Torsion points in families of Drinfeld modules

Abstract

Let $\Phi^\l$ be an algebraic family of Drinfeld modules defined over a field $K$ of characteristic $p$, and let $\bfa,\bfb\in K[\l]$. Assume that neither $\bfa(\l)$ nor $\bfb(\l)$ is a torsion point for $\Phi^\l$ for all $\l$. If there exist infinitely many $\l\in\Kbar$ such that both $\bfa(\l)$ and $\bfb(\l)$ are torsion points for $\Phi^\l$, then we show that for each $\l\in\Kbar$, we have that $\bfa(\l)$ is torsion for $\Phi^\l$ if and only if $\bfb(\l)$ is torsion for $\Phi^\l$. In the case $\bfa,\bfb\in K$, then we prove in addition that $\bfa$ and $\bfb$ must be $\Fpbar$-linearly dependent.