On \mathbbF_p-roots of the Hilbert class polynomial modulo p

2022/02/09 by Mingjie Chen, Chen, Mingjie, Jiangwei Xue +1
Mathematics · #11G15 #11G20 #11R52 #14H52 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2202.04317

openalex publication_date 2022/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Hilbert class polynomial H_O(x)∈ ℤ[x] attached to an order O in an imaginary quadratic field K is the monic polynomial whose roots are precisely the distinct j-invariants of elliptic curves over ℂ with complex multiplication by O. Let p be a prime inert in K and strictly greater than |disc(O)|. We show that the number of \mathbbF_p-roots of H_O(x) \pmodp is either zero or |Pic(O)[2]| by exhibiting a free and transitive action of Pic(O)[2] on the set of \mathbbF_p-roots of H_O(x) \pmod p whenever it is nonempty. We also provide a concrete criterion for the nonemptiness of the set of \mathbbF_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.

Related

• Constructing elliptic curves with a known number of points over a prime field
• Ordinary elliptic curves of high rank over F_p(x) with constant j-invariant
• Factorization of Hilbert class polynomials over prime fields
• Generalized class polynomials
• Hilbert's tenth problem for families of ℤ_p -extensions of imaginary quadratic fields