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

TL;DR

This paper extends a known result about elliptic curves to Jacobians of genus two curves with complex multiplication, showing non-degeneracy of pairings over certain field extensions.

Contribution

It generalizes a result on roots of unity and torsion points from elliptic curves to genus two Jacobians with complex multiplication.

Findings

Weil- and Tate-pairings are non-degenerate over the same field extension.

The generalization applies to Jacobians of genus two curves with complex multiplication.

The result links roots of unity containment with rationality of torsion points.

Abstract

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.