Downloaded from856 Wayne Aitken et al.In this work, we discuss a method for studying finitely ramified extensions of number fields via arithmetic dynamical systems on P 1 . At least conjecturally, this method provides a vista on a part of G K,S invisible to p-adic representations. We now sketch the construction, which is quite elementary. Suppose ϕ ∈ K[x] is a polynomial of degree d ≥ 1. 1 For each n ≥ 0, let ϕ •n be the n-fold iterate of ϕ, that is, ϕ •0 (x) = x and ϕ •n+1 (x) = ϕ(ϕ •n (x)) = ϕ •n (ϕ(x)) for n ≥ 0. Let t be a parameter for P 1 /K with function field F = K(t). We are interested in the tower of branched covers of P 1 given by(1.1)as well as extensions of K obtained by adjoining roots of its specializations at arbitraryFix an algebraic closure F of F, and let K be the algebraic closure of K determined by this choice, that is, the subfield of F consisting of elements algebraic over K. For n ≥ 0, let T ϕ,n be the set of roots in F of Φ n (x, t); it has cardinality d n . We denote by T ϕ the d-regular rooted tree whose vertex set is ∪ n≥0 T ϕ,n , and whose edges point from v to w exactly when ϕ(v) = w; its root (at ground level) is t.We choose and fix an end ξ = (ξ 0 , ξ 1 , ξ 2 , . . .) of this tree; in other words, we choose a compatible system of preimages of t under the iterates of ϕ: ϕ(ξ 1 ) = ξ 0 = t and ϕ(ξ n+1 ) = ξ n for n ≥ 1. For each n ≥ 1, we consider the field F n = F(ξ n ) F[x]/(Φ n ) andits Galois closure F n = F(T ϕ,n ) over F. Let O Fn be the integral closure of K[t] in F n . Corresponding to each t 0 ∈ K, we may fix compatible specialization maps σ n,t 0 : O Fn → K with image K n,t 0 , a normal extension field of K, and put ξ n | t 0 = σ n,t 0 (ξ n ) for the corresponding compatible system of roots of Φ n (x, t 0 ). We denote by K n,t 0 the image of the restriction of σ n,t 0 to O Fn . We refer the reader to Section 2.2 for more details, but we should emphasize here that Φ n (x, t 0 ) is not necessarily irreducible over K; hence, although K n,t 0 depends only on ϕ, n, and t 0 , the isomorphism class of K n,t 0 depends a priori on the choice of ξ as well as on the choice of compatible σ n,t 0 . Also, the Galois closure of K n,t 0 /K is contained in, but possibly distinct from, K n,t 0 .Taking the compositum over all n ≥ 1, we obtain the iterated extension F ϕ = ∪ n F n attached to ϕ, with Galois closure F ϕ = ∪ n F n over F. Similarly for each t 0 ∈ K, we obtain a specialized iterated extension K ϕ,t 0 = ∪ n K n,t 0 with Galois closure over K contained in K ϕ,t 0 = ∪ n K n,t 0 . We put M ϕ = Gal(F ϕ /F) for the iterated monodromy group of ϕ and for t 0 ∈ K, we denote by M ϕ,t 0 = Gal(K ϕ,t 0 /K) its specialization at t 0 .The group M ϕ has a natural and faithful action on the tree T ϕ , hence comes equipped 1 This construction actually works for any perfect K as long as the derivative ϕ is not identically zero in K[x].
We call a pair of polynomials f, g ∈ F q [T ] a Davenport pair (DP) if their value sets are equal, V f (F q t ) = V g (F q t ), for infinitely many extensions of F q . If they are equal for all extensions of F q (for all t ≥ 1), then we say (f, g) is a strong Davenport pair (SDP). Exceptional polynomials and SDP's are special cases of DP's. Monodromy/Galois-theoretic methods have successfully given much information on exceptional polynomials and SDP's. We use these methods to study DP's in general, and analogous situations for inclusions of value sets.For example, if (f, g) is an SDP then f (T ) − g(S) ∈ F q [T, S] is known to be reducible. This has interesting consequences. We extend this to DP's (that are not pairs of exceptional polynomials) and use reducibility to study the relationship between DP's and SDP's when f is indecomposable. Additionally, we show that DP's satisfy (deg f, q t − 1) = (deg g, q t − 1) for all sufficiently large t with V f (F q t ) = V g (F q t ). This extends Lenstra's theorem (Carlitz-Wan conjecture) concerning exceptional polynomials.
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