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

FINDING: Classification of ℓ-adic Galois images for CM elliptic curves over imaginary quadratic fields of class number 2 is being actively developed, with explicit models and modularity results. | MATH: CM elliptic curves have endomorphism ring larger than ℤ; over imaginary quadratic fields, the ℓ-adic Galois representation image is contained in a normalizer of a Cartan subgroup. For class number 2, the possible images are constrained by the CM field's Hilbert class field and the ℓ-adic Lie algebra structure. Key constants: ℓ (prime), class number h=2, discriminant D of imaginary quadratic field. | CONNECTION: The Cartan subgroups and their normalizers relate to root systems of type A₁ (for CM by a single quadratic order) and to the binary quadratic forms of discriminant D. The class number 2 implies a specific lattice structure (ideal class group of order 2), which mirrors the symmetry of the golden ratio's continued fraction [1;1,1,…] in the sense of quadratic irrationals of discrimi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com