Abstract
Numerous examples of functional relations for multiple polylogarithms are known. For elliptic polylogarithms, however, tools for the exploration of functional relations are available, but only very few relations are identified. Starting from an approach of Zagier and Gangl, which in turn is based on considerations about an elliptic version of the Bloch group, we explore functional relations between elliptic polylogarithms and link them to the relations which can be derived using the elliptic symbol formalism. The elliptic symbol formalism in turn allows for an alternative proof of the validity of the elliptic Bloch relation. While the five-term identity is the prime example of a functional identity for multiple polylogarithms and implies many dilogarithm identities, the situation in the elliptic setup is more involved: there is no simple elliptic analogue, but rather a whole class of elliptic identities.
Highlights
[1,2,3,4,5] while abelian differentials on a genus-one Riemann surface are the starting point for the elliptic polylogarithms [6, 7] to be discussed in this article
Starting from an approach of Zagier and Gangl, which in turn is based on considerations about an elliptic version of the Bloch group, we explore functional relations between elliptic polylogarithms and link them to the relations which can be derived using the elliptic symbol formalism
This article is structured in the following way: in section 2 we present some of the well-known results for functional relations of the Bloch-Wigner function and in particular the construction of the Bloch group and the Bloch relation
Summary
The description of functional relations of polylogarithms and in particular of the single-valued dilogarithm – the Bloch-Wigner function – can be formalised using the concept of (higher) Bloch groups. These are certain (abelian) groups Bm which capture functional relations satisfied by single-valued polylogarithms of order m. Afterwards, in subsection 2.2 we introduce the Bloch relation of the Bloch-Wigner function, which generates functional identities such as the five-term identity. In the subsequent section this Bloch relation will be generalised to the elliptic curve and will be used to define the elliptic analogue of B2, the elliptic Bloch group, which is discussed in subsection 3.4
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