Regulators in the Arithmetic of Function Fields
Quentin Gazda
TL;DR
The paper develops a function-field analogue of Beilinson regulators for rigid analytically trivial $A$-motives by constructing a Hodge-Pink regulator derived from the exact Hodge-Pink realization functor. It proves finiteness for the regulated $A$-motive Ext groups and, under a negative-weight hypothesis, establishes a rank-dimension equality between $A$-motivic extensions and Hodge-Pink extensions with infinite Frobenius. The work highlights that, unlike the number-field case, the regulator image need not be full, signaling a fundamental subtlety in formulating Beilinson-type conjectures in function-field arithmetic. Central techniques revolve around shtuka models on $(\operatorname{Spec} A)\times C$ and $C\times C$, enabling cohomological computations that relate $A$-motivic data to Hodge-Pink invariants and Galois cohomology of $G_{\infty}$. These methods illuminate the arithmetic of $A$-motives and provide a robust framework for finiteness results and regulator maps in the function-field setting.
Abstract
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.