Improved CRT Algorithm for Class Polynomials in Genus 2

We present a generalization to genus 2 of the probabilistic algorithm in Sutherland for computing Hilbert class polynomials. The improvement over the algorithm presented in [BGL] for the genus 2 case, is that we do not need to find a curve in the isogeny class with endomorphism ring which is the maximal order: rather we present a probabilistic algorithm for “going up” to a maximal curve (a curve with maximal endomorphism ring), once we find any curve in the right isogeny class. Then we use the structure of the Shimura class group and the computation of isogenies to compute all isogenous maximal curves from an initial one.