Effective rigid analytic trivializations for Drinfeld modules

TL;DR

This paper introduces a novel method for constructing rigid analytic trivializations of Drinfeld modules using infinite products, enabling efficient computation of periods and quasi-periods with new formulas and connections to existing results.

Contribution

It develops a new approach to construct rigid analytic trivializations via infinite Frobenius twist products, simplifying calculations and deriving new formulas for periods and quasi-periods.

Findings

Infinite product formulas for periods and quasi-periods.

Efficient computation from finite initial data.

Connections to Gekeler and Maurischat's results on period fields.

Abstract

We develop tools for constructing rigid analytic trivializations for Drinfeld modules as infinite products of Frobenius twists of matrices, from which we recover the rigid analytic trivialization given by Pellarin in terms of Anderson generating functions. One advantage is that these infinite products can be obtained from only a finite amount of initial calculation, and consequently we obtain new formulas for periods and quasi-periods, similar to the product expansion of the Carlitz period. We further link to results of Gekeler and Maurischat on the $\infty$-adic field generated by the period lattice.