Non-Cyclic Subgroups of Jacobians of Genus Two Curves

TL;DR

This paper extends a known result about elliptic curves to Jacobians of genus two curves, providing insights into the rationality of torsion points and the structure of their subgroups over field extensions.

Contribution

It generalizes the relation between roots of unity and torsion points from elliptic curves to genus two Jacobians, including a complete description of l-torsion subgroups.

Findings

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

l-torsion points are rational over extensions of degree at most 24 for l>3.

Provides a classification of l-torsion subgroups for supersingular genus two Jacobians.

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.