  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.
