Journal of the London Mathematical Society Advance Access originally published online on August 31, 2009
Journal of the London Mathematical Society 2009 80(3):567-584; doi:10.1112/jlms/jdp043
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© 2009 London Mathematical Society
Cyclicity of finite Drinfeld modules
Department of Pure Mathematics
Faculty of Mathematics
University of Waterloo
Waterloo, ON
Canada N2L 3G1
Department of Pure Mathematics
Faculty of Mathematics
University of Waterloo
Waterloo, ON
Canada N2L 3G1
yrliu@math.uwaterloo.ca
Let A = 
q[T] be the polynomial ring over the finite field q, let k = q(T) be the rational function field, and let K be a finite extension of k. For a prime 
of K, we denote by 
the valuation ring of , by 
the maximal ideal of , and by the residue field /. Let 
be a Drinfeld A-module over K of rank r. If has good reduction at , let 
denote the reduction of at and let () denote the A-module ( )(). If is of rank 2 with End
() = A, then we obtain an asymptotic formula for the number of primes of K of degree x for which () is cyclic. This result can be viewed as a Drinfeld module analogue of Serre's cyclicity result on elliptic curves. We also show that when is of rank r 
3 a similar result follows.
2000 Mathematics Subject Classification 11G09 (primary), 11R45, 11N36 (secondary).
The research of the first author was supported by an NSERC discovery grant. The research of the second author was supported by an NSERC discovery grant.
Received April 23, 2007; revised April 9, 2009; published online August 31, 2009.