Cyclicity of elliptic curves over function fields

Abstract

Let K be a global function field over a finite field F containing q elements. Let E be an elliptic curve defined over K. For a prime P in K we can reduce the elliptic curve mod P and get an elliptic curve over a finite extension of F. The group of points on the reduced elliptic curve is either a cyclic group or it is a product of two cyclic groups. We determine the Dirichlet density of the primes in K such that the reduced curve has a cyclic group structure.