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 \(\mathcal{O}_K\), the real period \(\Omega_E\) satisfies \(\Omega_E = \frac{(2\pi)^{k}}{\Gamma(\text{rational})} \cdot (\text{algebraic number})\) via Chowla–Selberg; specifically \(\Omega_E \propto \prod_{j=1}^{h} \Gamma(a_j)^{w_j}\) with \(a_j \in \mathbb{Q}\), \(w_j \in \mathbb{Z}\). The arXiv paper (1503.07971) computes periods of \(X_0^6(1)/W_6\) via hypergeometric \({}_2F_1\left(\frac{1}{12},\frac{5}{12};1;\lambda\right)\) with \(\lambda\) algebraic, yielding \(\Omega\) as algebraic multiples of \(\frac{\Gamma(1/12)\Gamma(5/12)}{\Gamma(1/2)}\). Sexagesimal approximation: \(\Omega_E \approx 0.6180339887\ldots\) (for specific CM curves, the real period normalized by \(2\pi\) approaches \(1/\varphi = 0.618\ldots\) in certain families — e. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com