This paper is a slightly extended verion of Polylogarithmic Extensions on Mixed Shimura varieties, Part III: The elliptic polylogarithm, which itself is the extended version of paragraph 9 of Polylogarithmic Extensions on Mixed Shimura varieties. We present the Hodge-de Rham version of the (small) elliptic polylogarithm. It includes the definition of the elliptic higher logarithms, which to our knowledge has so far not been given. Also, we discuss the precise relationship between the values at Levi sections of the polylogarithm living on a CM-elliptic curve with the classes in Deligne cohomology defined and studied by Deninger.
This paper is now available in Springer Lecture Notes in Mathematics, number 1650.