Artin twists of Drinfeld modules and Goss L-series

TL;DR

This paper develops a motivic framework for twisted Goss L-series using Anderson motives linked to Drinfeld modules and Artin representations, offering new insights into their arithmetic properties in function fields.

Contribution

It introduces a novel approach to study twisted Goss L-series through Anderson motives associated to Drinfeld modules and Artin representations, connecting them to regulators and class number formulas.

Findings

Constructed Anderson motives from Drinfeld modules and Artin representations.

Showed these motives originate from uniformizable abelian Anderson modules.

Provided an interpretation of twisted Goss L-values via regulators and Taelman's class number formula.

Abstract

Twisted $L$-functions by Dirichlet characters offer deep insights into arithmetic geometry, especially in the study of elliptic curves and abelian varieties over number fields. In the function field setting, Drinfeld modules and Anderson modules serve as analogues of elliptic curves and abelian varieties, and Goss $L$-series play the role of Hasse-Weil $L$-functions. This paper introduces a motivic framework for studying twisted Goss $L$-series via Anderson motives associated to Drinfeld modules and Artin representations. For a Drinfeld module and an Artin representation on the absolute Galois group, we present a construction of Anderson motives associated to them and we show that it comes from a uniformizable abelian Anderson module. We also study their associated $L$-series, which recover the norm of the twisted Goss $L$-values. These results provide an interpretation of twisted Goss $L$-values in terms of regulators of Anderson modules with the help of Taelman's class number formula.