Dynamical Galois groups of trinomials and Odoni's conjecture

Looper, Nicole

doi:10.1112/blms.12227

Cited by 17 publications

(23 citation statements)

References 31 publications

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“…Looper [153] recently proved Odoni's conjecture (Conjecture 5.4) for K = Q and all prime degrees d, and a number of researchers [23,130,220] independently extended Looper's work. In particular, Specter [220] proved Odoni's conjecture for all number fields, and more generally, for all algebraic extensions K/Q that are unramified outside of a finite set of primes.…”

Section: Arboreal Representationsmentioning

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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Section: Arboreal Representationsmentioning

confidence: 99%

Current trends and open problems in arithmetic dynamics

Benedetto¹,

Ingram²,

Jones³

et al. 2019

Bull. Amer. Math. Soc.

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“…Taking the inverse limit φ = lim ←− φ n , one obtains a homeomorphism (34), which satisfies (35) for all u ∈ P d and all g ∈ Φ(G) since every φ n has property (39). This extends to the condition (35) on the closures of Φ(G) and H.…”

Section: Asymptotic Discriminant For Arboreal Representationsmentioning

confidence: 95%

“…Since then, a flurry of results for polynomials of degree

d = 2

have been obtained, we refer to Jones for an overview. Examples of polynomials satisfying the Odoni conjecture in all prime degrees

d ⩾ 3

for

K = Q

were obtained by Looper . Juul studied this question for rational functions over fields of positive characteristic.…”

Section: Introductionmentioning

confidence: 99%

Arboreal Cantor actions

Lukina

2018

J. London Math. Soc.

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In this paper, we consider minimal equicontinuous actions of discrete countably generated groups on Cantor sets, obtained from the arboreal representations of absolute Galois groups of fields. In particular, we study the asymptotic discriminant of these actions. The asymptotic discriminant is an invariant obtained by restricting the action to a sequence of nested clopen sets, and studying the isotropies of the enveloping group actions in such restricted systems. An enveloping (Ellis) group of such an action is a profinite group. A large class of actions of profinite groups on Cantor sets is given by arboreal representations of absolute Galois groups of fields. We show how to associate to an arboreal representation an action of a discrete group, and give examples of arboreal representations with stable and wild asymptotic discriminant.

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“…If d = 2, then f is of the form of Theorem 3.2, and the relations (8) show that f satisfies condition (i) of Theorem 3.1. Similarly, if d 4, then f is of the form of Theorem 3.1, and relations (8) show that f satisfies conditions (i) and (ii) of that Theorem. In both cases, we claim that for each n 1, the quantity F n of equation (4) is not a square modulo p = p 2 .…”

Section: Proof Of Odoni's Conjecture For D Evenmentioning

confidence: 98%

“…Stoll [13] produced infinitely many such polynomials for F = Q and d = 2. In 2016, Looper showed that Odoni's conjecture holds for F = Q and d = p a prime [8].…”

Section: Introductionmentioning

confidence: 99%