Let q be a power of the prime number p, let $K={\mathbb{F}}_q(t)$, and let $r⩾2$ 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 $\ell ⩾N_0$ and for each generator τ 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 $\mathbf{a}(\tau )$ and $\mathbf{b}(\tau )$ cannot have orders of degrees less than $c_0\log \log \ell$ as torsion points for the Drinfeld module ${\mathrm{\Phi }}^{(\tau ,\lambda )}:{\mathbb{F}}_q[T]⟶{\mathrm{End}}_{\overline{{\mathbb{F}}_q}}({\mathbb{G}}_a)$ (where ${\mathbb{G}}_a$ is the additive group scheme), given by ${\mathrm{\Phi }}_T^{(\tau ,\lambda )}(x)=\tau x+\lambda x^q+x^{q^r}$.

中文翻译：

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用于减少 Drinfeld 模块族的线性无关点的扭转顺序

设q是素数p的幂，令$钾={\mathbb{F}}_q(吨)$， 然后让 $r⩾2$是一个整数。对于积分$\mathbf{一个},\mathbf{乙}\in 钾$ 哪个是 ${\mathbb{F}}_q$-线性无关，我们证明存在正常数 $N_0$ 和 $C_0$ 使得对于每个整数 $\ell ⩾N_0$并且对于每个发生器τ的${\mathbb{F}}_{q^{\ell }}/{\mathbb{F}}_q$，除了 $N_0$ 价值观 $\lambda \in \overline{{\mathbb{F}}_q}$, 相应的专业 $\mathbf{一个}(\tau )$ 和 $\mathbf{乙}(\tau )$ 度数不能小于 $C_0\mathrm{日志}\mathrm{日志}\ell$ 作为 Drinfeld 模块的扭力点 ${\mathrm{\Phi }}^{(\tau ,\lambda )}：{\mathbb{F}}_q[吨]⟶{\mathrm{结尾}}_{\overline{{\mathbb{F}}_q}}({\mathbb{G}}_{\mathrm{一个}})$ （在哪里 ${\mathbb{G}}_{\mathrm{一个}}$ 是附加组方案），由下式给出 ${\mathrm{\Phi }}_{吨}^{(\tau ,\lambda )}(X)=\tau X+\lambda X^q+X^{q^r}$.