Title: Canonicalizing zeta generators: genus zero and genus one Show Chinese title

Title: 规范ζ生成元：零亏格和一亏格

Abstract: 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. Show Chinese abstract

Abstract: 泽塔生成元是与奇数黎曼泽塔值相关的导子，它们自由作用于带标记点的黎曼曲面基本群李代数上。 泽塔生成元的亏格零形式是某些由两个生成元定义的李多项式的伊藤导子，可以从德林费尔德关联子中得到。 我们通过形式和动机多重泽塔值对偶空间的作用，刻画了这些多项式的一种规范选择，以及它们在偶数次下的非李对应物$w\geq 2$。 基于这些规范多项式，我们提出了一种规范同构，将动机多重泽塔值映射到$f$字母表中。 亏格零设定中的规范李多项式决定了亏格一中的规范泽塔生成元，它们作用于恩里克斯椭圆关联子的两个生成元上。 在固定次数下只有一个贡献的情况下，亏格一中的泽塔生成元系统地展现在宫胁敏广义的几何导子中，后者与全纯爱森斯坦级数对偶，这导致了许多明确的高阶计算。 通过施加新的表示论条件，解决了亏格一泽塔生成元非几何部分的早期歧义。 这项工作中揭示的亏格零和亏格一泽塔生成元之间的紧密相互作用将球面上单值多元对数的构造与模图形式的迭代爱森斯坦积分表示联系起来。