Torsion points in families of Drinfeld modules

Abstract

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

arXiv comment17 pages. arXiv admin note: text overlap with arXiv:1102.2769