Computing Igusa class polynomials

TL;DR

This paper provides the first proven bounds on the running time and correctness of an algorithm for computing genus-two class polynomials of primitive quartic CM-fields, using complex analytic methods and height bounds.

Contribution

It introduces the first rigorous time complexity bounds and correctness proof for an algorithm computing genus-two class polynomials, with improvements and detailed analysis.

Findings

Algorithm runs in time Otilde(Delta^(7/2)) where Delta is the discriminant.

Provides bounds on polynomial heights combining existing and new estimates.

Includes a complete correctness proof and rounding error analysis.

Abstract

We bound the running time of an algorithm that computes the genus-two class polynomials of a primitive quartic CM-field K. This is in fact the first running time bound and even the first proof of correctness of any algorithm that computes these polynomials. Essential to bounding the running time is our bound on the height of the polynomials, which is a combination of denominator bounds of Goren and Lauter and our own absolute value bounds. The absolute value bounds are obtained by combining Dupont's estimates of theta constants with an analysis of the shape of CM period lattices. The algorithm is basically the complex analytic method of Spallek and van Wamelen, and we show that it finishes in time Otilde(Delta^(7/2)), where Delta is the discriminant of K. We give a complete running time analysis of all parts of the algorithm, and a proof of correctness including a rounding error analysis. We also provide various improvements along the way.