A large arboreal Galois representation for a cubic postcritically finite polynomial

Benedetto, Robert L.; Faber, Xander; Hutz, Benjamin; Juul, Jamie; Yasufuku, Yu

doi:10.1007/s40993-017-0092-8

Cited by 16 publications

(36 citation statements)

References 18 publications

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“…Our Conditions 3.1.5 generalize Condition ( †) in that paper. The general strategy for proving our results is similar to that of [3]. However, the details of the proofs are quite different.…”

Section: Theorem 1 (Theorem 231) Let F Be a Normalized Belyi Map Amentioning

confidence: 99%

“…Theorem 2 is a generalization of the main result (Theorem 1.1) of [3], in which the authors consider the special case that f (x) = −2x 3 + 3x 2 is a concrete polynomial of degree 3. Our Conditions 3.1.5 generalize Condition ( †) in that paper.…”

Section: Theorem 1 (Theorem 231) Let F Be a Normalized Belyi Map Amentioning

confidence: 99%

“…[12,15]. In [3], the authors consider the case of the cubic polynomial f (x) = −2x 3 + 3x 2 , providing the first completely explicit result on its Galois theory. This cubic polynomial is the easiest example of a normalized Belyi map.…”

Section: Introductionmentioning

confidence: 99%

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Dynamical Belyi maps and arboreal Galois groups

2020

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We consider a large class of so-called dynamical Belyi maps and study the Galois groups of iterates of such maps. From the combinatorial invariants of the maps, we construct a useful presentation of the geometric Galois groups as subgroups of automorphism groups of regular trees, in terms of iterated wreath products. Using results on the reduction of dynamical Belyi maps modulo certain primes, we obtain results on the corresponding arithmetic Galois groups of iterates. These lead to results on the behavior of the arithmetic Galois groups under specialization, with applications to dynamical sequences.

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Section: Theorem 1 (Theorem 231) Let F Be a Normalized Belyi Map Amentioning

confidence: 99%

Section: Theorem 1 (Theorem 231) Let F Be a Normalized Belyi Map Amentioning

confidence: 99%

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Dynamical Belyi maps and arboreal Galois groups

2020

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“…It is a very hard problem, in general, to compute explicitly the Galois groups G n = Gal(f (n) ) for a given sequence {f k } ⊆ O K [x], even when such sequence is constant (see [2], [4] or [20] for examples in degree 2 and 3). It is natural to ask what is the generic behaviour of a sequence of fixed spherical index.…”

Section: The Generic Casementioning

confidence: 99%

“…In the recent years, there has been a growing interest in the field of arithmetic dynamics (see for example [2], [3], [8], [9], [10], [11], [12], [14], [15]). One of its main objects of study is the arithmetic of dynamical systems given by a pair (P 1 (K), f ), where K is a global field and f is a rational function on P 1 .…”

Section: Introductionmentioning

confidence: 99%

The set of stable primes for polynomial sequences with large Galois group

Ferraguti

2018

Proc. Amer. Math. Soc.

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Let K be a number field with ring of integers OK , and let {f k } k∈N ⊆ OK [x] be a sequence of monic polynomials such that for every n ∈ N, the composition f (n) = f1 • f2 • . . . • fn is irreducible. In this paper we show that if the size of the Galois group of f (n) is large enough (in a precise sense) as a function of n, then the set of primes p ⊆ OK such that every f (n) is irreducible modulo p has density zero. Moreover, we prove that the subset of polynomial sequences such that the Galois group of f (n) is large enough has density 1, in an appropriate sense, within the set of all polynomial sequences.

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Finite index theorems for iterated Galois groups of cubic polynomials

Bridy

Tucker

2018

Math. Ann.

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Let K be the function field of a smooth, irreducible curve defined over Q. Let f ∈ K[x] be of the form f (x) = x q + c where q = p r , r ≥ 1, is a power of the prime number p, and let β ∈ K. For all n ∈ N ∪ {∞}, the Galois groups Gn(β) = Gal(K(f −n (β))/K(β)) embed into [Cq] n , the n-fold wreath product of the cyclic group Cq. We show that if f is not isotrivial, then [[Cq] ∞ : G∞(β)] < ∞ unless β is postcritical or periodic. We are also able to prove that if f1(x) = x q + c1 and f2(x) = x q + c2 are two such distinct polynomials, then the fields ∞ n=1 K(f −n 1 (β)) and ∞ n=1 K(f −n 2 (β)) are disjoint over a finite extension of K.

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A large arboreal Galois representation for a cubic postcritically finite polynomial

Cited by 16 publications

References 18 publications

Dynamical Belyi maps and arboreal Galois groups

Dynamical Belyi maps and arboreal Galois groups

The set of stable primes for polynomial sequences with large Galois group

Finite index theorems for iterated Galois groups of cubic polynomials

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