Computing Igusa class polynomials

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.

31 pages (Various improvements to the exposition suggested by the referee. For the most detailed exposition, see Chapter II of the author's thesis http://hdl.handle.net/1887/15572)

References in corpus (5)

• Computing Arakelov class groups
• Primes in the denominators of Igusa Class Polynomials
• The $p$-adic CM-method for genus 2
• Computing Hilbert Class Polynomials
• Arithmetic Intersection on a Hilbert Modular Surface and the Faltings Height

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