A motivic Weil height machine for curves

TL;DR

This paper develops a motivic height machine for curves, aiming to relate rational points to motivic algebra augmentations, and proves finiteness results for certain curves, advancing the understanding of the motivic approach to Diophantine geometry.

Contribution

It extends the Weil height machine to motivic augmentations and establishes finiteness results, providing a new perspective on rational points via motivic algebra.

Findings

Established a motivic height machine for augmentations.

Proved a finiteness theorem for motivic augmentations on specific curves.

Determined the structure of the motive of a $\

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.