Supersingular $j$-invariants and the Class Number of $\mathbb{Q}(\sqrt{-p})$

TL;DR

This paper investigates the roots of class polynomials modulo p for imaginary quadratic orders, leading to a deterministic algorithm with sublinear complexity for computing class numbers of imaginary quadratic fields.

Contribution

It introduces a new analysis of supersingular j-invariants and class polynomial roots modulo p, and presents a novel deterministic algorithm for class number computation.

Findings

Analysis of solutions of class polynomial mod p

Identification of common roots of class polynomials

A deterministic algorithm with complexity O(p^{3/4+ε})

Abstract

For a prime $p>3$, let $D$ be the discriminant of an imaginary quadratic order with $|D|< \frac{4}{\sqrt{3}}\sqrt{p}$. We research the solutions of the class polynomial $H_D(X)$ mod $p$ in $\mathbb{F}_p$ if $D$ is not a quadratic residue in $\mathbb{F}_p$. We also discuss the common roots of different class polynomials in $\mathbb{F}_p$. As a result, we get a deterministic algorithm (Algorithm 3) for computing the class number of $\mathbb{Q}(\sqrt{-p})$. The time complexity of Algorithm 3 is $O(p^{3/4+\epsilon})$.