Computing class polynomials for abelian surfaces

TL;DR

This paper introduces a quasi-linear algorithm for computing Igusa class polynomials of genus 2 Jacobians using complex approximations, theta constants, and Newton iterations, with practical experiments and a large class number example.

Contribution

It presents a novel quasi-linear time algorithm for computing Igusa class polynomials, improving efficiency in complex multiplication computations for abelian surfaces.

Findings

Algorithm achieves quasi-linear complexity

Successful computation of a class with number 17608

Implementation demonstrates practical feasibility

Abstract

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 $\theta$- 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.