Good reduction and Shafarevich-type theorems for dynamical systems with portrait level structures

Silverman, Joseph H.

doi:10.2140/pjm.2018.295.145

Cited by 5 publications

(20 citation statements)

References 27 publications

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“…The following gives a strong Dynamical Shafarevich Theorem for P 1 with orbit points; see also earlier work of Szpiro-Tucker [227, (2008)], Szpiro-West [225, (2017)], and Petsche-Stout [183, (2015)]. [203,Proposition 11]. However, if one further specifies the exact orbit structure of the map f : Y → X, then some configurations with n = 2d permit infinitely many good reduction triples, while other configurations allow only finitely many.…”

Section: Good Reduction Of Maps and Orbitsmentioning

confidence: 86%

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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: Good Reduction Of Maps and Orbitsmentioning

confidence: 86%

“…See[203] for details, as well as for a complete list of the values of ShafDim 1 2 [P] for the 35 portraits that contain at most 4 points and are allowable for degree 2 maps.…”

mentioning

confidence: 99%

Current trends and open problems in arithmetic dynamics

Benedetto¹,

Ingram²,

Jones³

et al. 2019

Bull. Amer. Math. Soc.

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“…R d,N [m](K, S) admits only finitely many PL N +1 (O S )-orbits. Theorem 1.6 recovers [8,Theorem 2] in the case N = 1, restricting the latter to triples (f, X, Y ) for which f has multiplicity 1 on Y . As in [8] we could allow multiplicities to be taken into account when N = 1, but there does not seem to be an appropriate higher dimensional analogue of this aspect that follows from our proof.…”

Section: Introductionmentioning

confidence: 57%

“…It is easy to generalize this construction to higher dimensions, so that the most obvious dynamical counterpart of the Shafarevich conjecture fails for all choices of d, N, K, and S. In order to recover an appropriate statement, one must impose more structure on the space of maps. As has been done under various guises in [5], [6], [8], [9], and [10], we study pairs (f, X) consisting of a map f having good reduction outside S and an appropriate finite Gal(K/K)-invariant set X ⊆ P N (K) also having good reduction outside S. Naturally, we would expect X to be dynamically related to f -in particular, to have X = Y ∪ f (Y ) for some Y ⊆ P N (K)and for the points of X to lie in a certain kind of general position modulo reduction by all primes p / ∈ S.…”

Section: Introductionmentioning

confidence: 99%

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A dynamical Shafarevich theorem for endomorphisms of $\mathbb{P}^N$

Juul¹,

Krieger²,

Looper³

et al. 2021

Preprint

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Dynamical modular curves for quadratic polynomial maps

Doyle

2018

Trans. Amer. Math. Soc.

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Motivated by the dynamical uniform boundedness conjecture of Morton and Silverman, specifically in the case of quadratic polynomials, we give a formal construction of a certain class of dynamical analogues of classical modular curves. The preperiodic points for a quadratic polynomial map may be endowed with the structure of a directed graph satisfying certain strict conditions; we call such a graph admissible. Given an admissible graph G, we construct a curve X1(G) whose points parametrize quadratic polynomial maps -which, up to equivalence, form a one-parameter family -together with a collection of marked preperiodic points that form a graph isomorphic to G. Building on work of Bousch and Morton, we show that these curves are irreducible in characteristic zero, and we give an application of irreducibility in the setting of number fields. We end with a discussion of the Galois theory associated to the preperiodic points of quadratic polynomials, including a certain Galois representation that arises naturally in this setting.

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Good reduction and Shafarevich-type theorems for dynamical systems with portrait level structures

Cited by 5 publications

References 27 publications

Current trends and open problems in arithmetic dynamics

Current trends and open problems in arithmetic dynamics

A dynamical Shafarevich theorem for endomorphisms of $\mathbb{P}^N$

Dynamical modular curves for quadratic polynomial maps

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