Lecture Notes on Arithmetic Dynamics Arizona Winter School March 13–17, 2010
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Distribution of Rational Points on Toric Varieties: a Multi-Height Approach
Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Arda H. Demirhan University of Illinois at Chicago March 19, 2021 Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Height Machine associates counting devices to divisor classes. ( ) ( ) geometric facts arithmetic facts , given by divisor relations given by height relations Height Machine is useful for other fields too. How to Count Rational Points? X: an algebraic variety in Pn over a number field Main Idea: We use height functions to count rational points. Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Height Machine is useful for other fields too. How to Count Rational Points? X: an algebraic variety in Pn over a number field Main Idea: We use height functions to count rational points. Height Machine associates counting devices to divisor classes. ( ) ( ) geometric facts arithmetic facts , given by divisor relations given by height relations Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach How to Count Rational Points? X: an algebraic variety in Pn over a number field Main Idea: We use height functions to count rational points. Height Machine associates counting devices to divisor classes. ( ) ( ) geometric facts arithmetic facts , given by divisor relations given by height relations Height Machine is useful for other fields too. Arda H. Demirhan Distribution of Rational Points on Toric Varieties: A Multi-Height Approach Counting Function: N(U(K); B) = cardfx 2 U(K): HK (x) ≤ Bg Let U ⊆ X be a Zariski open with some rational points. -
ARTIN-MAZUR ZETA FUNCTIONS in ARITHMETIC DYNAMICS How Should We Define the Zeta Function of a Rational Map F : P 1(K) → P 1(K)
ARTIN-MAZUR ZETA FUNCTIONS IN ARITHMETIC DYNAMICS EVAN WARNER How should we define the zeta function of a rational map f : P1(k) ! P1(k), k a field? All we need is a sequence of numbers, and it makes sense to consider the sequence jPern(f)j, where 1 n Pern(f) = fx 2 P (k): f (x) = xg: Then we construct a zeta function in the usual way: 1 ! X tn ζ(f; t) = exp jPer (f)j : n n n=1 This is a special case of an Artin-Mazur zeta function, which is defined for certain dynamical systems (and in general counts the number of isolated fixed points). Note that we are, as usual, not counting fixed points with multiplicity, which in this case would be something like the Lefschetz index. In fact, if we did count with n multiplicity and deg f = d, then jPern(f)j = d + 1 for all n, and after summing the geometric series we would get 1 ζ(f; t) = : (1 − t)(1 − dt) n In general, we at least have jPern(f)j ≤ d + 1, so the zeta function is certainly defined as a formal power series and even converges on a positive radius around the origin if k is a normed field. The usual formal manipulations give rise to an Euler product over cycles, and in particular ζ(f; t) 2 Z[[t]]. Over C, these zeta functions aren't too hard to calculate: Theorem 1 (Hinkkanen). If f : P1(C) ! P1(C) has degree d ≥ 2, then ζ(f; t) is a rational function. -
Heights, Algebraic Dynamics and Berkovich Analytic Spaces
HEIGHTS, ALGEBRAIC DYNAMICS AND BERKOVICH ANALYTIC SPACES JORGE PINEIRO Abstract. The present paper is an exposition on heights and their importance in the modern study of algebraic dynamics. We will explain the idea of canonical height and its surprising relation to algebraic dynamics, invariant measures, arithmetic intersection theory, equidistribution and p-adic analytic geometry. 1. Introduction In many situations, solving mathematical problems involves study the solution set of a system of polynomial equations. The language of algebraic varieties is developed for that purpose. An algebraic variety is a topological locally ringed space whose underlying topological space \behave locally like" the zero set of a system of polynomials in an a±ne space. A formal de¯nition can be found for example in Chapter 1 of [18]. A projective variety is an algebraic variety that can be embedded into some projective space. Many number theoretic questions are nat- urally expressed as diophantine problems. Suppose that the algebraic variety X is de¯ned over the number ¯eld F . An algebraic dynamical system ' : X ! X is a ¯nite map from the algebraic variety X to itself. One of the tools to study algebraic dynamics over number ¯elds is the de¯nition of height functions. A (canonical) height function as- ^ ¹ sociated to ' is a function h' : X(F ) ! R, that attempts to compute the complexity of the point P 2 X(F¹) relative to the map '. For in- stance the points of ¯nite forward orbits for ' will be exactly the points ^ of height h'(P ) = 0. The notion of height can be also generalized to subvarieties Y ½ X, still measuring the behavior under iteration. -
