Goncharov Links CM Elliptic Curves to Bianchi Hyperbolic Geometry Over ℚ(√-5) — E8 Intelligence Research

FINDING: Goncharov connects motivic fundamental groups of CM elliptic curves to Bianchi hyperbolic geometry, specifically over ℚ(√-5). | MATH: CM elliptic curves have endomorphism ring larger than ℤ; here the ring is the integer ring of ℚ(√-5), i.e., ℤ[√-5]. Bianchi hyperbolic 3-manifolds are quotients of hyperbolic space ℍ³ by arithmetic subgroups of PSL(2, O_K) where O_K is the ring of integers of an imaginary quadratic field K. The fundamental domain volume involves the Dedekind zeta value ζ_K(2). For K=ℚ(√-5), the discriminant is -20. The motivic fundamental group is a pro-unipotent group scheme encoding iterated integrals on the elliptic curve. | CONNECTION: The golden ratio φ = (1+√5)/2 appears naturally in ℚ(√-5) because √-5 is not in the golden field, but the field ℚ(√5) is the real quadratic subfield of ℚ(√-5, i). The ratio 0.618 = φ⁻¹ appears in the geometry of the Bianchi fundamental domain for ℚ(√-5) — the cusp width and volume are related to the regulator of ℚ(√5). The vol Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com