Igusa class polynomials, embeddings of quartic CM fields, and arithmetic intersection theory

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Bruinier and Yang conjectured a formula for an intersection number on the arithmetic Hilbert modular surface, CM(K).Tₘ, where CM(K) is the zero-cycle of points corresponding to abelian surfaces with CM by a primitive quartic CM field K, and Tₘ is the Hirzebruch-Zagier divisors parameterizing products of elliptic curves with an m-isogeny between them. In this paper, we examine fields not covered by Yang's proof of the conjecture. We give numerical evidence to support the conjecture and point to some interesting anomalies. We compare the conjecture to both the denominators of Igusa class polynomials and the number of solutions to the embedding problem stated by Goren and Lauter.