Summary: A NOTE ON ELLIPTIC CURVES
AND GALOIS MODULE STRUCTURE
IN GLOBAL FUNCTION FIELDS
Version of June 17, 1995
x0. Introduction and statement of results. The purpose of this paper is to
study the Galois module structure of certain Kummer orders obtained by dividing
torsion points on an elliptic curve defined over a global function field.
For any field L, we shall write L c for a separable closure of L,
and\Omega L for
Gal(L c =L). Let r be a prime, and let F r denote the finite field containing r elements.
Let k be a field such that either k ` F
r or k ` C . Suppose that C is a smooth,
geometrically irreducible curve defined over k. Set F = k(C), the function field of
C over k. Let S = fv 1 ; : : : ; v t g be a fixed, nonempty, finite set of places of F , and
let OF;S = OF denote the ring of functions in F which are regular away from S.
OF is the function field analogue of the ring of integers of a number field. Write
O c for the integral closure of OF in F c . If L=F is any finite extension of F , then
we shall write OL for the integral closure of OF in L.