Galois theory of iterated endomorphisms

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 $(α\bmod{\p})$ 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 $\alpha \notin 2A(F(A[4]))$, then the density of $\mathfrak{p}$ with ($α\bmod{\p}$) having odd order is 11/21.

33 pages; The appendix has been updated, several examples have been redone, and a number of typos corrected. The paper has been accepted for publication in Proceedings of the London Mathematical Society

References in corpus (3)

• The density of prime divisors in the arithmetic dynamics of quadratic polynomials
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Cited by in corpus (14)

• $\ell$-adic images of Galois for elliptic curves over $\mathbb{Q}$
• Prescribing valuations of the order of a point in the reductions of abelian varieties and tori
• Reductions of points on algebraic groups
• Explicit Kummer Theory for Elliptic Curves
• Reductions of points on algebraic groups, II
• Some uniform bounds for elliptic curves over $\mathbb{Q}$
• Divisibility properties of the Fibonacci entry point
• Galois theory of quadratic rational functions
• On the reduction of points on abelian varieties and tori
• The 1-eigenspace for matrices in ${\mathrm{GL}}_2({\mathbb{Z}}_{\ell })$
• The density of odd order reductions for elliptic curves with a rational point of order 2
• Characterizing Abelian Varieties by the Reductions of the Mordell-Weil Group
• Elliptic curves over $\mathbb{Q}$ and 2-adic images of Galois
• On the local-global divisibility of torsion points on elliptic curves and ${\mathrm{GL}}_2$-type varieties