Regularization of elliptic multiple zeta values
Taichi Katayama
TL;DR
The paper proves that regularized elliptic multiple zeta values can be expressed as polynomials in EMZVs with admissible indices together with a finite set of special EMZVs whose indices are in {0,1}. By extending Fay relations from length-2 EMZVs to the general case and employing regularization at tangential base points, it shows that the full EMZV algebra $\mathcal{E}^A\mathcal{Z}$ is generated by $\mathcal{E}^A\mathcal{Z}_{\mathrm{adm}}$ together with $I^A(k_1,\ldots,k_r;\tau)$ for $k_i\in\{0,1\}$. The work builds on elliptic polylogarithms, the KZB-elliptic associator framework, and the Fay identities for Eisenstein–Kronecker series to derive a coherent regularization mechanism and a precise structural description of EMZVs. This provides a concrete, computable basis for EMZVs and advances understanding of their algebraic relations and monodromy interpretations.
Abstract
In this paper, we show that regularized elliptic multiple zeta values are given by polynomials in elliptic multiple zeta values with admissible indices and special ones whose indices consist of 0 and 1.