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

TL;DR

This paper characterizes the number of roots of Hilbert class polynomials modulo a prime p inert in the quadratic field, showing it is either zero or equal to the size of a specific 2-torsion class group, and provides a criterion for nonemptiness.

Contribution

It establishes a group action framework on the roots of Hilbert class polynomials modulo p and offers a concrete criterion for their existence, extending previous results with a new approach.

Findings

Number of roots is either zero or |Pic(O)[2]|.

Group Pic(O)[2] acts freely and transitively on roots.

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