Extending Singular Moduli to Real Quadratic Fields via Rigid Cocycles — E8 Intelligence Research

FINDING: Vonk's work extends singular moduli (special values of the modular j-invariant at imaginary quadratic points) to real quadratic fields via rigid cocycles, replacing classical Heegner points with real quadratic analogues and connecting to mock modular forms and ring class fields. | MATH: Classical singular moduli: \(j(τ)\) for \(τ ∈ H\) with \([Q(τ):Q]=2\), \(τ\) imaginary quadratic → algebraic integers generating ring class fields. Vonk's real quadratic case: uses **rigid cocycles** (cohomology classes in \(H^1(SL_2(Z[1/p]), C_p)\)) evaluated at real quadratic \(τ\), producing \(p\)-adic numbers whose logarithms/values relate to **Darmon cycles** and **mock modular forms** (weight 2, shadow weight 3/2). Gross–Zagier formula analog: differences of singular moduli \(j(τ_1)-j(τ_2)\) encode heights of Heegner points; Vonk's real quadratic version replaces heights with **rigid analytic theta cocycl Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com