Non-Cyclic Subgroups of Jacobians of Genus Two Curves with Complex Multiplication

Details Non-Cyclic Subgroups of Jacobians of Genus Two Curves with Complex Multiplication Non-Cyclic Subgroups of Jacobians of Genus Two Curves with Complex Multiplication

2008-01-18

arxiv.org/abs/0801.2828v1

Mathematics

Algebraic Geometry

The paper was presented at AGCT 11, november 2007

Scientific paper

Let E be an elliptic curve defined over a finite field. Balasubramanian and Koblitz have proved that if the l-th roots of unity m_l is not contained in the ground field, then a field extension of the ground field contains m_l if and only if the l-torsion points of E are rational over the same field extension. We generalize this result to Jacobians of genus two curves with complex multiplication. In particular, we show that the Weil- and the Tate-pairing on such a Jacobian are non-degenerate over the same field extension of the ground field.

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