Shafarevich–Tate group

Shafarevich–Tate group
Name	Shafarevich–Tate group
Field	Algebraic number theory
Known for	Arithmetic of abelian varieties

Shafarevich–Tate group The Shafarevich–Tate group is an arithmetic invariant attached to an abelian variety over a number field, measuring failures of the local–global principle for principal homogeneous spaces; it plays a central role in the arithmetic of elliptic curves, modular forms, Galois representations and in conjectures of Birch–Swinnerton-Dyer. Introduced in the work of Igor Shafarevich and John Tate, the group governs obstructions visible in the study of Selmer groups, Weil–Châtelet groups, and the arithmetic of Jacobian varieties over global fields such as Q and its extensions.

Definition and basic properties

For an abelian variety A over a number field K, the Shafarevich–Tate group is defined as the kernel of the localization map from the Weil–Châtelet group H^1(K,A) to the product of local cohomology groups H^1(K_v,A) for all places v of K, a construction that uses ingredients from the theories of Galois cohomology, Tate cohomology, local fields and global duality theorems such as those of Tate duality. It is commonly denoted by a symbol honoring Shafarevich and Tate and is equipped with an action of the absolute Galois group of K, interacts with the Tamagawa numbers of A, and fits into exact sequences with the Selmer group and the Mordell–Weil theorem data for A over K. Basic properties include torsion behavior under isogenys between abelian varieties, functoriality under base change to finite extension (algebra)s of K, and relation to Cassels–Tate pairings which give it a bilinear structure akin to pairings studied by Heegner and Gross–Zagier.

Relationship to the Birch and Swinnerton-Dyer conjecture

In the context of an elliptic curve E over K, the Birch–Swinnerton-Dyer conjecture predicts an explicit formula relating the leading term of the L-function L(E,s) at s=1 to arithmetic invariants including the regulator from the Mordell–Weil group, the product of Tamagawa numbers, the order of the torsion subgroup, and the order of the Shafarevich–Tate group. This conjectural identity links analytic objects studied by Atkin and Serre with arithmetic objects appearing in the work of Faltings, Mazur, and Kolyvagin, and it implies finiteness of the Shafarevich–Tate group if the L-series has the expected zero order. Partial results connecting nonvanishing of Heegner point constructions, the Gross–Zagier theorem, and Euler system techniques of Kolyvagin have established the finiteness of Shafarevich–Tate groups in many cases treated by Skinner–Urban and Wiles-style modularity lifting.

Computation and examples

Concrete computations of Shafarevich–Tate groups appear in tables of elliptic curves over Q compiled by projects associated to Cremona and in explicit studies by Cassels, Tate, and later computational arithmetic geometers; these computations exploit descent via Galois cohomology, 2-descent, 3-descent, and higher descent methods linked to Selmer group calculations. Examples over quadratic fields, cyclotomic extensions related to Iwasawa theory, and CM cases informed by Complex multiplication show varied behavior, including nontrivial divisible parts predicted by Poitou–Tate duality and explicit nontrivial elements arising from principal homogeneous spaces constructed by techniques of Rubin and Silverman. Computational evidence supports conjectural distributions connected to statistical heuristics proposed by Delaunay and patterns visible in databases maintained by researchers affiliated with L-functions and Modular Forms Database projects.

Cohomological interpretation

Cohomologically, the Shafarevich–Tate group is expressed via first and second Galois cohomology groups H^1 and H^2 with coefficients in A and its dual A^∨, and it features in duality theorems like Poitou–Tate duality and the Tate local duality framework developed in the studies of Artin and Tate. The Cassels–Tate pairing provides an alternating pairing on the Shafarevich–Tate group of an abelian variety and its dual, rooted in cup product constructions and the theory of Brauer groups; this places the group at the intersection of arithmetic duality, etale cohomology and the arithmetic of Néron models. These cohomological descriptions have links to deformation-theoretic perspectives used by researchers like Mazur and to categorical viewpoints encountered in modern work of Grothendieck.

Finiteness conjectures and results

The finiteness of the Shafarevich–Tate group for abelian varieties over number fields is a longstanding conjecture implied by the full Birch–Swinnerton-Dyer conjecture and connected to modularity results of Wiles, Taylor, and Breuil. Significant partial results include Kolyvagin's proof of finiteness for many elliptic curves of analytic rank 0 or 1 under hypotheses used by Gross–Zagier and later extensions by Kato using Euler systems, while analytic nonvanishing results of Iwaniec and Sarnak inform average-case finiteness heuristics. Open problems remain for higher rank situations, for abelian varieties without modular parametrizations as in the work of Faltings and for the behavior in infinite towers studied in Iwasawa theory by Greenberg.

Analogues and generalizations

Analogues of the Shafarevich–Tate group appear for motivic cohomology classes, for higher-dimensional Calabi–Yau motives, and in the study of principal homogeneous spaces under reductive groups treated by Harder and Langlands program perspectives; there are parallels in the function field setting over F_q where techniques of Drinfeld and Weil conjectures provide more accessible finiteness results. Generalizations include Selmer groups for Galois representations, fine Selmer groups studied in Coates and Suwa-type contexts, and obstructions in the Brauer–Manin obstruction framework used in rational point problems investigated by Skorobogatov and Manin.

Category:Algebraic number theory