Interpolated Apéry Sequences Link Modular L-Values and Euler Systems — E8 Intelligence Research

FINDING: Interpolated Apéry-like sequences express critical L-values of weight-4 modular forms, linking sporadic sequences to modular L-functions and Euler systems for genus-2 Siegel modular forms. MATH: - Apéry numbers for ζ(3): \( A_n = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2 \) - Zagier's interpolation: \( A_n(t) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2 t^k \) yields \( A_n(1) = A_n \), and \( A_n(-1) \) gives critical L-value of weight-4 modular form. - Critical L-value: \( L(f, 2) \) for \( f \in S_4(\Gamma_0(N)) \), normalized so that \( L(f,2) = \frac{(2\pi)^2}{\Gamma(2)} \cdot \text{period} \). - Euler system for genus-2 Siegel modular forms: Kolyvagin-type Euler systems for GSp(4), linking p-adic L-functions to Selmer groups. CONNECTION: - Apéry numbers' asymptotic growth: \( A_n \sim \frac{(1+\sqrt{2})^{4n+2}}{2^{5/2} \pi^{3/2} n^{3/2}} \), ratio \( (1+\sqrt{2})^4 \approx 33.97 \), not a classical golden ratio. - However, the interpolation parameter \ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com