# Moduli interpretation of Hirzebruch-Zagier divisors

In their famous 1976 paper, Hirzebruch and Zagier define certain divisors $$T_N$$ on the Hilbert modular surface corresponding to the group $$\text{SL}_2(\mathcal{O}_F)$$ for $$F=\mathbb{Q}(\sqrt{p})$$. They then prove the generating $$q$$-series of the $$T_N$$ is modular. The definition of these cycles is given analytically, via the uniformization of the surface by the squared upper half-plane $$\mathcal{H}^2$$ with coordinates $$z_1,z_2$$. In particular, the cycle is given by the image of the locus of the equation $$a\sqrt{p} z_1z_2+\lambda z_2 -\lambda' z_1 +b\sqrt{p}=0$$ for $$a,b\in\mathbb{Z},\lambda \in \mathcal{O}_F$$ and $$\lambda'$$ its Galois conjugate, satisfying the equation $$\mathbf{N}(\lambda)+abp=N$$.

I'm aware there are more algebro-geometric definitions, e.g. in some papers by Yifeng Liu, but they work in a slightly different, sometimes more general setting, and I'm unable to reconcile them or see precisely how it corresponds. So my question is: what is an interpretation of the original cycles $$T_N$$ of Hirzebruch-Zagier, in terms of embeddings of modular/Shimura curves via moduli of elliptic curves/abelian varieties with extra structure? A reference which spells out transparently the relation between the more algebro-geometric definitions and the formula-based ones would be especially appreciated.

• The paper 'exceptional splittings of abelian surfaces' should be relevant arxiv.org/abs/1706.08154 – Jef Sep 23 at 20:56
• this looks super helpful, thanks jef! – xir Sep 23 at 21:27