Summary: 1
x1. Introduction.
In [T1] M.J.Taylor began the study of the Galois module structure of certain Kum­
mer orders arising from group laws of abelian varieties defined over number fields
(see also [ST], [CN­S] and [CN­T]). The purpose of this paper is to study similar
Kummer orders which are derived from CM abelian varieties over global function
fields.
For any field F we shall write F c for a separable closure of F
and\Omega F for
Gal(F c =F ). Let C be a smooth, geometrically irreducible curve defined over a field
k ` F c
p . Set L = k(C), the function field of C over k. Let S = fv 1 ; : : : ; v t g be a
fixed nonempty set of places of L and let OL = O L;S denote the ring of functions
in L which are regular away from S. OL is the function field analogue of the ring
of integers of a number field. Write O c for the integral closure of OL in L c .
Let A=L be a simple abelian variety defined over L with complex multiplication.
This implies (see [M] p.220) that A is either a constant or a twisted constant variety
over L. In what follows, we shall always assume that S contains all places of bad
reduction of A. We shall also suppose that all endomorphisms of A that we consider
are defined over L. The endomorphism ring D =End(A) of A is an order in a finite­

Source: Agboola, Adebisi - Department of Mathematics, University of California at Santa Barbara

Collections: Mathematics