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

Abstract

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 $|\mathrm{disc}(\mathcal{O})|$. We show that the number of ${\mathbb{F}}_p$-roots of $H_{\mathcal{O}}(x)(\mathrm{mod}p)$ is either zero or $|\mathrm{Pic}(\mathcal{O})[2]|$ by exhibiting a free and transitive action of $\mathrm{Pic}(\mathcal{O})[2]$ on the set of ${\mathbb{F}}_p$-roots of $H_{\mathcal{O}}(x)(\mathrm{mod}p)$ 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.

9 pages