model and results of Ulmer.
Techniques include descent and Selmer group computations influenced by Cayley-style arithmetic, Cassels–Tate pairings building on Cassels and Tate duality, visibility and congruence methods linked to Mazur and Ribet, Euler system methods introduced by Kolyvagin and refined by Rubin, and p-adic L-function and Iwasawa-theoretic approaches stemming from Iwasawa and Coleman. Étale and flat cohomology tools from Grothendieck and Serre underpin structural results, while computational algebra systems and databases propagated by Cremona and L-Functions and Modular Forms Database facilitate explicit examples.
Variants of Sha appear in broader contexts: for tori and linear algebraic groups reflecting reciprocity laws like Artin and Tate–Nakayama duality contexts, in Bloch–Kato Selmer groups within the framework of the Bloch–Kato conjecture linking motivic cohomology to special L-values, and in noncommutative Iwasawa theory associated with Coates and Neukirch-style generalisations. Geometric analogues occur in the study of principal bundles over curves in the spirit of Grothendieck and Mumford, and in arithmetic duality for complexes as developed by Deligne and Bloch.