Proof of Ramanujan and Sato-Tate for Bianchi Modular Forms over Imaginary Quadratic Fields — E8 Intelligence Research

FINDING: Proof of the Ramanujan and Sato-Tate conjectures for Bianchi modular forms (weight ≥2) over imaginary quadratic fields, via potential automorphy and modularity lifting for CM fields. MATH: - Bianchi modular forms: cuspidal automorphic forms on GL₂(A_K), K imaginary quadratic. - Ramanujan conjecture: for a cuspidal automorphic representation π of GL₂(A_F) (F CM), the Satake parameters α_p, β_p at unramified p satisfy |α_p| = |β_p| = 1 (normalized). - Sato-Tate: equidistribution of the angle θ_p ∈ [0,π] where a_p = 2·N(p)1/2·cos θ_p, with measure (2/π) sin²θ dθ. - Key tool: potential automorphy — proving automorphy after base change to a solvable extension, using the Taylor–Wiles method and Khare–Wintenberger style lifting. - The paper (arXiv:2309.15880v3) extends to all regular algebraic cuspidal automorphic representations of GL₂(A_F) of parallel weight, F any CM field. - No new constants appear; the result is qualitative (bounds and equidistribution), not a n Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com