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An arithmetic intersection formula for denominators of Igusa class polynomials. (English) Zbl 1392.11033
Summary: In this paper we prove an explicit formula for the arithmetic intersection number $$({\mathrm CM}(K).{\mathrm G}_1)_{\ell}$$ on the Siegel moduli space of abelian surfaces, generalizing the work of J. H. Bruinier and T. Yang [Invent. Math. 163, No. 2, 229–288 (2006; Zbl 1093.11041)] and T. Yang [Am. J. Math. 132, No. 5, 1275–1309 (2010; Zbl 1206.14049)]. 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 [loc. cit.] conjectured a formula for intersection numbers on an arithmetic Hilbert modular surface, and as a consequence obtained a conjectural formula for the intersection number $$({\mathrm CM}(K).{\mathrm G}_1)_{\ell}$$ under strong assumptions on the ramification of the primitive quartic CM field $$K$$. Yang later proved this conjecture assuming that $$\mathcal{O}_K$$ is freely generated by one element over the ring of integers of the real quadratic subfield. In this paper, we prove a formula for $$({\mathrm CM}(K).{\mathrm G}_1)_{\ell}$$ 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 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 E. Z. Goren and the first author [Int. Math. Res. Not. 2012, No. 5, 1068–1142 (2012; Zbl 1236.14033)] and counting solutions using our previous article [Int. Math. Res. Not. 2015, No. 19, 9206–9250 (2015; Zbl 1392.11088)] that generalized work of Gross-Zagier and Dorman to arbitrary discriminants.

##### MSC:
 11G15 Complex multiplication and moduli of abelian varieties 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 14K22 Complex multiplication and abelian varieties
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