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Computing Igusa class polynomials


Author: Marco Streng
Journal: Math. Comp. 83 (2014), 275-309
MSC (2010): Primary 11G15; Secondary 14K22, 11Y40
DOI: https://doi.org/10.1090/S0025-5718-2013-02712-3
Published electronically: May 20, 2013
MathSciNet review: 3120590
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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 (Section 8).

The algorithm is basically the complex analytic method of Spallek and van Wamelen, and we show that it finishes in time $ \widetilde {O}(\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.


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Additional Information

Marco Streng
Affiliation: Department of Mathematics, VU University Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands
Email: marco.streng@gmail.com

DOI: https://doi.org/10.1090/S0025-5718-2013-02712-3
Received by editor(s): February 9, 2011
Received by editor(s) in revised form: January 23, 2012
Published electronically: May 20, 2013
Additional Notes: The results in this paper were part of the author’s Ph.D. thesis at Universiteit Leiden
The author was partially supported by EPSRC grant number EP/G004870/1
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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