The 𝔭-primary uniform boundedness conjecture for Drinfeld modules

Abstract

In this paper, we study a Drinfeld module analogue of the Uniform Boundedness Conjecture on the torsion of abelian varieties. As a result, we prove the [Formula: see text]-primary Uniform Boundedness Conjecture for one-dimensional families of Drinfeld modules of arbitrary rank, which extends a result of Poonen. This result can be regarded as a Drinfeld module analogue of the Cadoret–Tamagawa’s result on the [Formula: see text]-primary Uniform Boundedness Conjecture for one-dimensional families of abelian varieties.