On the de Rham and p-adic realizations of the Elliptic Polylogarithm for CM elliptic curves

TL;DR

This paper provides explicit descriptions of de Rham and p-adic elliptic polylogarithms for CM elliptic curves, linking their specializations to p-adic Eisenstein-Kronecker numbers and extending previous complex-analytic results.

Contribution

It introduces a new explicit framework for p-adic elliptic polylogarithms on CM elliptic curves, including supersingular cases, connecting them to Eisenstein-Kronecker numbers.

Findings

Explicit formulas for p-adic elliptic polylogarithms using Kronecker theta functions

Specializations at torsion points relate to p-adic Eisenstein-Kronecker numbers

Results hold even for supersingular reduction at p

Abstract

In this paper, we give an explicit description of the de Rham and p-adic polylogarithms for elliptic curves using the Kronecker theta function. We prove in particular that when the elliptic curve has complex multiplication and good reduction at p, then the specializations to torsion points of the p-adic elliptic polylogarithm are related to p-adic Eisenstein-Kronecker numbers, proving a p-adic analogue of the result of Beilinson and Levin expressing the complex elliptic polylogarithm in terms of Eisenstein-Kronecker-Lerch series. Our result is valid even if the elliptic curve has supersingular reduction at p.