FINDING SINGULAR MODULI ON A COMPLEX LINE

Abstract

Let (α, β, γ) be three non-zero coprime integers. Using Baker's method and classical technique of reduction, we describe an algorithm which computes every point (x1, x2) on the ℂ-line αX + βY + γ = 0 such that x1 and x2 are both singular moduli. We illustrate our algorithm by writing down some examples, in particular we prove that there are non-complex multiplication points on a line such that |α|, |β| and |γ| are less than 10.