English

Torsion points in families of Drinfeld modules

Number Theory 2012-07-02 v1 Dynamical Systems

Abstract

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

Keywords

Cite

@article{arxiv.1206.7047,
  title  = {Torsion points in families of Drinfeld modules},
  author = {Dragos Ghioca and Liang-Chung Hsia},
  journal= {arXiv preprint arXiv:1206.7047},
  year   = {2012}
}

Comments

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