# On field extensions given by periods of Drinfeld modules

## TL;DR

This paper investigates the degrees of field extensions generated by periods of rank 2 Drinfeld modules, demonstrating that unlike rank 1, no universal upper bound exists, and providing bounds based on module coefficients.

## Contribution

It proves the non-existence of a universal upper bound for extension degrees of rank 2 Drinfeld modules and generalizes the result to higher ranks, with bounds depending on coefficients.

## Key findings

- No universal upper bound for extension degrees in rank 2 Drinfeld modules.
- Extension degree bounds depend on valuations of defining coefficients.
- The result extends to higher rank Drinfeld modules.

## Abstract

In this short note, we answer a question raised by M. Papikian on a universal upper bound for the degree of the extension of $K_\infty$ given by adjoining the periods of a Drinfeld module of rank 2. We show that contrary to the rank 1 case such a universal upper bound does not exist, and the proof generalises to higher rank. Moreover, we give an upper and lower bound for the extension degree depending on the valuations of the defining coefficients of the Drinfeld module. In particular, the lower bound shows the non-existence of a universal upper bound.