Periods of $t$-modules as special values

In this article we show that all periods of uniformizable $t$-modules (resp. their coordinates) can be obtained via specializing a rigid analytic trivialization of a related dual $t$-motive at $t=\theta$. The proof is even constructive. The central object in the construction is a subset $H$ of the Tate algebra points of $E$ which turns out to be isomorphic to the period lattice of $E$ via kind of generating series in one direction and residues in the other. This isomorphism even holds for arbitrary $t$-modules $E$, even non-abelian ones.