Paper 2012/614

An arithmetic intersection formula for denominators of Igusa class polynomials

Kristin Lauter and Bianca Viray

Abstract

In this paper we prove an explicit formula for an arithmetic intersection number on the Siegel moduli space of abelian surfaces, generalizing the work of Bruinier-Yang and Yang. These intersection numbers allow one to compute the denominators of Igusa class polynomials, which has important applications to the construction of genus 2 curves for use in cryptography. Bruinier and Yang conjectured a formula for intersection numbers on an arithmetic Hilbert modular surface, and as a consequence obtained a conjectural formula for the intersection number relevant to denominators of Igusa class polynomials under strong assumptions on the ramification of the primitive quartic CM field K. Yang later proved this conjecture assuming that the ring of integers is freely generated by one element over the ring of integers of the real quadratic subfield. In this paper, we prove a formula for the intersection number for more general primitive quartic CM fields, and we use a different method of proof than Yang. We prove a tight bound on this intersection number which holds for all primitive quartic CM fields. As a consequence, we obtain a formula for a multiple of the denominators of the Igusa class polynomials for an arbitrary primitive quartic CM field. Our proof entails studying the Embedding Problem posed by Goren and Lauter and counting solutions using our previous article that generalized work of Gross-Zagier and Dorman to arbitrary discriminants.

Metadata
Available format(s)
PDF
Category
Public-key cryptography
Publication info
Published elsewhere. Unknown where it was published
Keywords
hyperelliptc curvescomplex multiplication
Contact author(s)
klauter @ microsoft com
History
2012-11-01: received
Short URL
https://ia.cr/2012/614
License
Creative Commons Attribution
CC BY

BibTeX

@misc{cryptoeprint:2012/614,
      author = {Kristin Lauter and Bianca Viray},
      title = {An arithmetic intersection formula for denominators of Igusa class polynomials},
      howpublished = {Cryptology ePrint Archive, Paper 2012/614},
      year = {2012},
      note = {\url{https://eprint.iacr.org/2012/614}},
      url = {https://eprint.iacr.org/2012/614}
}
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