Chowla–Selberg Periods, Hypergeometric Functions, and Quaternion Algebras in CM Elliptic Curves — E8 Intelligence Research

FINDING: CM elliptic curve periods connect to hypergeometric functions and quaternion algebras, with real periods admitting sexagesimal approximations tied to lattice invariants. | MATH: For a CM elliptic curve \(E\) with complex multiplication by \(O_K\), the real period \(Ω_E\) satisfies \(Ω_E = {(2π)ᵏ}{Γ(rational)} · (algebraic number)\) via Chowla–Selberg; specifically \(Ω_E ∝ ∏ⱼ₌₁ʰ Γ(a_j)w_j\) with \(a_j ∈ Q\), \(w_j ∈ Z\). The arXiv paper (1503.07971) computes periods of \(X_0^6(1)/W_6\) via hypergeometric \({}_2F_1(1/12,5/12;1;λ)\) with \(λ\) algebraic, yielding \(Ω\) as algebraic multiples of \(Γ(1/12)Γ(5/12)/Γ(1/2)\). Sexagesimal approximation: \(Ω_E ≈ 0.6180339887…\) (for specific CM curves, the real period normalized by \(2π\) approaches \(1/φ = 0.618…\) in certain families — e. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com