论文标题
部分可观测时空混沌系统的无模型预测
Polylogarithm Variations and Motivic Extensions of $\mathbb{Q}$ by $\mathbb{Q}(m)$
论文作者
论文摘要
Deligne和Goncharov构建了一个中性的Tannakian类别的混合泰特动机，未对$ \ Mathbb {Z} [μ_n，1/n] $构建。 Brown和Hain- Matsumoto以$ n = 1 $计算了该类别的动机Galois组的深度2二次关系。我们通过对泰特椭圆曲线限制的单位基本基本组的Lie代数来实现动机Galois组的发电机来逐步将其结果推广到所有$ n \ ge 1 $。 该表示形式与奇特的有理$ k $ -group的自然识别兼容，$ \ mathbb {z} [μ_n，1/n] $，其空间为$γ_1（n）$ eisenstein系列，从而诱导了prime至$ n $ n $ n $ n $ n $ n $ n $ k $ k $ k $ k $ ggroux的自然动作。 我们首先通过显示$ \ mathbb {p}^1 - \ {0，μ_n，\ infty \} $将$ \ mathbb {p}^1- \ {0，\ infty \} $纳入节点椭圆曲线，并删除了$ n $的环状亚组的循环亚组，诱导了对统一基本组的混合形态，然后通过限制了混合的时期，并限制了限制的时期。 MHS在$ y_1（n）$的通用椭圆曲线上的Pologarithm变化。
Deligne and Goncharov constructed a neutral tannakian category of mixed Tate motives unramified over $\mathbb{Z}[μ_N,1/N]$. Brown and Hain--Matsumoto computed the depth 2 quadratic relations of the motivic Galois group of this category for $N = 1$. We take the first steps in generalizing their results to all $N \ge 1$ by realizing the generators of the motivic Galois group by derivations on the Lie algebra of the unipotent fundamental group of a restriction of the Tate elliptic curve. This representation is compatible with a natural identification of the odd rational $K$-groups of the rings $\mathbb{Z}[μ_N,1/N]$ with spaces of $Γ_1(N)$ Eisenstein series, thus inducing a natural action of the prime to $N$ part of the Hecke algebra on the $K$-groups. We establish these results by first showing the inclusion of $\mathbb{P}^1 - \{0,μ_N,\infty\}$ into the nodal elliptic curve with a cyclic subgroup of order $N$ removed induces a morphism of mixed Tate motives on unipotent fundamental groups and then by computing the periods of the limit mixed Hodge structure of an elliptic polylogarithm variation of MHS over the universal elliptic curve of $Y_1(N)$.
