Galois theory of iterated endomorphisms

Given an abelian algebraic group $A$ over a global field $F$, $\alpha \in A(F)$,andaprime$\ell$,thesetofallpreimagesof$\alpha$ under some iterate of $[\mathrm{\ell }]$ generates an extension of $F$ that contains all $\mathrm{\ell }$-power torsion points as well as a Kummer-type 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 $\text{p}$ in the ring of integers of $F$ such that the order of $(\alpha \text{p})$ is prime to $\mathrm{\ell }$. We compute this density in the general case for several classes of $A$, including elliptic curves and one-dimensional tori. For example, if $F$ is a number field, $A\mathrm{/}F$ is an elliptic curve with surjective 2-adic representation and $\alpha \in A(F)$ with $\alpha \in ̸2A(F(A[4]))$, then the density of $\mathfrak{p}$ with ($\alpha \text{p}$) having odd order is 11/21.