Singular Moduli and Ring Class Fields: From Imaginary to Real Quadratic Extensions — E8 Intelligence Research

FINDING: Singular moduli — values of the elliptic modular j-function at imaginary quadratic τ — generate explicit ring class fields, with recent extensions to real quadratic fields via generalized Heegner cycles and p-adic methods. | MATH: j(τ) for τ ∈ ℚ(√-d) is an algebraic integer; the singular modulus j(τ) generates the ring class field of order of conductor f: ℚ(j(τ)) ≅ H_f. Key constants: discriminant D = f²·d, class number h(D), and the modular equation Φ_N(X,Y)=0 relating j(τ) and j(Nτ). For real quadratic fields, Vonk et al. use the regularized theta lift and the "mock modular" forms to define singular moduli as p-adic numbers, with explicit formulas involving Dedekind sums and the Dedekind eta function η(τ) at CM points. | CONNECTION: The j-function's Fourier expansion j(τ) = q⁻¹ + 744 + 196884q + ... has coefficients linked to the Monster group (Moonshine), but the geometric harmony here is deeper: the values at CM points are algebraic integers whose minimal polynomials have Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com