FINITENESS of COMMUTABLE MAPS of BOUNDED DEGREE 1. Introduction a Dynamical System (V,F) Consists of a Set V and a Self Map
Bull. Korean Math. Soc. 52 (2015), No. 1, pp. 45–56 http://dx.doi.org/10.4134/BKMS.2015.52.1.045 FINITENESS OF COMMUTABLE MAPS OF BOUNDED DEGREE Chong Gyu Lee and Hexi Ye Abstract. In this paper, we study the relation between two dynamical systems (V,f) and (V,g) with f◦g = g◦f. As an application, we show that an endomorphism (respectively a polynomial map with Zariski dense, of bounded Preper(f)) has only finitely many endomorphisms (respectively polynomial maps) of bounded degree which are commutable with f. 1. Introduction A dynamical system (V,f) consists of a set V and a self map f : V → V . If V is a subset of a projective space Pn defined over a finitely generated field K over Q, then we have arithmetic height functions on V , which make a huge contribution to the study of (V,f). In this paper, we show the following results by studying arithmetic relations between two dynamical systems (V,f) and (V,g) with f ◦ g = g ◦ f. n n Main Theorem (Theorems 3.3 and 4.2). (1) Let φ : PC → PC be an en- n domorphism on PC, of degree at least 2. Then there are only finitely many endomorphisms of degree d which are commutable with φ: n Com(φ, d) := {ψ ∈ End(PC) | φ ◦ ψ = ψ ◦ φ, deg ψ = d} is a finite set. n n (2) Let f : AC → AC be a quasi-projective endomorphism (i.e., polyno- n mial maps) on AC. Suppose that Preper(f), the set of preperiodic points of f, is of bounded height and Zariski dense. -
Small Dynamical Heights for Quadratic Polynomials and Rational Functions.1.1
Experimental Mathematics, 23:433–447, 2014 Copyright C Taylor & Francis Group, LLC ISSN: 1058-6458 print / 1944-950X online DOI: 10.1080/10586458.2014.938203 Small Dynamical Heights for Quadratic Polynomials and Rational Functions Robert L. Benedetto1, Ruqian Chen1, Trevor Hyde2, Yordanka Kovacheva3, and Colin White1 1Department of Mathematics and Statistics, Amherst College, Amherst, Massachusetts, USA 2Department of Mathematics, University of Michigan, Ann Arbor, Michigan, USA 3Department of Mathematics, University of Chicago, Chicago, Illinois, USA CONTENTS Let φ ∈ Q(z) be a polynomial or rational function of degree 2. A spe- 1. Introduction cial case of Morton and Silverman’sdynamical uniform boundedness 2. Background conjecture states that the number of rational preperiodic points of φ 3. Quadratic Polynomials is bounded above by an absolute constant. A related conjecture of 4. Quadratic Rational Functions Silverman states that the canonical height hˆφ (x) of a nonpreperiodic Acknowledgments rational point x is bounded below by a uniform multiple of the height Funding of φ itself. We provide support for these conjectures by computing References the set of preperiodic and small-height rational points for a set of degree-2 maps far beyond the range of previous searches. 1. INTRODUCTION In this paper, we consider the dynamics of a rational func- tion φ(z) ∈ Q(z) acting on P1(Q). The degree of φ = f/g is deg φ := max{deg f, deg g}, where f, g ∈ Q[z]haveno common factors. Define φ0(z) = z, and for every n ≥ 1, let Downloaded by [] at 10:38 03 August 2015 φn(z) = φ ◦ φn−1(z); that is, φn is the nth iterate of φ un- der composition. -
Computing Points of Small Height for Cubic Polynomials
COMPUTING POINTS OF SMALL HEIGHT FOR CUBIC POLYNOMIALS ROBERT L. BENEDETTO, BENJAMIN DICKMAN, SASHA JOSEPH, BENJAMIN KRAUSE, DANIEL RUBIN, AND XINWEN ZHOU Abstract. Let φ Q[z] be a polynomial of degree d at least two. The associated ∈ canonical height hˆφ is a certain real-valued function on Q that returns zero precisely at preperiodic rational points of φ. Morton and Silverman conjectured in 1994 that the number of such points is bounded above by a constant depending only on d. A related conjecture claims that at non-preperiodic rational points, hˆφ is bounded below by a positive constant (depending only on d) times some kind of height of φ itself. In this paper, we provide support for these conjectures in the case d = 3 by computing the set of small height points for several billion cubic polynomials. Let φ(z) Q[z] be a polynomial with rational coefficients. Define φ0(z) = z, and for every n ∈ 1, let φn(z) = φ φn−1(z); that is, φn is the n-th iterate of φ under composition.≥ A point x is said to◦ be periodic under φ if there is an integer n 1 such that φn(x) = x. In that case, we say x is n-periodic; the smallest such positive≥ integer n is called the period of x. More generally, x is preperiodic under φ if there are integers n>m 0 such that φn(x)= φm(x); equivalently, φm(x) is periodic for some m 0. In 1950,≥ using the theory of arithmetic heights, Northcott proved that if deg≥φ 2, then φ has only finitely many preperiodic points in Q. -
