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 \(\mathcal{O}_K\), \(p\geq 5\) a prime of good reduction. The classical Eisenstein-Kronecker series: \(H_k(z,s) = \sum_{(m,n)\neq(0,0)} \frac{\overline{(mz+n)}^k}{|mz+n|^{2s}}\). The p-adic analogue \(\mathcal{E}_{k,p}(z)\) is constructed via Coleman power series, satisfying \(\mathcal{E}_{k,p}(z) \equiv H_k(z,0) \pmod{p^N}\) for suitable \(N\). The Kronecker limit formula gives \(\lim_{s\to 1} (H_0(z,s) - \frac{\pi}{s-1}) = \log|\Delta(z)|\), and its p-adic version yields a p-adic regulator. The interpolation property: for Hecke characters \(\chi\) of \(K\), \(L_p(1-k,\chi) = (1-\chi(p)p^{k-1})L(1-k,\chi)\) up to explicit periods. | CONNECTION: The imaginary quadratic field \(K\) has a lattice \(\Lambda \subset \mathbb{C}\) with \(\m Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com