Torsion points in families of Drinfeld modules
Volume 161 / 2013
Abstract
Let $\varPhi^\lambda$ be an algebraic family of Drinfeld modules defined over a field $K$ of characteristic $p$, and let ${\bf a},{\bf b}\in K[\lambda]$. Assume that neither ${\bf a}(\lambda)$ nor ${\bf b}(\lambda)$ is a torsion point for $\varPhi^{\lambda}$ for all $\lambda$. If there exist infinitely many $\lambda\in\bar{K}$ such that both ${\bf a}(\lambda)$ and ${\bf b}(\lambda)$ are torsion points for $\varPhi^{\lambda}$, then we show that for each $\lambda\in\overline K$, ${\bf a}(\lambda)$ is torsion for $\varPhi^{\lambda}$ if and only if ${\bf b}(\lambda)$ is torsion for $\varPhi^{\lambda}$. In the case ${\bf a},{\bf b}\in K$, we prove in addition that ${\bf a}$ and ${\bf b}$ must be $\overline{\mathbb{F}_p}$-linearly dependent.
