Arboreal Representations for Rational Maps with Few Critical Points

Juul, Jamie; Krieger, Holly; Looper, Nicole; Thompson, Bianca; Walton, Laura

doi:10.1007/978-3-030-19478-9_6

Cited by 7 publications

(8 citation statements)

References 27 publications

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“…It is worth remarking that one direction of Conjecture 1.1 is already known to be true: if f is PCF or 0 is periodic for f , then [Ω ∞ : Im(ρ f,0 )] = ∞ (see [20,Section 3]). The converse statement remains unproven, although there are partial results towards it [21], conditional on abc and Vojta's conjectures.…”

Section: Introductionmentioning

confidence: 96%

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Constraining Images of Quadratic Arboreal Representations

Ferraguti

Pagano

2020

International Mathematics Research Notices

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In this paper, we prove several results on finitely generated dynamical Galois groups attached to quadratic polynomials. First, we show that, over global fields, quadratic post-critically finite (PCF) polynomials are precisely those having an arboreal representation whose image is topologically finitely generated. To obtain this result, we also prove the quadratic case of Hindes’ conjecture on dynamical non-isotriviality. Next, we give two applications of this result. On the one hand, we prove that quadratic polynomials over global fields with abelian dynamical Galois group are necessarily PCF, and we combine our results with local class field theory to classify quadratic pairs over ${ {\mathbb{Q}}}$ with abelian dynamical Galois group, improving on recent results of Andrews and Petsche. On the other hand, we show that several infinite families of subgroups of the automorphism group of the infinite binary tree cannot appear as images of arboreal representations of quadratic polynomials over number fields, yielding unconditional evidence toward Jones’ finite index conjecture.

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Section: Introductionmentioning

confidence: 96%

“…This is still a rather mysterious topic even in the first non-trivial case, which is that of quadratic polynomials. A great amount of interest on the topic has risen in recent years, as witnessed by the numerous papers on the topic, such as [1,3,4,7,9,10,11,12,13,20,21].…”

Section: Introductionmentioning

confidence: 99%

Constraining Images of Quadratic Arboreal Representations

Ferraguti

Pagano

2020

International Mathematics Research Notices

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“…It is worth remarking that one direction of Conjecture 1.1 is already known to be true: if f is PCF or 0 is periodic for f , then [Ω ∞ : Im(ρ f,0 )] = ∞ (see [19,Section 3]). The converse statement remains unproven, although there are partial results towards it [20], conditional on abc and Vojta's conjectures.…”

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Constraining images of quadratic arboreal representations

Ferraguti¹,

Pagano²

2020

Preprint

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In this paper, we prove several results on finitely generated dynamical Galois groups attached to quadratic polynomials. First we show that, over global fields, quadratic post-critically finite polynomials are precisely those having an arboreal representation whose image is topologically finitely generated. To obtain this result, we also prove the quadratic case of Hindes' conjecture on dynamical non-isotriviality. Next, we give two applications of this result. On the one hand we show that several infinite families of subgroups of the automorphism group of the infinite binary tree cannot appear as images of arboreal representations of quadratic polynomials over number fields, yielding unconditional evidence towards Jones' finite index conjecture. On the other hand, we prove that quadratic polynomials over global fields with abelian dynamical Galois group are necessarily post-critically finite. Finally, we combine our results with local class field theory to prove that if f ∈ Q[x] is quadratic and α ∈ Q, then the dynamical Galois group attached to the pair (f, α) is abelian if and only if (f, α) is Q-conjugated to either (x 2 , ±1) or (x 2 − 2, β), where β ∈ {±2, ±1, 0}. This improves on recent results of Andrews and Petsche.

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“…2 Under Vojta's conjectures for number fields there is substantial evidence for Jones' conjecture [14], and over function fields one can, in some cases, use these techniques to prove unconditionally that the representation is surjective [9]. Under the abc conjecture [18,Theorem 3] shows Jones' conjecture in the case of eventually stable pairs (f, α). Here eventually stable means that f N − α factorizes in a number of irreducibles that is eventually constant in N .…”

Section: Introductionmentioning

confidence: 99%

The size of arboreal images, I: exponential lower bounds for PCF and unicritical polynomials

Pagano¹

2021

Preprint

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Let f be a polynomial over a global field K. For each α in K and N in Z ≥0 denote by K N (f, α) the arboreal field K(f −N (α)) and by D N (f, α) its degree over K.It is conjectured that D N (f, α) should grow as a double exponential function of N , unless f is post-critically finite (PCF), in which case there are examples like D N (x 2 , α) ≤ 4 N . There is evidence conditionally on Vojta's conjecture. However, before the present work, no unconditional non-trivial lower bound was known for post-critically infinite f . In the case f is PCF, no non-trivial lower bound was known, not even under Vojta's conjecture.In this paper we give two simple methods that turn the finiteness of the critical orbit into an exploitable feature, also in the post-critically infinite case. First, assuming GRH for number fields, we establish for all PCF polynomials f of degree at least 2 and all α outside of the critical orbits of f , the existence of a positive constant c(f, α) such thatfor all N , which is sharp up to, possibly, improve the constant c(f, α).Second, we show unconditionally that if f is post-critically infinite over any number field K and unicritical, then, for each α in K, there exists a positive constant c(f, α) such thatfor all N . The main input here is to work modulo a suitably chosen prime and use a construction available for PCF unicritical polynomials with periodic critical orbit.

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Arboreal Representations for Rational Maps with Few Critical Points

Cited by 7 publications

References 27 publications

Constraining Images of Quadratic Arboreal Representations

Constraining Images of Quadratic Arboreal Representations

Constraining images of quadratic arboreal representations

The size of arboreal images, I: exponential lower bounds for PCF and unicritical polynomials

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