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In this paper we describe a connection between realizations of the action ofthe motivic Galois group on the motivic fundamental groups of Gaussian andEisenstein elliptic curves punctured at the $\mathfrak{p}$-torsion points,$\pi_1^{\rm Mot}(E-E[\mathfrak{p}],v_0)$, and the geometry of the Bianchihyperbolic threefolds $\Gamma_1(\mathfrak{p})\setminus\mathbb{H}^3$, where$\Gamma_1(\mathfrak{p})$ is a congruence subgroup of ${\rm GL}_2({\rmEnd}(E))$. The first instance of such a connection was found by A.Goncharov(arXiv:math/0510310).  In particular, we study the Hodge realization of the image of the aboveaction in the fundamental Lie algebra, a pronilpotent Lie algebra carrying afiltration by depth. The depth-1 associated graded quotient of the image isfully described by Beilinson and Levin's elliptic polylogarithms. In thispaper, we consider the depth-2 associated graded quotient. One of our mainresults is the construction of a homomorphism from the complex computing thecohomology of a certain local system on the Bianchi threefold to thisquotient's standard cochain complex. This result generalizes those ofGoncharov, as well the connection between modular manifolds and the motivicfundamental group of $\mathbb{G}_m$ punctured at roots of unity(arXiv:1105.2076, arXiv:1910.10321).  Our construction uses the mechanism of Hodge correlators, canonicalgenerators in the image that were defined in arXiv:0803.0297. Our second mainresult is a system of double shuffle relations on the canonical real periods ofthese generators, the Hodge correlator integrals. These relations deform therelations on the depth-2 Hodge correlators on the projective line, previouslyfound by the author (arXiv:2003.06521).

Motivic fundamental groups of CM elliptic curves and geometry of Bianchi hyperbolic threefolds