# What should motives for $L(E,n)$ look like?

Goncharov and Manin showed in this paper that the zeta values $\zeta(n)$ can be realized as periods of framed mixed Tate motives constructed from moduli spaces $\overline{\mathcal{M}}_{0,n+3}$ of rational curves. (See also this MO-question for a discussion.) I would like to know if analogous statements are expected (or maybe even known?) for special values of $L$-functions of elliptic curves.

So let $E$ be an elliptic curve over a number field $K$ and $L(E,s)$ its associated Hasse-Weil $L$-function. If $E$ is modular, then Beilinson's theorem tells us that the special values $L(E,n)$ for $n\geq 2$ are periods, see e.g. this MO-question. So it is, in principle, possible to write $L(E,n)$ as integral of a rational differential form over some cycle in an algebraic variety (roughly).

• Can the corresponding variety be made more explicit (ideally as explicit as in the work of Goncharov and Manin)? Has there been any work in this direction?

• What should framed motives realizing $L(E,n)$ look like?

(I know there is a paper by Brown and Levin on multiple elliptic polylogarithms, based on configuration spaces of marked points on the elliptic curve. However, motives realizing elliptic polylogarithms do not seem to be explicitly discussed in the paper.)

• Could we expect framed motives realizing special values $L(E,n)$ to be constructed from configuration spaces of points on $E$?

Let's consider first an analogous case of a number field $$F$$. Then it is conjectured that special values of the Dedekind zeta function $$\zeta_{F}(n)$$ is a linear combination of multiple polylogarithms, evaluated at some configurations of points of $$\mathbb{P}^1_\mathbb{C}$$ defined over $$F$$. Multiple polylogarithms are periods of the pro-unipotent completion of a punctured projective line. This is a weakened version of Zagier conjecture, known for $$n\leq 4.$$
I think that similar statement is expected to hold in your case, with $$\mathbb{P}^1$$ substituted by the elliptic curve. For $$n=3$$ it is known for modular elliptic curves. The relation between the generalized Eisenstein-Kronecker and pro-unipotent completion of the fundamental group of punctured elliptic curve is explained here (theorem 11.9).