LAUSR

Let q be a power of the prime number p, let K = Fq(t), and let r > 2 be an integer. For points a,b ∈ K which are Fq-linearly independent, we show that there exist positive constants N0 and c0 such that for each integer l > N0 and for each generator τ of Fql/Fq, we have that for all except N0 values λ ∈ Fq, the corresponding specializations a(τ ) and b(τ ) cannot have orders of degrees less than c0 log log l as torsion points for the Drinfeld module Φ : Fq[T ] −→ EndFq (Ga) (where Ga is the additive group scheme), given by Φ (τ,λ) T (x) = τx + λx + x r .