Non-Cyclic Subgroups of Jacobians of Genus Two Curves

Let E be an elliptic curve de ned over a nite eld. Balasubramanian and Koblitz have proved that if the ` roots of unity μ` is not contained in the ground eld, then a eld extension of the ground eld contains μ` if and only if the `-torsion points of E are rational over the same eld extension. We generalize this result to Jacobians of genus two curves. In particular, we show that the Weiland the Tate-pairing are non-degenerate over the same eld extension of the ground eld. From this generalization we get a complete description of the `-torsion subgroups of Jacobians of supersingular genus two curves. In particular, we show that for ` > 3, the `-torsion points are rational over a eld extension of degree at most 24.