Torsion points on elliptic curves over function fields and a theorem of Igusa

TL;DR

This paper provides an alternative proof of Igusa's theorem on torsion points of elliptic curves over function fields, using Tate's uniformization, and discusses finiteness results and applications to rational torsion points.

Contribution

It offers a new proof of Igusa's theorem using Tate's uniformization and extends classical finiteness results to the function field setting.

Findings

Alternative proof of Igusa's theorem using Tate's theory

Finiteness of isomorphism classes of elliptic curves with restricted bad reduction

Application to torsion points over abelian extensions

Abstract

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.