Torsion subgroups of elliptic curves over function fields of genus 0

Abstract

Let K=Fq(T) be the function field of a finite field of characteristic p, and E/K be an elliptic curve. It is known that E(K) is a finitely generated abelian group, and that for a given p, there is a finite, effectively calculable, list of possible torsion subgroups which can appear. For p≠2,3, a minimal list of prime-to-p torsion subgroups has been determined by Cox and Parry. In this article, we extend this result to the case when p=2,3, and determine the complete list of possible full torsion subgroups which can appear, and appear infinitely often, for a given p.