Galois criterion for torsion points of Drinfeld modules

2020/09/26 by Chen, Chien-Hua
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2009.12598

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 ϕ over \F_q(T) contains a rational \fl-torsion point if the reduction of ϕ 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.

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