A duality result about special functions in Drinfeld modules of arbitrary rank
Giacomo Hermes Ferraro
TL;DR
This work develops a comprehensive, functorial duality framework for Drinfeld modules of arbitrary rank by introducing Anderson eigenvectors and their duals, ω_φ and ζ_φ, and proving their universal representability. It generalizes Pellarin zeta functions through the dual theory, establishes residue-duality and lattice-analytic tools (notably Poonen duality) to control coefficients and convergence, and proves a rank-1 conjecture of Gazda–Maurischat while connecting to Hartl–Juschka pairings. The paper also provides concrete computational methods, including an algorithm to compute the differential forms ζ_φ · ω_φ^{(i)} on genus-0 and hyperelliptic curves, and yields explicit identities that unify special-function theory across genera with existing results in the elliptic (GP18) setting. Overall, it broadens the scope of Anderson-type special functions in positive characteristic, linking A-motive theory, dual A-motives, and explicit arithmetic-geometric identities.
Abstract
In the setting of a Drinfeld module $φ$ 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 Anglès, Ngo Dac and Tavares Ribeiro, and prove the existence of a universal object $ω_φ$. We adopt an analogous approach with the dual Drinfeld module $φ^*$ to define $\textit{dual Anderson eigenvectors}$. The universal object of this functor, denoted by $ζ_φ$, 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 $ζ_φ\cdotω_φ^{(i)}$ 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 $ζ_φ\cdotω_φ^{(i)}$ when $X=\mathbb{P}^1$, and prove a conjecture of Gazda and Maurischat about the invertibility of special functions for Drinfeld modules of rank $1$.