Abstract
We describe a quasilinear 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 quasilinear time using Newton iterations on the Borchardt mean. We report on experiments with our implementation and present an example with class number 20 016.
Notes
2Such a quadratic, yet efficient, implementation was used by T. Houtmann to compute class polynomials of degree up to 500 (personal communication; no reference exists).