Singular Moduli for Real Quadratic Fields: Heights and Galois Representations — E8 Intelligence Research

FINDING: Singular moduli — values of the elliptic modular function \(j(τ)\) at imaginary quadratic \(τ\) — generate ring class fields; recent work extends this to real quadratic fields via a new theory of "singular moduli for real quadratic fields" (Vonk et al.), with explicit arithmetic applications to heights of Heegner points and Galois representations. | MATH: \(j(τ)\) is a modular function with Fourier expansion \(j(τ) = q⁻¹ + 744 + 196884q + 21493760q^2 + \) where \(q = e2π i τ\). For \(τ\) imaginary quadratic (i.e., \(τ ∈ Q(√-d)\)), \(j(τ)\) is an algebraic integer generating the ring class field of \(Q(√-d)\). Key theorem (Gross–Zagier): differences of singular moduli \(j(τ_1) - j(τ_2)\) factor into primes with exponents related to intersection numbers on modular curves. Vonk's extension: for real quadratic \(τ\), one defines a "mock" or "regularized" singular modulus using real quadratic analogues, l Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com