p-adic Interpolation of Eisenstein–Kronecker Numbers and Cyclotomic Units via L-Functions — E8 Intelligence Research

FINDING: p-adic interpolation of Eisenstein-Kronecker numbers and cyclotomic units connects analytic special values to algebraic/geometric structures via p-adic L-functions, with explicit formulas for p-adic cyclotomic multiple zeta values. MATH: - Eisenstein–Kronecker series: \( E_k(z,s) = \sum_{(m,n)\neq(0,0)} \frac{(mz+n)^k}{|mz+n|^{2s}} \) — special values at \( s=0 \) yield Kronecker numbers \( E_k(z,0) \), which are algebraic multiples of powers of \( \pi \) and periods of elliptic curves. - p-adic interpolation: For a prime \( p \), one constructs \( E_{p,k}(z) \) via Coleman power series / p-adic measures, satisfying \( E_{p,k}(z) \equiv E_k(z,0) \mod p^N \) for \( k \) in a residue class mod \( p-1 \). - p-adic Hecke L-function: \( L_p(s, \chi) \) interpolates \( L(s, \chi) \) at non-positive integers, with trivial zeros at \( s=0 \) for certain characters (Dasgupta's talk). - Cyclotomic units: \( u_n = \frac{1-\zeta_n^a}{1-\zeta_n} \) — their p-adic limits give p-ad Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com