A motivic Weil height machine for curves

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.