An arithmetic intersection formula for denominators of Igusa class polynomials

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.