Definition

Elliptic multiple zeta values (eMZVs) are defined by [arXiv:1110.6917, arXiv:1301.3042, arXiv:1412.5535]

\omega(n_1,n_2,\ldots,n_r;\tau) \equiv \int \limits_{0\leq z_i \leq z_{i+1} \leq 1} f^{(n_1)}(z_1,\tau) \mathrm{d} z_1\, f^{(n_2)}(z_2,\tau)\mathrm{d} z_2\, \ldots f^{(n_r)}(z_r,\tau) \mathrm{d} z_r

on an elliptic curve of modular parameter \tau. The number of integrations herein is called the length whereas \sum_{i=1}^r n_i is referred to as the weight of an eMZV. The weighting functions f^{(n)}(z,\tau) arise as expansion coefficients of

\alpha \Omega(z,\alpha,\tau) \equiv \sum_{n=0}^{\infty}f^{(n)}(z,\tau)\alpha^{n}\,,

whereas the generating function \Omega(z,\alpha,\tau) is a proper periodic version of the pseudo-periodic Eisenstein-Kronecker series F(z,\alpha,\tau)

\Omega(z,\alpha,\tau)\equiv \exp \bigg( 2\pi i \alpha \frac{\Im(z)}{\Im(\tau)} \bigg) F(z,\alpha,\tau)= \exp \bigg( 2\pi i \alpha \frac{\Im(z)}{\Im(\tau)} \bigg) \frac{\theta_1'(0,\tau)\theta_1(z+\alpha,\tau)}{\theta_1(z,\tau)\theta_1(\alpha,\tau)}\,.

In the above equation, \theta_1(z,\tau) is the odd Jacobi-Theta-function, whereas a tick denotes a derivative with respect to the first argument.

Relations

In addition to the reflection identity

\omega(n_1,n_{2} ,\ldots ,n_{r-1} ,n_r;\tau)=(-1)^{n_1+n_2+\ldots+n_r}\omega(n_r,n_{r-1} ,\ldots ,n_2 ,n_1;\tau)

elliptic multiple zeta values are subject to shuffle relations

\omega(n_1,n_2,\ldots,n_r;\tau)\,\omega(k_1,k_2,\ldots,k_s;\tau) = \omega\big( (n_1,n_2,\ldots,n_r) \sqcup\mkern-3mu\sqcup\, (k_1,k_2,\ldots,k_s);\tau \big)

because of their definition as iterated integrals. Moreover, the Eisenstein-Kronecker series F(z,\alpha,\tau) is subject to the so-called Fay identities (Fay trisecant equation). If written in terms of the generating function \Omega(z,\alpha,\tau) they read

\Omega(z_1,\alpha_1,\tau)\,\Omega(z_2,\alpha_2,\tau)=
\Omega(z_1,\alpha_1+\alpha_2,\tau)\,\Omega(z_2-z_1,\alpha_2,\tau) +\Omega(z_2,\alpha_1+\alpha_2,\tau)\,\Omega(z_1-z_2,\alpha_1,\tau)

Upon expanding the above equation in f^{(n)}'s, it results in identities which can be applied to the integrand.

Starting from the two classes of relations above, one can identify indecomposable eMZVs for every length and weight. Accordingly, one can express any decomposable zeta value as a rational combination of indecompoasable eMZVS as well as products of shorter eMZVs and usual MZVs. For a (non-canonical) choice of indecomposable eMZVs, rules allowing to convert any other eMZV into this language are available at the datamine page.

Iterated Eisenstein integrals and derivation algebra

Alternatively, eMZVs can be expressed in terms of iterated Eisenstein integrals

\gamma(k_1,k_2,\ldots,k_n;q) \equiv \frac{1}{4\pi^2} \int\limits_{0 \leq q'\leq q} \mathrm{d} \log q' \ \gamma(k_1,\ldots,k_{n-1};q') \mathrm{G}_{k_n}(q')

where \mathrm{G}_0=-1 and \mathrm{G}_k(q) with k\neq0 are Eisenstein series. Furthermore, q=\exp(2\pi i \tau). Only certain convergent linear combinations of iterated Eisenstein integrals show up in rewritings of eMZVs: these combinations are implied by relations in a derivation algebra. While these relations are known in principle, eMZVs in turn can be used to find them. Our collection of relations can be obtained here.