### Torsion in Rank 1 Drinfeld Modules and the Uniform Boundedness Conjecture, by Bjorn Poonen

This paper has appeared in Math. Ann. 308 (1997), no. 4, 571-586, and so
the dvi version has been removed.
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
Lbar-isomorphism class, there are only finitely many Drinfeld modules
up to isomorphism over L which have nonzero torsion. For the case
A=Fq[T], r=1, and L=Fq(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
Fq[T]-modules over a fixed function field.

Bjorn Poonen <poonen@msri.org>