Periods of $t$-modules as special values

TL;DR

This paper demonstrates that all periods of uniformizable t-modules can be explicitly obtained through specialization of a rigid analytic trivialization, providing a constructive approach applicable to both abelian and non-abelian t-modules.

Contribution

It introduces a method to explicitly compute periods of t-modules via specialization, extending to non-abelian cases and establishing a new isomorphism with Tate algebra points.

Findings

All periods can be obtained via specialization of trivializations.

The isomorphism holds for arbitrary t-modules, including non-abelian.

A constructive proof method is provided.

Abstract

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.