DOCSLIB.ORG

Heights, Algebraic Dynamics and Berkovich Analytic Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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. The existence of a map from an algebraic variety to a projective space and the construction of the height associated to a self map, both relate to 2000 Mathematics Subject Classi¯cation. Primary: 14G40; Secondary: 11G50, 28C10, 14C17. Key words and phrases. height function, algebraic dynamics, analytic spaces. I am very thankful to the organizers of the \XVII Coloquio latinoamericano de Algebra" for the opportunity to participate and present this work. The author was partially supported by PSC-CUNY Award 69333-00 38. 1 2 JORGE PINEIRO the concept of line bundle. A line bundle is a variety L together with a map ¼ : L! X, in such a way that for some open covering fUig of X, ¡1 1 the restriction LjUi = ¼ (Ui) is isomorphic to Ui £ AF . The existence of a morphism from X to a projective space can be expressed in terms of line bundles. A line bundle L generated by global sections s0; :::; sr, determines a map to the some projective space Pr. Su±cient conditions for the construction of a height relative to a dynamical system can also be expressed in terms of the existence of certain line bundles on X. A line bundle L that satis¯es an equation of the sort '¤L »= L­® for ^ ® > 1, allows to build a height h' relative to the self map ' : X ! X. The existence of a line bundle as before is called a polarization for the dynamical system '. A polarized dynamical system has associ- ^ ated a canonical function h' and a canonical invariant measure d¹';v an on the analytic space Xv , for each place v of F . For places at in¯n- ity the canonical measure will be the product of positive currents on an Xv = Xσ = X £σ C. For ¯nite places, the canonical measure will be an v de¯ned over the Berkovich analytic space Xv = XBer [5] correspond- ^ ing to the place in question. A Mahler formula for the height h' will compute the height using integration against the canonical measure. The general Mahler formula for a map ' : X ! X is a consequence of the work of Chambert-Loir and Thuillier in theorem 1.3 of [11]. The Mahler formula over curves was ¯rst considered in [24] without the tools of Berkovich spaces. On the Riemann sphere the formula reads: 1 1 1 Theorem 1.1. Suppose that ' : P ! P is a map on PF and f(T ) 2 ^ OF (T ) is a polynomial equation. The height h'(Div(f)) can be ex- pressed as sum of integrals over all places v of F of the log of f(x) against the invariant measure d¹';v, that is, X Z ^ h'(Div(f)) = log jf(x)jd¹';v: 1 v Pv;an An important property of the measures d¹';v is the equidistribution property. Let's ¯x a place v of F and an embedding Fv ,! Cp. A se- ¹ quence fPng of points in X(F ) is generic if for any subvariety V ½ X there exist N such that Pn 2= V for n > N. Let (X; L; '; ®) be a po- larized dynamical system de¯ned over the number ¯eld F . A sequence ¹ of points fPng 2 X(F ) is said to be small if h'(Pn) converges to zero. Theorem 1.2. (Yuan [29]) Let (X; L; '; ®) be a polarized dynamical system on the projective variety X de¯ned over the number ¯eld F . ¹ Let fPng a sequence of points on X(F ) which is generic and small, then for any place v of F the Galois orbits of the sequence fPng are HEIGHTS, ALGEBRAIC DYNAMICS AND ANALYTIC SPACES 3 an equidistributed in the analytic space Xv Pwith respect to the invariant measure ¹ , in the sense that 1 ± converges weakly ';v #(O(Pn)) P 2O(Pn) P an to d¹';v on Xv . 1.1. Notation. Throughout this paper, X will denote a projective al- gebraic variety of dimension n and ' : X ! X a ¯nite self-map. F will denote a number ¯eld and K an algebraically closed ¯eld which is complete with respect to a non-archimedean absolute value j:j. L will denote a line bundle on the algebraic variety X. A metrized line bundle ¹ (L; k:k) on X will be denoted by L. The variety Xv will denote the variety obtained from X by extension of F to the algebraically closed and complete ¯eld Cp. The Berkovich analytic space attached to Xv v for a place v of F will be denoted by XBer. 2. Canonical heights and canonical measures Height functions are a tool to study dynamics over number ¯elds. As our ¯rst approach to height functions we take a look at the naive height on the projective space. The naive height computes the com- n plexity of the point P 2 PQ¹ . Suppose that P 2 PF . The naive height represents a variation of the function log max(jx0j; :::; jxnj), weighted over all absolute values on F . De¯nition 2.1. The naive height hnv(P ) of a point P = (x0; :::; xn) 2 n PQ¹ is given by 1 X h (x ; :::; x ) = log max(jx j ; :::; jx j )Nv ; nv 0 n [F : Q] 0 v n v v where v is running over all places of F , Fv denotes the completion at v and Nv = [Fv : Qp] for v=p. Let X be a non-singular algebraic variety de¯ned over F and L a line bundle on X. Assume that the global sections s0; :::; sr 2 ¡(X; L) r de¯ne a map ÁL = (s0; :::; sr): X ! P . A height hL associated to L ¹ could be de¯ned by hX;L(P ) = hnv(ÁL(P )) for all points P 2 X(F ). This de¯nition however depend on the selection of the sections. We can construct a height function hL associated to every element of L 2 Pic(X) ­ R. It will be unique up a bounded function O(1) on X and it will satisfy: (i) hL is R¡linear. n (ii) If X = P and L = OPn (1), then hL = hnv + O(1). (iii) If f : X ! Y is a morphism of non-singular algebraic varieties and L 2 Pic(Y ) ­ R, then hX;f ¤L = hY;L ± f + O(1). 