A motivic Weil height machine for curves
L. Alexander Betts, Ishai Dan-Cohen
TL;DR
This work develops a motivic analogue of rational points on curves by introducing augmentation spaces Aug(X/k) arising from the motivic cochain algebra C^*(X). It builds a Weil height machine in the motivic setting, establishing a Northcott-type finiteness result (motivic Manin–Demjanenko) for Augmentations and computing the motivic cochain algebras for abelian varieties and G_m-torsors, including a G_m-bundle description via relatively free E_infty-algebras. The paper also develops local-global height decompositions, relates motivic augmentations to realizations (étale and Hodge), and formulates a principled link to non-abelian Chabauty/Kim-type obstructions, connecting motivic heights to questions about rational points on curves. The framework culminates in a robust structure for augmentations under realization functors and paves a path toward a motivic analogue of Grothendieck's section conjecture with concrete finiteness criteria and positivity properties. This approach highlights the potential of motivic homotopy theory to inform arithmetic questions about rational points via height and finiteness phenomena.
Abstract
The rational points of a smooth curve $X$ over a number field $k$ map to the set of augmentations of the associated motivic algebra. An expectation, related to Kim's conjecture, is that for $X$ hyperbolic, the set of augmentations which come locally at each place of $k$ from a point is equal to the set of rational points. Our view is that this should provide a relative of the Grothendieck section conjecture which may be both more accessible, and more directly applicable, than the latter. As a first step in this direction, we extend aspects of the ``Weil height machine'' to the set of such augmentations, and use this to prove a Manin--Dem'janenko-style finiteness result for motivic augmentations for particular curves. Along the way, we determine the structure of the cohomological motive of a $\mathbb{G}_m$-bundle over an algebraic variety as a highly structured algebra in the derived $\infty$-category of mixed motives with rational coefficients.