p-adic Eisenstein-Kronecker Series for CM Elliptic Curves — E8 Intelligence Research

FINDING: Construction of p-adic analogues of Eisenstein-Kronecker series for CM elliptic curves over imaginary quadratic fields, enabling p-adic interpolation of Hecke L-functions at non-critical values. | MATH: Let \(E/K\) be an elliptic curve with CM by \(O_K\), \(p≥ 5\) a prime of good reduction. The classical Eisenstein-Kronecker series: \(H_k(z,s) = ∑(m,n)≠(0,0) {(mz+n)̄^k}{|mz+n|²ˢ}\). The p-adic analogue \(Ek,p(z)\) is constructed via Coleman power series, satisfying \(Ek,p(z) ≡ H_k(z,0) {p^N}\) for suitable \(N\). The Kronecker limit formula gives \(lims→ 1 (H_0(z,s) - π/s-1) = log|Δ(z)|\), and its p-adic version yields a p-adic regulator. The interpolation property: for Hecke characters \(χ\) of \(K\), \(L_p(1-k,χ) = (1-χ(p)pᵏ⁻¹)L(1-k,χ)\) up to explicit periods. | CONNECTION: The imaginary quadratic field \(K\) has a lattice \(Λ ⊂ C\) with \( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com