Order of torsion for reduction of linearly independent points for a family of Drinfeld modules

Abstract

Let $q$ be a power of the prime number $p$, let $K={\mathbb F}_q(t)$, and let $r\ge 2$ be an integer. For points ${\mathbf a}, {\mathbf b}\in K$ which are $\mathbb{F}_q$-linearly independent, we show that there exist positive constants $N_0$ and $c_0$ such that for each integer $\ell\ge N_0$ and for each generator $\tau$ of ${\mathbb F}_{q^\ell}/{\mathbb F}_q$, we have that for all except $N_0$ values $\lambda\in{\overline{\mathbb{F}_q}}$, the corresponding specializations ${\mathbf a}, {\mathbf b}(\tau)$ and ${\mathbf b}(\tau)$ cannot have orders of degrees less than $c_0\log\log\ell$ as torsion points for the Drinfeld module $\Phi^{(\tau,\lambda)}:\mathbb{F}_q[T] {\longrightarrow} {\mathrm{End}}_{\overline{\mathbb{F}_q}}({\mathbb G}_a)$ (where ${\mathbb G}_a$ is the additive group scheme), given by $\Phi^{(\tau,\lambda)}_T(x)=\tau x+\lambda x^q + x^{q^r}$.