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

TL;DR

This paper investigates the order of torsion points in specializations of a family of Drinfeld modules over function fields, establishing lower bounds on torsion degrees for generic parameters.

Contribution

It provides new bounds on torsion point orders in specialized Drinfeld modules, extending understanding of their arithmetic properties in function field settings.

Findings

Torsion point orders grow at least logarithmically with the degree of field extension.

Most specializations avoid small torsion orders, except for finitely many.

Lower bounds depend on the logarithm of the logarithm of the extension degree.

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 $\Phi^{(\tau,\lambda)}_T(x)=\tau x+\lambda x^q + x^{q^r}$.