Galois theory of iterated endomorphisms

TL;DR

This paper studies the Galois groups arising from iterated preimages of points on abelian algebraic groups over global fields, characterizing when these groups are maximal and computing related prime density results.

Contribution

It provides a new analysis of the Galois groups associated with iterated endomorphisms on abelian varieties, including explicit density calculations for primes.

Findings

Galois groups are maximal under certain conditions.

Explicit density formulas for primes with specific properties.

Application to elliptic curves and tori over number fields.

Abstract

Given an abelian algebraic group $A$ over a global field $F$, $\alpha \in A(F)$, and a prime $\ell$, the set of all preimages of $\alpha$ under some iterate of $[\ell]$ generates an extension of $F$ that contains all $\ell$-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 $\p$ in the ring of integers of $F$ such that the order of $(\alpha \bmod{\p})$ is prime to $\ell$. 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, $A/F$ is an elliptic curve with surjective 2-adic representation and $\alpha \in A(F)$ with $\alpha \not\in 2A(F(A[4]))$, then the density of $\mathfrak{p}$ with ($\alpha \bmod{\p}$) having odd order is 11/21.