ℓ-adic Galois Image Classification for CM Elliptic Curves Over Class Number 1-2 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, the possible ℓ-adic images are determined for curves with CM by orders of discriminant −3, −4, −7, −8, −11, −12, −16, −19, −27, −28, −43, −67, −163 (class number 1) and −15, −20, −24, −32, −35, −36, −40, −48, −51, −52, −60, −64, −75, −88, −91, −99, −100, −112, −115, −123, −147, −148, −187, −232, −235, −267, −403, −427 (class number 2). The ℓ-adic representation is the inverse limit of mod ℓⁿ representations; its image is a subgroup of GL₂(ℤ_ℓ). For CM curves, the image is always contained in a Cartan subgroup or its normalizer, with explicit exceptions for ℓ dividing the conductor. | CONNECTION: The class number 1 discriminants include the golden ratio field ℚ(√5) (discriminant −20, class number 2). The singular moduli (j-invariants) for these CM curves are algebraic integers; for discriminant −3, j=0; for −4, j=1728; Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com