The Image of an Arboreal Galois Representation

Abstract

Much is known regarding images of p-adic Galois representations coming from subquotients ofétale cohomology groups of varieties over number fields. In particular, the Mumford-Tate conjecture gives them up to subgroups of finite index, and has been proved in many cases by Serre and others. In the analogous situation of arboreal Galois representations little is known. In this paper we give a conjectural description of their images in the case that they arise via iteration of a given polynomial. We also discuss in depth the case of iteration of a quadratic polynomial with integer coefficients.

Local Arboreal Representations

“…There is a large literature studying the image of ρ ∞ for various polynomials over global fields [Odo85a,Odo85b,Sto92,Odo97,BJ07,Jon08,BJ09,Jon13,Hin16], and occasionally also for rational functions [JM14].…”

Iterates of generic polynomials and generic rational functions

“…Since clearly, f n (x) − α has the same cycle pattern as f n (x + α) − α, we see that f (x) ∈ E(α, π) if and only if f n (x) − α is squarefree with cycle pattern π. Hence, for fixed f (x) equation (6.2) implies, #{α ∈ F q : f (x) ∈ E(α, π)} = qρ(π) + O(q 1 2 ), for all f (x) ∈ B. Remark 6.9.…”

“…The group [S d ] n has a nice interpretation as the automorphism group of the d-ary rooted tree up to the n-th level [17], [1]. Suppose ϕ n (x) − α has d n distinct roots for each n ≥ 1.…”

Galois theory of iterated endomorphisms

“…Suppose that α ∈ ℓA(F ) and the ℓ-adic Galois representation associated to E surjects onto GL 2 (Z ℓ ). If ℓ = 2, suppose in addition that α ∈ 2A(F (A [4])). Then the density of primes p ⊂ O F with α mod p having order prime to ℓ is ℓ 5 − ℓ 4 − ℓ 3 + ℓ + 1 ℓ 5 − ℓ 3 − ℓ 2 + 1 .…”