\lambda' & b\end{smallmatrix}\right) };\; a,b\in {\mathbb{Q}},\; \lambda\in F\},$$ equipped with the quadratic form $M\mapsto \det(M)$. Here ${}^tM$ is the transpose of $M$. The group ${\operatorname{SL}}_2(F)$ acts on $V$ via $$\gamma . M = \gamma' M \, {}^t\gamma$$ leaving the quadratic form invariant. The lattice $$\begin{aligned}
\label{def:L}
L=\{{\left(\begin{smallmatrix} a & \lambda \\ \lambda' & b\end{smallmatrix}\right) }\in V;\; a,b\in {\mathbb{Z}},\; \lambda\in
\partial^{-1}_F\}\end{aligned}$$ is stable under the action of $\Gamma$. Recall that $M\in L$ is called [*primitive*]{}, if $\frac{1}{d} M\notin L$ for every integer $d>1$.

If $M={\left(\begin{smallmatrix} a & \lambda \\ \lambda' & b\end{smallmatrix}\right) }$ is a vector in $L$, we let $$M^\perp=\{(z_1,z_2)\in {\mathbb{H}}^2; \quad az_1 z_2 +\lambda z_1 + \lambda' z_2 +b=0\}$$ be the corresponding analytic divisor on ${\mathbb{H}}^2$. If $m$ is a positive integer, then $$\begin{aligned}
F_m&=\sum_{\substack{\text{$M\in L/\{\pm 1\}$ primitive}\\
\det(M)=m/p}} M^\perp\end{aligned}$$ is a $\Gamma$-invariant divisor on ${\mathbb{H}}^2$, which descends to an algebraic divisor on $X$, also denoted by $F_m$. It is well known that the Hirzebruch-Zagier divisors $T_m$ on $X$ can be written as $T_m=\sum_{d^2|m} F_{m/d^2}$ (see [@Ge], [@HZ]).

The divisor $F_m$ on $X$ is irreducible if and only if $p^2\nmid m$. If $p^2|m$ then it decomposes into two irreducible components $F_m=F_m^++F_m^-$. To distinguish these components we define a function 