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

Abstract

In this paper we describe a connection between realizations of the action of the motivic Galois group on the motivic fundamental groups of Gaussian and Eisenstein elliptic curves punctured at the $\mathfrak{p}$-torsion points, $\pi_1^{\rm Mot}(E-E[\mathfrak{p}],v_0)$, and the geometry of the Bianchi hyperbolic threefolds $\Gamma_1(\mathfrak{p})\setminus\mathbb{H}^3$, where $\Gamma_1(\mathfrak{p})$ is a congruence subgroup of ${\rm GL}_2({\rm End}(E))$. 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 above action in the fundamental Lie algebra, a pronilpotent Lie algebra carrying a filtration by depth. The depth-1 associated graded quotient of the image is fully described by Beilinson and Levin's elliptic polylogarithms. In this paper, we consider the depth-2 associated graded quotient. One of our main results is the construction of a homomorphism from the complex computing the cohomology of a certain local system on the Bianchi threefold to this quotient's standard cochain complex. This result generalizes those of Goncharov, as well the connection between modular manifolds and the motivic fundamental group of $\mathbb{G}_m$ punctured at roots of unity (arXiv:1105.2076, arXiv:1910.10321). Our construction uses the mechanism of Hodge correlators, canonical generators in the image that were defined in arXiv:0803.0297. Our second main result is a system of double shuffle relations on the canonical real periods of these generators, the Hodge correlator integrals. These relations deform the relations on the depth-2 Hodge correlators on the projective line, previously found by the author (arXiv:2003.06521).