TORSION IN RANK-1 DRINFELD MODULES AND

THE UNIFORM BOUNDEDNESS CONJECTURE

Bjorn Poonen

Abstract. It is conjectured that for fixed A, r ≥ 1, and d ≥ 1, there is a uniform bound on the size of the torsion submodule of a Drinfeld A-module of rank r over a degree d extension L of the fraction field K of A. We verify the conjecture for r = 1, and more generally for Drinfeld modules having potential good reduction at some prime above a specified prime of K. Moreover, we show that within an L̄- isomorphism class, there are only finitely many Drinfeld modules up to isomorphism over L which have nonzero torsion. For the case A = F_{q}[T], r = 1, and L = F_{q}(T), we give an explicit description of the possible torsion submodules. We present three methods for proving these cases of the conjecture, and explain why they fail to prove the conjecture in general. Finally, an application of the Mordell conjecture for characteristic p function fields proves the uniform boundedness for the 𝔭-primary part of the torsion for rank 2 Drinfeld F_{q}[T]-modules over a fixed function field.

Typeset by AMS-TEX