Computing Class Polynomials for Abelian Surfaces

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.