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Torsion points in families of Drinfeld modules

Volume 161 / 2013

Dragos Ghioca, Liang-Chung Hsia Acta Arithmetica 161 (2013), 219-240 MSC: Primary 37P05; Secondary 37P10. DOI: 10.4064/aa161-3-2

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.

Authors

  • Dragos GhiocaDepartment of Mathematics
    University of British Columbia
    Vancouver, BC V6T 1Z2, Canada
    e-mail
  • Liang-Chung HsiaDepartment of Mathematics
    National Taiwan Normal University
    Taiwan, ROC
    e-mail

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