A note on elliptic curves and Galois module structure in global function fields

In this paper we study the Galois module structure of certain Kummer orders obtained by dividing torsion points on an elliptic curve defined over a global function field. We prove that such Kummer orders are globally free as Galois modules. This is the analogue over function fields of a conjecture first stated by M. J. Taylor for CM elliptic curves defined over number fields.