  If F is a global function field of characteristic p>3, we employ Tate's
theory of analytic uniformization to give an alternative proof of a theorem of
Igusa describing the image of the natural Galois representation on torsion
points of non-isotrivial elliptic curves defined over F. Along the way, using
basic properties of Faltings heights of elliptic curves, we offer a detailed
proof of the function field analogue of a classical theorem of Shafarevich
according to which there are only finitely many F-isomorphism classes of
admissible elliptic curves defined over F with good reduction outside a fixed
finite set of places of F. We end the paper with an application to torsion
points rational over abelian extensions of F.
