A FUNCTION FIELD ANALOGUE OF THE RASMUSSEN-TAMAGAWA CONJECTURE: THE DRINFELD MODULE CASE

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 constraints 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 the base field is not divisible by the rank of Drinfeld modules. Conversely if the rank divides the inseparable degree, then we prove the existence of a Drinfeld module satisfying Rasmussen-Tamagawa type conditions.