Torsion Points on Elliptic Curves over Function Fields and a Theorem of Igusa

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 .