Modules de Drinfeld formels et algébricité

Abstract

Let P ( t ) ∈ F q [ t ] be a monic prime polynomial of degree n and let F q [ t ] P be the completion of F q [ t ] for the P(t)-adic valuation. For each formal Drinfeld module Φ : F q [ t ] P → F q [ t ] P { { σ } } of rank 1, we can define the reduced module (when the reduction is not trivial) Φ ¯ : F q [ t ] P → F q n { { σ } } , where F q n = F q [ t ] P / ( P ) . Let F q n = F q [ t ] P / ( P ) . If Φ ¯ R ( T ) is the power series which represents the action Φ ¯ R ∈ E n d F q n ( Φ ¯ ) on a transcendantal element T, we establish the following result: Φ ¯ R ( T ) is algebraic over F q n ( T ) if and only if R ( t ) ∈ F q ( t ) .