Current Trends and Open Problems in Arithmetic Dynamics
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 56, Number 4, October 2019, Pages 611–685 https://doi.org/10.1090/bull/1665 Article electronically published on March 1, 2019 CURRENT TRENDS AND OPEN PROBLEMS IN ARITHMETIC DYNAMICS ROBERT BENEDETTO, PATRICK INGRAM, RAFE JONES, MICHELLE MANES, JOSEPH H. SILVERMAN, AND THOMAS J. TUCKER Abstract. 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 geom- etry 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. Contents 1. Introduction 612 2. Abstract dynamical systems 613 3. Background: Number theory and algebraic geometry 615 4. Uniform boundedness of (pre)periodic points 617 5. Arboreal representations 619 6. Dynatomic representations 622 7. Intersections of orbits and subvarieties 624 8. (Pre)periodic points on subvarieties 626 9. Dynamical (dynatomic) modular curves 627 10. Dynamical moduli spaces 630 11. Unlikely intersections in dynamics 634 12. Good reduction of maps and orbits 636 13. Dynamical degrees of rational maps 642 14. Arithmetic degrees of orbits 645 15. Canonical heights 649 16. Variation of the canonical height 653 17. p-adic and non-archimedean dynamics 656 18. Dynamics over finite fields 661 19. Irreducibilty and stability of iterates 665 20. Primes, prime divisors, and primitive divisors in orbits 668 21. Integral points in orbits 671 Received by the editors June 30, 2018. -
UC Berkeley UC Berkeley Electronic Theses and Dissertations
UC Berkeley UC Berkeley Electronic Theses and Dissertations Title The arithmetic Hodge-index theorem and rigidity of algebraic dynamical systems over function fields Permalink https://escholarship.org/uc/item/4xb7w3f8 Author Carney, Alexander Publication Date 2019 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California The arithmetic Hodge-index theorem and rigidity of algebraic dynamical systems over function fields by Alexander Carney Adissertationsubmittedinpartialsatisfactionofthe requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Associate Professor Xinyi Yuan, Chair Associate Professor Burkhard Militzer Associate Professor Sug Woo Shin Spring 2019 The arithmetic Hodge-index theorem and rigidity of algebraic dynamical systems over function fields Copyright 2019 by Alexander Carney 1 Abstract The arithmetic Hodge-index theorem and rigidity of algebraic dynamical systems over function fields by Alexander Carney Doctor of Philosophy in Mathematics University of California, Berkeley Associate Professor Xinyi Yuan, Chair In one of the fundamental results of Arakelov’s arithmetic intersection theory, Faltings and Hriljac (independently) proved the Hodge-index theorem for arithmetic surfaces by relating the intersection pairing to the negative of the Neron-Tate height pairing. More recently, Moriwaki and Yuan–Zhang generalized this to higher dimension. In this work, we extend these results to projective varieties over transcendence degree one function fields. The new challenge is dealing with non-constant but numerically trivial line bundles coming from the constant field via Chow’s K/k-image functor. As an application of the Hodge-index theorem to heights defined by intersections of adelic metrized line bundles, we also prove a rigidity theorem for the set height zero points of polarized algebraic dynamical systems over function fields. -
Rationality of Dynamical Canonical Height 11
RATIONALITY OF DYNAMICAL CANONICAL HEIGHT LAURA DE MARCO AND DRAGOS GHIOCA Abstract. We present a dynamical proof of the well-known fact that the N´eron-Tate canonical height (and its local counterpart) takes rational values at points of an elliptic curve over a function field k = C(X), where X is a curve. More generally, we investigate the mechanism for which the local canonical height for a map f : P1 ! P1 defined over a function field k can take irrational values (at points in a local completion of k), providing examples in all degrees deg f ≥ 2. Building on Kiwi's classification of non-archimedean Julia sets for quadratic maps [Ki1], we give a complete answer in degree 2 characterizing the existence of points with irrational local canonical heights. As an application we prove that if the heights bhf (a); bhg(b) are rational and positive, for maps f and g of multiplicatively 1 ¯ n m independent degrees and points a; b 2 P (k), then the orbits ff (a)gn≥0 and fg (b)gm≥0 intersect in at most finitely many points, complementing the results of [GTZ1]. 1. Introduction Throughout this article, we let K be an algebraically closed field of characteristic 0, X a smooth projective curve defined over K, and k = K(X) its function field. We examine the values of the canonical height function associated to dynamical systems 1 1 f : P ! P defined over k, of degree deg(f) ≥ 2. The canonical height bhf was introduced in [CS] for general polarized dynamical systems f on projective varieties. -
