# On the field generated by the periods of a Drinfeld module

## TL;DR

This paper investigates the properties of the field generated by the periods of a Drinfeld module over a local function field, showing it can be arbitrarily large while its residue class degree remains bounded.

## Contribution

It generalizes Maurischat's results by demonstrating the potential unboundedness of the period field over $K_{
olinebreak}_
olinebreak{
olinebreak}_
olinebreak_
olinebreak$ and establishes a bound on the residue class degree depending only on the rank.

## Key findings

- The period field $K_{ ext{infty}}( ext{Lambda})$ can be arbitrarily large over $K_{ ext{infty}}$.
- The residue class degree $f(K_{ ext{infty}}( ext{Lambda})|K_{ ext{infty}})$ is bounded by a constant depending only on the rank $r$.
- Generalization of Maurischat's results to broader classes of Drinfeld modules.

## Abstract

Generalizing the results of Maurischat in \cite{Maurischatxx}, we show that the field $K_{\infty}(\Lambda)$ of periods of a Drinfeld module $\phi$ of rank $r$ defined over $K_{\infty} = \mathds{F}_{q}((T^{-1}))$ may be arbitrarily large over $K_{\infty}$. We also show that, in contrast, the residue class degree $f( K_{\infty}(\Lambda) | K_{\infty})$ remains bounded by a constant that depends only on $r$.