规范化ζ生成元：零亏格与一亏格情形
Canonicalizing zeta generators: genus zero and genus one
AI总结：
本文针对零亏格与一亏格的ζ生成元开展规范化研究，通过多重ζ值对偶空间确定规范多项式，构造对应同构，解决一亏格生成元非几何部分的定义歧义，建立了两类亏格下ζ生成元的关联，连通了两类数学对象的构造。
摘要：
ζ生成元是与奇黎曼ζ值相关的导子，可自由作用在带标记点黎曼曲面基本群的李代数上。零亏格下的ζ生成元是关于两个生成元的特定李多项式的Ihara导子，这类多项式可从Drinfeld结合子得到。我们通过形式多重ζ值与动形多重ζ值的对偶空间作用，刻画了这些多项式的一种规范选取，以及次数w≥2的偶次非李对应项。基于这些规范多项式，我们提出了一个将动形多重ζ值映射到f字母表的规范同构。从零亏格设定得到的规范李多项式确定了一亏格下的规范ζ生成元，这类生成元作用在Enriquez椭圆结合子的两个生成元上。除固定次数的单个贡献项外，一亏格ζ生成元可系统地展开为与全纯艾森斯坦级数对偶的Tsunogai几何导子的形式，由此可开展大量显式高阶计算。我们通过施加一个新的表示论条件，解决了此前定义一亏格ζ生成元非几何部分时存在的歧义问题。本工作揭示的零亏格与一亏格ζ生成元之间的紧密关联，将球上单值多重对数的构造与模图形式的迭代艾森斯坦积分表示联系了起来。
英文摘要：
Zeta generators are derivations associated with odd Riemann zeta values that act freely on the Lie algebra of the fundamental group of Riemann surfaces with marked points. The genus-zero incarnation of zeta generators are Ihara derivations of certain Lie polynomials in two generators that can be obtained from the Drinfeld associator. We characterize a canonical choice of these polynomials, together with their non-Lie counterparts at even degrees $w\geq 2$, through the action of the dual space of formal and motivic multizeta values. Based on these canonical polynomials, we propose a canonical isomorphism that maps motivic multizeta values into the $f$-alphabet. The canonical Lie polynomials from the genus-zero setup determine canonical zeta generators in genus one that act on the two generators of Enriquez' elliptic associators. Up to a single contribution at fixed degree, the zeta generators in genus one are systematically expanded in terms of Tsunogai's geometric derivations dual to holomorphic Eisenstein series, leading to a wealth of explicit high-order computations. Earlier ambiguities in defining the non-geometric part of genus-one zeta generators are resolved by imposing a new representation-theoretic condition. The tight interplay between zeta generators in genus zero and genus one unravelled in this work connects the construction of single-valued multiple polylogarithms on the sphere with iterated-Eisenstein-integral representations of modular graph forms.
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