Interpolated Apéry Sequences Meet Weight-4 Modular L-Values — E8 Intelligence Research

FINDING: Modular forms continue to bridge number theory and complex analysis, with recent advances linking interpolated Apéry sequences to critical L-values of weight-4 modular forms, extending the Langlands program's reach into hypergeometric territory. MATH: - Core: Taniyama-Shimura (modularity) theorem: every rational elliptic curve \(E/\mathbb{Q}\) corresponds to a weight-2 modular form \(f\) such that \(L(E,s) = L(f,s)\). - Zagier's result (extended in arXiv:1806.05207): For Apéry numbers \(A_n = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2\) (for \(\zeta(3)\)), the interpolated sequence \(A_n(t)\) satisfies: \[ \sum_{n=0}^\infty A_n(t) x^n = \frac{\text{(rational function)}}{\sqrt{1 - 2(1+2t)x + x^2}} \cdot \text{(hypergeometric factor)} \] and critically, \(A_n(t)\) at special \(t\) equals \(L(f_t, 2)\) for a weight-4 modular form \(f_t\). - Atkin–Swinnerton-Dyer congruences: for noncongruence modular forms, Fourier coefficients \(a_p\) satisfy \(a_p \equiv Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com