Untitled

Abstract

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].

Galois representations from pre-image trees: an arboreal survey

“…One readily checks that w n = w 2 n−1 − w n−1 + 1, and so Sylvester's sequence is the orbit of 2 under iteration of f (x) = x 2 − x + 1. Odoni proves the highly non-trivial result that P f (2) has density zero in the set of all primes by establishing isomorphisms (2) G n (f ) ∼ = Aut(T n ) for all n ≥ 1,…”

“…Thanks to several generalizations of the discriminant formula (9), it is known that these primes must belong to a very restricted set. First W. Aitken, F. Hajir, and C. Maire gave a generalization to polynomials of arbitrary degree [2], and recently J. Cullinan and Hajir [6] as well as the author and M. Manes [25,Theorem 3.2] have produced further generalizations to rational functions. In each case, the formulae show that the only primes of K that can ramify in the extensions K n /K are those dividing φ i (c) for some critical point c of φ (aside from a finite set of primes that does not grow with n, such as the primes dividing the resultant of φ).…”

“…Moreover, in [37,Section 4], he responded to a question of J. McKay by giving a powerful algorithm for deciding whether G n (f ) = Aut(T n ) for f (x) = x 2 + 1. J. Cremona [5] used this algorithm (2) to verify the assertion for n up to 5 • 10 7 . Note that for n = 5 • 10 7 ,…”

The density of prime divisors in the arithmetic dynamics of quadratic polynomials

“…Many well-known sequences belong to this class, such as the Fibonacci numbers (F 0 = F 1 = 1, F n = F n−1 + F n−2 ), the Lucas numbers (L 0 = 1, L 1 = 3, L n = L n−1 + L n−2 ), the Fermat numbers t n = 2 2 n + 1 (t 0 = 3, t n = (t n−1 − 1) 2 + 1), and the Mersenne numbers m n = 2 n − 1 (m 0 = 0, m n = 2m n−1 + 1). The set of prime divisors of a n , namely (1) P (a n ) = {p prime : p divides a i for some i ≥ 0 with a i = 0}…”

Discriminants of Chebyshev radical extensions

“…In this paper, we compute the index of the iterated extensions generated by T n (x) − t, where is an odd prime and T is the Chebyshev polynomial (of the first kind) of degree , with mild restrictions on t. Additionally in the case = 2, we provide an alternative proof of [1,Proposition 6.2] for when the index is equal to 1.…”