On ${\mathbb{F}}_p$-roots of the Hilbert class polynomial modulo $p$

The Hilbert class polynomial $H_{\mathcal{O}}(x)\in \mathbb{Z}[x]$ attached to an order $\mathcal{O}$ in an imaginary quadratic field $K$ is the monic polynomial whose roots are precisely the distinct $j$-invariants of elliptic curves over $\mathbb{C}$ with complex multiplication by $\mathcal{O}$. Let $p$ be a prime inert in $K$ and strictly greater than $\mathrm{\mid }\mathrm{disc}(\mathcal{O})\mathrm{\mid }$. We show that the number of ${\mathbb{F}}_p$-roots of $H_{\mathcal{O}}(x)\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}(\mathrm{m}\mathrm{o}\mathrm{d}p)$ is either zero or $\mathrm{\mid }\mathrm{Pic}(\mathcal{O})[2]\mathrm{\mid }$ by exhibiting a free and transitive action of $\mathrm{Pic}(\mathcal{O})[2]$ on the set of ${\mathbb{F}}_p$-roots of $H_{\mathcal{O}}(x)\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}(\mathrm{m}\mathrm{o}\mathrm{d}p)$ whenever it is nonempty. We also provide a concrete criterion for the nonemptiness of the set of ${\mathbb{F}}_p$-roots. A similar result was first obtained by Xiao et al.~[Int. J. Number Theory, DOI: 10.1142/S1793042122500555] and generalized much further by Li et al.~[arXiv:2108.00168] (that covers the current result) with a different approach.