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An arithmetic intersection formula for denominators of Igusa class polynomials
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 137, Number 2, April 2015
- pp. 497-533
- 10.1353/ajm.2015.0010
- Article
- Additional Information
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In this paper we prove an explicit formula for the arithmetic intersection
number $({\rm CM}(K).{\rm G}_1)_{\ell}$ 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 $({\rm CM}(K).{\rm G}_1)_{\ell}$ under strong assumptions on the
ramification of the primitive quartic CM field $K$. Yang later proved this
conjecture assuming that $\cal{O}_K$ is freely generated by one element
over the ring of integers of the real quadratic subfield. In this paper,
we prove a formula for $({\rm CM}(K).{\rm G}_1)_{\ell}$ 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
{\it 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.