**Non-Cyclic Subgroups of Jacobians of Genus Two Curves**

*Christian Robenhagen Ravnshoj*

**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. In particular, we show that the Weil- and the Tate-pairing are non-degenerate over the same field extension of the ground field.

From this generalization we get a complete description of the l-torsion subgroups of Jacobians of supersingular genus two curves. In particular, we show that for l>3, the l-torsion points are rational over a field extension of degree at most 24.

**Category / Keywords: **Jacobians, hyperelliptic genus two curves, pairings, embedding degree, supersingular curves

**Date: **received 22 Jan 2008

**Contact author: **cr at imf au dk

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**Version: **20080128:150638 (All versions of this report)

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