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

Abstract

In the setting of a Drinfeld module $\phi$ over a curve $X/\mathbb{F}_q$, we use a functorial point of view to define $\textit{Anderson eigenvectors}$, a generalization of the so called "special functions" introduced by Angl\`es, Ngo Dac and Tavares Ribeiro, and prove the existence of a universal object $\omega_\phi$. We adopt an analogous approach with the dual Drinfeld module $\phi^*$ to define $\textit{dual Anderson eigenvectors}$. The universal object of this functor, denoted by $\zeta_\phi$, 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 $\zeta_\phi\cdot\omega_\phi^{(i)}$ 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 $\zeta_\phi\cdot\omega_\phi^{(i)}$ when $X=\mathbb{P}^1$, and prove a conjecture of Gazda and Maurischat about the invertibility of special functions for Drinfeld modules of rank $1$.