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 \( S_k(\Gamma) \); eigenvalues \( \lambda_f(n) \) satisfy multiplicative relations: \( \lambda_f(m)\lambda_f(n) = \sum_{d|\gcd(m,n)} \lambda_f(mn/d^2) \). - For normalized eigenforms, \( \lambda_f(p) \in [-2,2] \) (Sato–Tate distribution). - Random matrix theory predicts spacing distribution of eigenvalues follows Wigner surmise: \( P(s) = \frac{\pi s}{2} e^{-\pi s^2/4} \). - Connection to \( L \)-functions: \( L(s,f) = \sum_{n\ge1} \lambda_f(n)n^{-s} \), with functional equation linking \( s \leftrightarrow 1-s \). 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 \( \frac{2}{\pi}\sin Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com