Computing class polynomials for abelian surfaces
This work addresses a computational bottleneck in number theory for researchers studying abelian surfaces, presenting an incremental improvement in efficiency.
The paper tackles the problem of computing Igusa class polynomials for genus 2 curves by developing a quasi-linear algorithm using complex floating-point approximations and Newton iterations on the Borchardt mean, achieving an example with class number 17608.
We describe a quasi-linear algorithm for computing Igusa class polynomials of Jacobians of genus 2 curves via complex floating-point approximations of their roots. After providing an explicit treatment of the computations in quartic CM fields and their Galois closures, we pursue an approach due to Dupont for evaluating $θ$- constants in quasi-linear time using Newton iterations on the Borchardt mean. We report on experiments with our implementation and present an example with class number 17608.