Galois criterion for torsion points of Drinfeld modules

TL;DR

This paper investigates when Drinfeld modules over function fields contain rational torsion points, establishing criteria based on Galois representations and classifying cases for rank 3 modules.

Contribution

It formulates a Galois criterion for torsion points of Drinfeld modules, paralleling classical results for abelian varieties, and classifies rank 3 cases where torsion points exist.

Findings

Positive for rank 2 Drinfeld modules

Negative for rank 3 Drinfeld modules

Classification of rank 3 cases with rational torsion points

Abstract

In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal $\fl$ of $\F_q[T]$, the question essentially asks whether, up to isogeny, a Drinfeld module $\phi$ over $\F_q(T)$ contains a rational $\fl$-torsion point if the reduction of $\phi$ at almost all primes of $\F_q[T]$ contains a rational $\fl$-torsion point. Similar to the case of abelian varieties, we show that the answer is positive if the rank of the Drinfeld module is $2$, but negative if the rank is $3$. Moreover, for rank $3$ Drinfeld modules we classify those cases where the answer is positive.