# A function field analogue of the Rasmussen-Tamagawa conjecture: The Drinfeld module case

## TL;DR

This paper explores a function field analogue of the Rasmussen-Tamagawa conjecture, establishing non-existence results for certain Drinfeld modules under specific conditions and providing examples when conditions are met.

## Contribution

It proves non-existence of Drinfeld modules with certain torsion constraints in specific cases and constructs examples when the conditions are satisfied.

## Key findings

- Non-existence of Drinfeld modules when inseparable degree is not divisible by rank.
- Existence of Drinfeld modules satisfying conditions when rank divides inseparable degree.
- Provides a new perspective on analogues of number field conjectures in function fields.

## Abstract

In the arithmetic of function fields, Drinfeld modules play the role that elliptic curves play in the arithmetic of number fields. The aim of this paper is to study a non-existence problem of Drinfeld modules with constrains on torsion points at places with large degree. This is motivated by a conjecture of Christopher Rasmussen and Akio Tamagawa on the non-existence of abelian varieties over number fields with some arithmetic constraints. We prove the non-existence of Drinfeld modules satisfying Rasmussen-Tamagawa type conditions in the case where the inseparable degree of base fields is not divisible by the rank of Drinfeld modules. Conversely if the rank divides the inseparable degree, then we give an example of Drinfeld modules satisfying Rasmussen-Tamagawa-type conditions.