Section: New Results

Computation of isogenies between abelian varieties

Following the work [11] of David Lubicz and Damien Robert (that has just been accepted for publication in Compositio Mathematica) about the explicit computation of isogenies using theta coordinates, Romain Cosset and Damien Robert [24] have developped further nice features. In the original paper, only $({\ell }^2,{\ell }^2)$-isogenies between abelian surfaces were available. It is now possible to handle $(\ell ,\ell )$-isogenies between genus 2 curves, thus providing a more precise tool. Two key elements were necessary: Romain Cosset gave explicit methods to transfer points between the classical representation with Mumford's coordinates and the theta functions. This is a generalisation of the work of Van Wamelen. And Romain Cosset and Damien Robert developped an explicit algorithm to change the level in the Theta coordinates that are used to represent the geometrical objects. Many details can be found in Cosset's thesis [2] .

Using the same kind of tools, Christophe Arène and Romain Cosset [23] have constructed the first complete addition law on abelian surfaces. Although they are not yet of any practical use, completeness is a feature that is in principal interesting for cryptographic applications.

The article by Faugère, Lubicz and Robert on computing modular correspondences with Theta constants has finally appeared in Journal of Algebra [9] .