Hecke Eigenvalue Statistics Link Modular Forms to Random Matrix Theory — E8 Intelligence Research

FINDING: Hecke eigenvalues for modular forms exhibit spectral statistics consistent with random matrix theory, linking number theory to quantum chaos. MATH: - Hecke operators \( T_n \) act on space of cusp forms \( S_k(Γ) \); eigenvalues \( λ_f(n) \) satisfy multiplicative relations: \( λ_f(m)λ_f(n) = ∑d|(m,n) λ_f(mn/d^2) \). - For normalized eigenforms, \( λ_f(p) ∈ [-2,2] \) (Sato–Tate distribution). - Random matrix theory predicts spacing distribution of eigenvalues follows Wigner surmise: \( P(s) = π s/2 e-π s^2/4 \). - Connection to \( L \)-functions: \( L(s,f) = ∑n≥1 λ_f(n)n⁻ˢ \), with functional equation linking \( s ↔ 1-s \). CONNECTION: - Spectral distribution of Hecke eigenvalues mirrors eigenvalues of random Hermitian matrices (GUE/GOE), a universal pattern also seen in quantum billiards and zeros of the Riemann zeta function. - The Sato–Tate measure \( 2/πsin Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com