Distinct differences of singular moduli

TL;DR

The authors prove that for CM elliptic curves $E_1,E_2$ over $C$, the pair $(E_1,E_2)$ is determined up to isomorphism by the difference of their $j$-invariants, i.e., if $x_1,x_2,x_3,x_4$ are singular moduli with $x_1-x_2=x_3-x_4$, then either $(x_1,x_2)=(x_3,x_4)$ or $(x_1,x_3)=(x_2,x_4)$. The argument combines CM theory, properties of ring class fields, and a detailed case analysis of dominant versus non-dominant singular moduli, supported by analytic bounds and effective control of discriminants via $2$-elementary structures and transfer fields. Key techniques include primitive-element results for differences of singular moduli, transfer-field automorphisms, and Tatuzawa-type effective class-number bounds. The result explicitly confirms André–Oort-type finiteness phenomena for the linear relation $x_1-x_2=x_3-x_4$ in the four-dimensional setting, with an explicit, constructive contradiction scheme that avoids non-feasible large discriminants. The work advances understanding of how singular moduli interact under linear relations and offers computational tools (PARI/GP scripts) for related investigations.

Abstract

Let $E_1, E_2 / \mathbb{C}$ be non-isomorphic elliptic curves with complex multiplication. We prove that the pair $(E_1, E_2)$ is characterised, up to isomorphism, by the difference $j(E_1) - j(E_2)$ of the respective $j$-invariants. In other words, we show that if $x_1, x_2, x_3, x_4$ are singular moduli such that $x_1 - x_2 = x_3 - x_4$, then either $(x_1, x_2) = (x_3, x_4)$ or $(x_1, x_3) = (x_2, x_4)$.

Theorems & Definitions (35)

• Theorem 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3: BiluLucaMasser17
• Lemma 2.4
• proof
• Lemma 3.1: AllombertBiluMadariaga15
• Lemma 3.2