发表机构
Stanford University（斯坦福大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了 Liu-Wang 猜想中的自守换位子关系，通过引入万有-局部叠并利用融合与纯性，移除了此前对曲线亏格和上同调消失的假设，从而对所有光滑射影曲线确立了该几何输入。
AI 中文摘要
我们证明了 Liu 和 Wang 在 arXiv:2504.00275 中猜想的一个自守换位子关系，该关系是他们框架中自守 Clifford 关系的几何输入，该框架将函数域上的高阶周期积分与 L 函数的高阶导数联系起来。此前，该关系仅在额外的上同调消失假设下以及对于亏格不等于一的曲线成立；我们移除了这些假设。主要的新成分是在局部与半局部情形之间引入了一个“万有-局部”阶段：我们引入了带有两个有序标记截面的形式多圆盘的分类叠，并构造了 Hecke 叠、单位对象、Hecke 作用以及特殊上同调对应的万有-局部类比。该叠上的融合与纯性产生了一个万有-局部换位子恒等式，其证明仅使用了进入该猜想的关于 Plancherel 代数的局部假设。将其特化到一条曲线并运用六函子形式体系，便给出了该猜想对每条光滑射影曲线成立。
英文摘要
We prove the automorphic commutator relation conjectured by Liu and Wang arXiv:2504.00275, which is the geometric input for the automorphic Clifford relations in their framework relating higher period integrals to higher derivatives of $L$-functions over function fields. The relation was previously established only under additional cohomological vanishing assumptions and for curves of genus different from one; we remove these assumptions. The main new ingredient is a "universal-local" stage between the local and the semi-local settings: we introduce the classifying stack of formal multidisks with two ordered marked sections, and we construct universal-local analogues of the Hecke stacks, unit objects, Hecke actions and special cohomological correspondences. Fusion and purity on this stack yield a universal-local commutator identity, which is proved using only the local assumptions on the Plancherel algebra that enter the conjecture. Specialization to a curve and the six-functor formalism then give the conjecture for every smooth projective curve.
