Regulators in the Arithmetic of Function Fields

As a natural sequel to the study of A-motivic cohomology initiated in "On the integral part of A-motivic cohomology", we develop a notion of regulator for rigid analytically trivial Anderson A-motives. In accordance with the conjectural picture over number fields, we define it as the morphism at the level of extension modules induced by the exactness of the Hodge-Pink realization functor. The purpose of this article is twofold: first, we prove a finiteness result for A-motivic cohomology; second, under a weight assumption, we show that the source and the target of the regulator have the same dimension. It came as a surprise to the author that the image of this regulator may fail to have full rank, thereby preventing an analogue of Beilinson's celebrated conjecture from holding in our setting.