A motivic Weil height machine for curves

The rational points of a hyperbolic curve over a number field map to the set of augmentations of the associated motivic algebra. An expectation, closely related to the Grothendieck section conjecture, is that the set of augmentations which are locally geometric is equal to the set of rational points. We provide evidence for this expectation by extending aspects of the "Weil height machine" to the set of locally geometric augmentations. As an application, we obtain a finiteness result which extends the classical Manin--Demjanenko theorem. This is ongoing joint work with L. Alexander Betts.