Torsion points in families of Drinfeld modules

Dragos Ghioca¹ ; Liang-Chung Hsia²

¹ Department of Mathematics University of British Columbia Vancouver, BC V6T 1Z2, Canada
² Department of Mathematics National Taiwan Normal University Taiwan, ROC

Acta Arithmetica, Tome 161 (2013) no. 3, pp. 219-240

Résumé

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.

Détail

DOI : 10.4064/aa161-3-2

Keywords: varphi lambda algebraic family drinfeld modules defined field characteristic lambda assume neither lambda nor lambda torsion point varphi lambda lambda there exist infinitely many lambda bar lambda lambda torsion points varphi lambda each lambda overline lambda torsion varphi lambda only lambda torsion varphi lambda prove addition overline mathbb linearly dependent