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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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