Torsion points in families of Drinfeld modules

TL;DR

This paper investigates the conditions under which pairs of points in families of Drinfeld modules are simultaneously torsion, revealing a linear dependence relation when certain infinite torsion conditions are met.

Contribution

It establishes a criterion linking the torsion properties of two points across a family of Drinfeld modules and proves their linear dependence over the algebraic closure of the base field.

Findings

If infinitely many parameters make both points torsion, then torsion status is equivalent for all parameters.

When points are in the base field, they must be linearly dependent over bar.

The results connect torsion behavior with linear relations 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.