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

Ferraguti, Andrea

doi:10.1090/proc/13958

Cited by 20 publications

(21 citation statements)

References 22 publications

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“…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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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: 99%

Constraining Images of Quadratic Arboreal Representations

Ferraguti

Pagano

2020

International Mathematics Research Notices

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“…As a specific example, Jones[122, Conjecture 6.3] has conjectured that x 2 + 1 is stable over F p if and only if p = 3, i.e., for f (x) = x 2 + 1, the set (19.3) is {3}. Ferraguti[76] has shown that under some Galois-theoretic assumptions, the set of stable primes (19.3) has density zero.…”

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confidence: 99%

Current trends and open problems in arithmetic dynamics

Benedetto¹,

Ingram²,

Jones³

et al. 2019

Bull. Amer. Math. Soc.

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Arithmetic dynamics is the study of number theoretic properties of dynamical systems. A relatively new field, it draws inspiration partly from dynamical analogues of theorems and conjectures in classical arithmetic geometry, and partly from p-adic analogues of theorems and conjectures in classical complex dynamics. In this article we survey some of the motivating problems and some of the recent progress in the field of arithmetic dynamics. 1 2 BENEDETTO, ET AL. 22. Local-Global Questions in Dynamics 61 References 63 TRENDS AND PROBLEMS IN ARITHMETIC DYNAMICS 3background on dynamical systems, number theory, and algebraic geometry that we will need. As noted at the end of Section 3, we have attempted to make this article accessible by generally restricting attention to maps of projective space, thereby obviating the need for more advanced material from algebraic geometry. Our survey of arithmetic dynamics then commences in Section 4 and concludes in Section 22. These nineteen sections are mostly independent of one another, albeit liberally sprinkled with instructive cross-references.A Brief Timeline of the Early Days. The study of Galois groups of iterated polynomials was initiated by Odoni in a series of papers [177,178,179] during the mid-1980s. Beyond this, it appears that the first paper to describe dynamical analogues for a wide range of classical problems in arithmetic geometry did so not for polynomial maps or projective space, but rather for K3 surfaces admitting an automorphism of infinite order. The 1991 paper of Silverman [204] discusses dynamical analogues of various problems, including: (1) Uniform boundedness of periodic points;(2) Periodic points on subvarieties;(3) Lower bounds for canonical heights; (4) Open image of Galois groups; (5) Integral points in orbits. The rest of the 1990s saw an explosion of papers in which numerous researchers began to study a wide variety of problems in arithmetic dynamics. The serious study of p-adic dynamics starts with a 1983 paper of Herman and Yoccoz [107], two of the world's leading complex dynamicists, in which they investigate a p-adic analogue of a classical problem in complex dynamics. Later in the 1980s and 90s, the physics literature includes several papers [6,7,21,230] that discuss p-adic dynamics for polynomial maps, but the deeper study of p-adic dynamics really took off with the Ph.D. theses of Benedetto [24, 25] in 1998 and Rivera-Letelier [190, 191] in 2000. The dynamics of iterated p-adic power series was investigated by Lubin [156] in a 1994 paper. We also mention that there is an extensive literature on ergodic theory over p-adic fields which, although not within the purview of this survey, dates back to at least 1975 [180]. Abstract Dynamical SystemsA discrete dynamical system consists of a set S and a self-map

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“…is surjective, and we denote by Ω N its image. More details on the action of Ω N on T N can be found for example in [5,16,22].…”

Section: The Tree Of Roots As a Cayley Graphmentioning

confidence: 99%

“…This gives that Ω ∞ is a pro-2-group. This description is of course just a rephrasing of the usual wreath product formulation used, for example, in [3,5,16,17].…”

Section: The Tree Of Roots As a Cayley Graphmentioning

confidence: 99%

The inverse problem for arboreal Galois representations of index two

Ferraguti¹,

Pagano²,

Casazza³

2019

Preprint

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This paper introduces a systematic approach towards the inverse problem for arboreal Galois representations of finite index attached to quadratic polynomials. Let F be a field of characteristic = 2 and f be a monic, quadratic polynomial in F [x]. Let ρ f be the arboreal Galois representation associated to f , taking values in the group Ω∞ of automorphisms of the infinite, rooted, regular binary tree. We give a complete description of the maximal closed subgroups of each closed subgroup of index at most two of Ω∞, and then we show how this description can be naturally given in terms of linear relations modulo squares between certain universal functions evaluated in elements of the post-critical orbit of f . We use such description in order to derive necessary and sufficient criteria for Im(ρ f ) to be a given index two subgroup of Ω∞. These criteria depend exclusively on the arithmetic of the critical orbit of f . Afterwards, we prove that if φ = x 2 + t ∈ Q(t)[x], then there exist exactly five distinct subgroups of index two of Ω∞ that can appear as images of ρ φt 0 for infinitely many t0 ∈ Q, where φt 0 is the specialized polynomial. We show that two of these groups appear infinitely often, by providing explicit examples, and if Vojta's conjecture over Q holds true, then so do the remaining three. Finally, we give an explicit description of the descending central series of each subgroup of index two. Using this, we introduce a sequence of combinatorial invariants for subgroups of index two. With a delicate use of these invariants we are able to establish that subgroups of index at most two of Ω∞ are pairwise non-isomorphic as topological groups, a result of independent interest. This implies, in particular, that the five aforementioned groups are pairwise distinct topological groups, and therefore yield five genuinely different instances of the infinite inverse Galois problem over Q.

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

Cited by 20 publications

References 22 publications

Constraining Images of Quadratic Arboreal Representations

Constraining Images of Quadratic Arboreal Representations

Current trends and open problems in arithmetic dynamics

The inverse problem for arboreal Galois representations of index two

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