A duality result about special functions in Drinfeld modules of arbitrary rank

Abstract

In the setting of a Drinfeld module $\ddot{I}•$ over a curve $X/{\mathbb{F}}_q$, we use a functorial point of view to define $\text{Anderson eigenvectors}$, a generalization of the so called "special functions" introduced by Anglès, Ngo Dac and Tavares Ribeiro, and prove the existence of a universal object $\ddot{I}{‰}_{\ddot{I}}•$. We adopt an analogous approach with the dual Drinfeld module $\ddot{I}{•}^{\ast }$ to define $\text{dual Anderson eigenvectors}$. The universal object of this functor, denoted by $\hat{I}{\P }_{\ddot{I}}•$, is a generalization of Pellarin zeta functions, can be expressed as an Eisenstein-like series over the period lattice, and its coordinates are analytic functions from $X({\mathbb{C}}_{\infty })\setminus \infty$ to ${\mathbb{C}}_{\infty }$. For all integers $i$ we define dot products $\hat{I}{\P }_{\ddot{I}}•\cdot \ddot{I}{‰}_{\ddot{I}}{•}^{(i)}$ as certain meromorphic differential forms over $X_{{\mathbb{C}}_{\infty }}\setminus \infty$, and prove they are actually rational. This amounts to a generalization of Pellarin's identity for the Carlitz module, and is linked to the pairing of the $A$-motive and the dual $A$-motive defined by Hartl and Juschka. Finally, we develop an algorithm to compute the forms $\hat{I}{\P }_{\ddot{I}}•\cdot \ddot{I}{‰}_{\ddot{I}}{•}^{(i)}$ when $X={\mathbb{P}}^1$, and prove a conjecture of Gazda and Maurischat about the invertibility of special functions for Drinfeld modules of rank $1$.

This version of the manuscript has been published by Res. Math. Sci. under the licensing agreement CC BY 4.0. 48 pages