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): \( A_n = ∑ₖ₌₀^n {n}{k}^2 {n+k}{k}^2 \) - Zagier's interpolation: \( A_n(t) = ∑ₖ₌₀^n {n}{k}^2 {n+k}{k}^2 t^k \) 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 \( f ∈ S_4(Γ_0(N)) \), normalized so that \( L(f,2) = (2π)^2/Γ(2) · period \). - 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: \( A_n ~ {(1+√2)⁴ⁿ⁺²}{25/2 π3/2 n3/2} \), ratio \( (1+√2)^4 ≈ 33.97 \), 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