Galois criterion for torsion points of Drinfeld modules.

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 of q [ T ] , the question essentially asks whether, up to isogeny, a Drinfeld module ϕ over q (T) contains a rational -torsion point if the reduction of ϕ at almost all primes of q [ T ] contains a rational -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. [ABSTRACT FROM AUTHOR]