Arithmetic Dynamics, Arithmetic Geometry, and Number Theory Joseph H
Arithmetic Dynamics, Arithmetic Geometry, and Number Theory Joseph H. Silverman Brown University MAGNTS (Midwest Arithmetic Geometry and Number Theory Series) October 12{13, 2019 0 Arithmetic Dynamics and Arithmetic Geometry 1 What is Arithmetic Dynamics? Arithmetic Geometry: Study solutions to polyno- mial equations (points on algebraic varieties) over non- algebraically closed fields. (Discrete) Dynamical Systems: Study orbits of points under iteration of a function. Arithmetic Dynamics: Study number theoretic prop- erties of orbits of points on algebraic varieties. A lot of arithmetic dynamics comes by analogy from arithmetic geometry. Sometimes the analogy is quite di- rect, sometimes less so, and there are parts of arithmetic geometry that still lack dynamical analogues. Today's talk will be a survey of what arithmetic dynamics is all about, with details on its connections to the arithmetic geometry that we all know and love. Then in tomorrow's talk I'll delve more deeply into some specific topics. Arithmetic Dynamics and Arithmetic Geometry 2 In Arithmetic Geometry We Study . Elliptic curves / higher dim'l abelian varieties Torsion points Torsion points defined over a fixed K. Fields generated by torsion points. Image of Galois G(K=K¯ ) ! Aut(Ators). Torsion points on subvarieties (dim A ≥ 2). Mordell{Weil groups Rank of K-rational points . for fixed A and varying K; for fixed K and varying A. Intersection with subvarieties (dim A ≥ 2). Moduli spaces of elliptic curve and abelian varieties Geometry of moduli spaces, e.g., X0(N) and Ag. Distribution of \special" points (CM moduli). Modular forms, L-series, Hecke operators, . Arithmetic Dynamics and Arithmetic Geometry 3 In Discrete Dynamics We Study . -
Small Dynamical Heights for Quadratic Polynomials and Rational Functions
SMALL DYNAMICAL HEIGHTS FOR QUADRATIC POLYNOMIALS AND RATIONAL FUNCTIONS ROBERT L. BENEDETTO, RUQIAN CHEN, TREVOR HYDE, YORDANKA KOVACHEVA, AND COLIN WHITE Abstract. Let φ 2 Q(z) be a polynomial or rational function of degree 2. A special case of Morton and Silverman's Dynamical Uniform Boundedness Conjecture states that the number of rational preperiodic points of φ is bounded above by an absolute ^ constant. A related conjecture of Silverman states that the canonical height hφ(x) of a non-preperiodic rational point x is bounded below by a uniform multiple of the height of φ itself. We provide support for these conjectures by computing the set of preperiodic and small height rational points for a set of degree 2 maps far beyond the range of previous searches. In this paper, we consider the dynamics of a rational function φ(z) 2 Q(z) acting on P1(Q). The degree of φ = f=g is deg φ := maxfdeg f; deg gg, where f; g 2 Q[z] have no common factors. Define φ0(z) = z, and for every n ≥ 1, let φn(z) = φ ◦ φn−1(z); that is, φn is the n-th iterate of φ under composition. In this context, the automorphism group PGL(2; Q) of P1(Q) acts on Q(z) by conjugation. The forward orbit of a point x 2 P1(Q) is the set of iterates n O(x) = Oφ(x) := fφ (x): n ≥ 0g: The point x is said to be periodic under φ if there is an integer n ≥ 1 such that φn(x) = x. -
Review: the Arithmetic of Dynamical Systems, by Joseph H
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0273-0979(XX)0000-0 REVIEW: THE ARITHMETIC OF DYNAMICAL SYSTEMS, BY JOSEPH H. SILVERMAN ROBERT L. BENEDETTO The field of dynamical systems traces its roots to Poincar´e’s qualitative study of solutions to differential equations in the late nineteenth century. The subfield of complex dynamics, which was initiated by Fatou and Julia in the late 1910s but which did not draw substantial attention until the 1980s, focuses mainly on the iteration of rational functions. In the 1990s, number theorists began to study such iterations as well, noting parallels to certain aspects of the theory of elliptic curves. Since then, number-theoretic dynamics has begun to emerge as a field in its own right, especially concerning the rationality properties of periodic points. 1. Rational preperiodic points A (discrete) dynamical system is a set X and a function φ : X → X; one considers the iterates φn := φ ◦ φ ◦···◦ φ for integers n ≥ 0. We say a point x ∈ X is periodic if φn(x) = x for some n ≥ 1; if n is the smallest such integer, the set {x,φ(x),...,φn−1(x)} is called an n-cycle. More generally, given x ∈ X, if there is an integer m ≥ 0 such that φm(x) is periodic, we say x is preperiodic. Complex dynamics is devoted mostly to the case that X is the complex projective line (i.e. Riemann sphere) X = P1(C) := C∪{∞}, and that φ is a rational function with coefficients in C.