动机增强的粗模空间

中文摘要 AI 辅助

一个在适当基$Z$上的簇$X$给出了$Z$上动机中的一个高度结构化的代数$C^*(X)$。反过来，一个$Z$-点给出了一个增广$C^*(X) \to 1$。这个赋值$X(Z) \to Aug(C^*(X))$在实现中分解了所谓的“单幂Kummer映射”到单幂基本群下的torsor。在一个方向上，这暗示了在没有等待动机t-结构的情况下“执行”Chabauty-Kim理论的动机版本的可能性。在另一个（很大程度上独立的）方向上，我们希望从超越$\pi_1$的完整有理同伦类型中提取算术信息，用于限制整点集。在这两个方向上，动机增广的粗空间$Aug(C^*(X))$将受益于有限型$\mathbb{Q}$-簇的结构，类似地，过滤$\phi$模块中增广的粗空间$Aug(C^*_{F\phi}(X))$将受益于有限型$\mathbb{Q}_p$-簇的结构。我们建立了两个可表示性准则，在几个例子中计算了这些空间，并与Selmer簇进行了比较。最后，我们通过一个例子展示了这些构造如何导致K理论有限性准则。

英文摘要

A variety $X$ over a suitable base $Z$ gives rise to a highly structured algebra $C^*(X)$ in motives over $Z$. In turn, a $Z$-point gives rise to an augmentation $C^*(X) \to 1$. This assignment $X(Z) \to Aug(C^*(X))$ factors the so-called ``unipotent Kummer map'' to torsors under the unipotent fundamental group in realizations. In one direction, this suggests the possibility of ``performing'' Chabauty-Kim theory motivically without waiting for a motivic t-structure. In a different (largely independent) direction, we may hope to extract arithmetic information for use in bounding sets of integral points from the full rational homotopy type going beyond $π_1$. In both directions, the coarse space for motivic augmentations $Aug(C^*(X))$ would benefit from a structure of finite type $\mathbb{Q}$-variety, and similarly, the coarse space $Aug(C^*_{Fϕ}(X))$ of augmentations in filtered $ϕ$ modules would benefit from a structure of finite type $\mathbb{Q}_p$-variety. We establish two criteria for representability, compute these spaces in several examples, and construct a comparison with Selmer varieties. Finally, we demonstrate how these constructions lead to K-theoretic finiteness criteria in an example.

发表机构

• Ben Gurion University of the Negev（内盖夫本-古里安大学）

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