Coarse moduli of motivic augmentations

Abstract

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 $\pi_1$. 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 $Aug(C^*_{F\phi}(X))$ of augmentations in filtered $\phi$ 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.