ℓ-adic Galois Images for CM Elliptic Curves Over Class Number 1 or 2 Orders — E8 Intelligence Research

FINDING: Complete classification of ℓ-adic Galois representations for CM elliptic curves over imaginary quadratic orders of class number 1 or 2. | MATH: For each prime ℓ, and each imaginary quadratic order O of class number h=1 or 2, the paper determines all possible ℓ-adic Galois images ρ_{E,ℓ^∞}(G_K) for elliptic curves E with CM by O over their minimal field of definition K. Key constants: class numbers 1, 2; ℓ-adic fields; singular moduli j-invariants (algebraic integers). | CONNECTION: Imaginary quadratic orders of class number 1 correspond to the 9 discriminants (-3, -4, -7, -8, -11, -19, -43, -67, -163). The j-invariants for these are singular moduli, often related to the golden ratio φ = (1+√5)/2 via modular equations (e.g., j( (1+√-163)/2 ) ≈ -640320³, a near-integer). The class number 2 orders (18 discriminants) yield j-values that are algebraic numbers of degree 2, often expressible in terms of φ (e.g., j( (1+√-15)/2 ) = -85995/2 + (76545/2)√5, involving φ). The ℓ-adic image Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com