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

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

2008-04-09

arxiv.org/abs/0804.1425v2

Mathematics

Number Theory

28 pages, final version, few minor changes

Scientific paper

10.1016/j.exmath.2008.06.001

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.

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