Given an algebraically closed complete valued field $K$ over $\mathbb{F}_q$, an Anderson $t$-module of dimension $d$ is given by the topological $\mathbb{F}_q$-vector space $K^d$, endowed with an $\mathbb{F}_q$-linear action $\phi_t=\sum_{i\geq0}T_i\tau^i\in M_{d\times d}(K)[\tau]$, where $\tau:K^d\to K^d$ sends $(v_1,\dots,v_d)$ to $(v_1^q,\dots,v_d^q)$.
In analogy with complex abelian varieties, there is an analytic map $\exp=\sum_{i\geq0}E_i\tau^i: K^d\to K^d$—which is not necessarily surjective—such that $\phi_t\exp=\exp T_0$.

The adjoint exponential, defined as the series $\exp^*:=\sum_{i\geq0}\tau^{-i}E_i^T$, determines a (non-analytic) continuous map $K^d\to K^d$. Using the factorization properties of $K[\![x]\!]$, Poonen proved that there is a perfect duality of topological $\mathbb{F}_q$-vector spaces $\ker(\exp)\times\ker(\exp^*)\to\mathbb{F}_q$ under the condition $d=1$.

In this talk, I explain that for an arbitrary \textit{abelian} Anderson $t$-module, we have a collection of perfect pairings $\ker(\phi_{t^n})\times\ker(\phi^*_{t^n})\to\mathbb{F}_q$, and that we can use them to obtain a canonical generating series $(F_\phi)_c\in M_{d\times d}(K)[\![\tau^{-1},\tau]\!]$ for all $c\in\mathbb{F}_q(\!(t^{-1})\!)/\mathbb{F}_q(t)$. The study of the properties of $F_\phi$ allows us to prove that, if $\exp$ is surjective, $\ker(\exp^*)$ is compact and isomorphic to the Pontryagin dual of $\ker(\exp)$. Moreover, we deduce an alternative explicit description of the Hartl–Juschka pairing, obtained by Gazda and Maurischat in a recent preprint.