Jacobians with Complex Multiplication

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

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 with complex multiplication. In particular, w...

Fast, Uniform Scalar Multiplication for Genus 2 Jacobians with Fast Kummers

We give oneand two-dimensional scalar multiplication algorithms for Jacobians of genus 2 curves that operate by projecting to Kummer surfaces, where we can exploit faster and more uniform pseudomultiplication, before recovering the proper “signed” output back on the Jacobian. This extends the work of López and Dahab, Okeya and Sakurai, and Brier and Joye to genus 2, and also to two-dimensional ...

Infinite Families of Pairs of Curves over Q with Isomorphic Jacobians

We present three families of pairs of geometrically non-isomorphic curves whose Jacobians are isomorphic to one another as unpolarized abelian varieties. Each family is parametrized by an open subset of P. The first family consists of pairs of genus-2 curves whose equations are given by simple expressions in the parameter; the curves in this family have reducible Jacobians. The second family al...

Generic Abelian Varieties with Real Multiplication Are Not Jacobians

Large Cyclic Subgroups of Jacobians of Hyperelliptic Curves

In this paper we obtain conditions on the divisors of the group order of the Jacobian of a hyperelliptic genus 2 curve, generated by the complex multiplication method described by Weng (2003) and Gaudry et al (2005). Examples, where these conditions imply that the Jacobian has a large cyclic subgroup, are given.