论文标题
CM椭圆曲线的动机基本组和Bianchi双曲线三倍的几何形状
Motivic fundamental groups of CM elliptic curves and geometry of Bianchi hyperbolic threefolds
论文作者
论文摘要
在本文中，我们介绍了在$ \ mathfrak {p} $ - torsion点（$π_1^{\ rm mot}（\ rm mot}（e-e-e ge_ ge_ ge_ ge_ ge_ ge_ ge_ ge_ ge_ v_ perfrak），我们介绍了动机Galois集团对高斯和Eisenstein椭圆曲线的动机基本群体的实现之间的联系。双曲线三倍$γ_1（\ Mathfrak {p}）\ setMinus \ Mathbb {h}^3 $，其中$γ_1（\ Mathfrak {p}）$是$ {\ rm gl} _2（\ rm gl} _2（\ rm rm end} {\ rm rm end}（e e e））的一致性子组。 A.Goncharov（Arxiv：Math/0510310）发现了这种连接的第一个实例。 特别是，我们研究了基本谎言代数中上述作用图像的杂物实现，这是一个载有过滤的倾向谎言代数。贝林森和莱文的椭圆形小聚集体完全描述了图像的深度1相关分级商。在本文中，我们考虑了DEPTH-2相关的分级商。我们的主要结果之一是从比安奇（Bianchi）在比安奇（Bianchi）上的特定局部系统的复杂计算中构建了同构的同态，这是该商的标准科班综合体的三倍。该结果概括了Goncharov的结果，以及模块化流形与$ \ Mathbb {G} _M $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $的连接。 我们的构造使用了hodge相关器的机制，图像中的典型发生器，这些发电机在Arxiv中定义：0803.0297。我们的第二个主要结果是在这些发电机（Hodge相关器积分）的规范真实时期内的双层混合关系系统。这些关系变形了Depth-2 Hodge相关因子在投影线上的关系，此前由作者（ARXIV：2003.06521）发现。
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, $π_1^{\rm Mot}(E-E[\mathfrak{p}],v_0)$, and the geometry of the Bianchi hyperbolic threefolds $Γ_1(\mathfrak{p})\setminus\mathbb{H}^3$, where $Γ_1(\mathfrak{p})$ 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).
