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Computing the Cassels-Tate pairing on the 3-Selmer group of an elliptic curve. (English) Zbl 1314.11042
The authors give a method of computing the Cassels-Tate pairing on $$p$$-Selmer groups of an elliptic curve $$E$$ for an odd prime number $$p$$ and especially compute the pairing explicitly for $$p=3$$. In [J. W. S. Cassels, J. Reine Angew. Math. 494, 101–127 (1998; Zbl 0883.11028)], the paring has been computed for the $$2$$-Selmer group. The argument in Cassels’ article cannot be directly applied to the odd prime cases. The Cassels-Tate pairing is expressed as a sum of local pairings and is computed by using a certain global function associated with principal homogeneous spaces under $$E$$. The authors express the local pairing in terms of Hilbert norm residue symbols. In the case $$p=3$$, the problem of computing the global function is reduced to a more general problem concerning $$3\times 3\times 3$$-cubes, and further the solution of this general problem is obtained by considering that of trivializing a $$3\times 3$$ matrix algebra over the field generated by the $$3$$-torsion points of $$E$$. As an example, they compute the pairing on the $$3$$-Selmer group of the elliptic curve $E: y^2+xy+y=x^3-x^2-19163564x-34134737802,$ which has no rational $$3$$-isogenies and the Tate-Shafarevich group of $$E$$ has non-trivial $$3$$-torsion subgroup.

##### MSC:
 11G05 Elliptic curves over global fields 11G07 Elliptic curves over local fields 16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
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