 
Summary: Submitted exclusively to the London Mathematical Society
doi:10.1112/0000/000000
Galois Theory of Iterated Endomorphisms
Rafe Jones and Jeremy Rouse
Abstract
Given an abelian algebraic group A over a global field F, A(F), and a prime , the set
of all preimages of under some iterate of [ ] generates an extension of F that contains all
power torsion points as well as a Kummertype extension. We analyze the Galois group of this
extension, and for several classes of A we give a simple characterization of when the Galois group
is as large as possible up to constraints imposed by the endomorphism ring or the Weil pairing.
This Galois group encodes information about the density of primes p in the ring of integers of
F such that the order of ( mod p) is prime to . We compute this density in the general case
for several classes of A, including elliptic curves and onedimensional tori. For example, if F is a
number field, A/F is an elliptic curve with surjective 2adic representation and A(F) with
2A(F(A[4])), then the density of p with ( mod p) having odd order is 11/21.
1. Introduction
Let F be a global field, A an abelian algebraic group defined over F, A(F), and a
prime. The tower of extensions F([ n
]1
()), n 1 contains all power torsion points for A,