4 JORGE PINEIRO De¯nition 2.2. Let ' : X ! X a self-map of X de¯ned over F and let L 2 Pic(X)­R be a line bundle on X such that for some number ® > 1 we have '¤L »= L­®. Such a dynamical system is called a polarized dynamical system (X; '; L; ®). The canonical height associated to a polarized dynamical system is given by n ^ hL(' (P )) h'(P ) = lim : n!1 ®n The condition ® > 1 ensures the existence of the limit. The proof can be found for example in [19], theorem B.4.1. The canonical height satisfy the following properties: ¹ (i) h'('(P )) = ®h'(P ) 8P 2 X(F ) ¹ (ii) jh'(P ) ¡ hL(P )j is bounded on X(F ). If L is ample we can also have the arithmetic properties: ¹ (iii) h' satis¯es Northcott's theorem: points with coordinates in F with bounded degree and bounded height are ¯nite in number. (iv) h' is a non-negative function. (v) h'(P ) = 0 if and only if P has a ¯nite forward orbit by the map '. Points with this property are called preperiodic points. (vi) Lehmer Question: Is there a constant c' > 0 such that if P is not preperiodic, then h'(P ) > c'= deg(P )? In a similar way as we did with the height, we can introduce the canon- ical invariant measure. Invariant measures were ¯rst studied by Brolin [8] and Lyubich [21] for maps on the Riemann sphere and later extended by Brien and Duval [7] to projective spaces of higher dimension. Sup- pose that (X; '; L; ®) is a polarized dynamical system de¯ned over a number ¯eld F . Suppose that σ : F,! C is a place of F over in¯nity.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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 .
    [Show full text]
  • 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.
    [Show full text]
  • Lecture Notes on Arithmetic Dynamics Arizona Winter School March 13–17, 2010
    Lecture Notes on Arithmetic Dynamics Arizona Winter School March 13{17, 2010 JOSEPH H. SILVERMAN [email protected] Contents About These Notes/Note to Students 1 1. Introduction 2 2. Background Material: Geometry 4 3. Background Material: Classical Dynamics 7 4. Background Material: Diophantine Equations 9 5. Preperiodic Points and Height Functions 12 6. Arithmetic Dynamics of Maps with Good Reduction 18 7. Integer Points in Orbits 22 8. Dynamical Analogues of Classical Results 28 9. Additional Topics 29 References 31 List of Notation 33 Index 34 Appendix A. Projects 36 About These Notes/Note to Students These notes are for the Arizona Winter School on Number Theory and Dynamical Systems, March 13{17, 2010. They include background material on complex dynamics and Diophantine equations (xx2{4) and expanded versions of lectures on preperiodic points and height func- tions (x5), arithmetic dynamics of maps with good reduction (x6), and integer points in orbits (x7). Two ¯nal sections give a brief description Date: February 8, 2010. 1991 Mathematics Subject Classi¯cation. Primary: 37Pxx; Secondary: 11G99, 14G99, 37P15, 37P30, 37F10. Key words and phrases. arithmetic dynamical systems. This project supported by NSF DMS-0650017 and DMS-0854755. 1 2 Joseph H. Silverman of dynamical analogues of classical results from the theory of Diophan- tine equations (x8) and some pointers toward other topics in arithmetic dynamics (x9). The study of arithmetic dynamics draws on ideas and techniques from both classical (discrete) dynamical systems and the theory of Diophantine equations. If you have not seen these subjects or want to do further reading, the books [1, 5, 15] are good introductions to complex dynamics and [2, 9, 12] are standard texts on Diophantine equations and arithmetic geometry.
    [Show full text]
  • Small Points and Adelic Metrics
    SMALL POINTS AND ADELIC METRICS Shouwu Zhang Department of Mathematics Princeton University Princeton, NJ 08540 [email protected] April 1993, Revised April 1994 1 Introduction Consider following generalized Bogomolov conjecture: Let A be an abelian variety over Q¹ , h : A(Q¹ ) ! R a N¶eron- Tate height function with respect to an ample and symmetric line bundle on A, Y a subvariety of A which is not a translate of an abelian subvariety by a torsion point; then there is a positive number ² such that the set Y² = fx 2 Y (Q¹ ): h(x) · ²g is not Zariski dense in Y . Replacing A by an abelian subvariety, we may assume that Y ¡ Y = fy1 ¡ y2 : y1; y2 2 Y (Q¹ )g generates A: A is the only abelian subvariety of A which contains Y ¡ Y . In this paper, we will prove that Y² is not Zariski dense if the map NS(A)R ! NS(Y )R is not injective. In spirit of Szpiro's paper [Sz], we will reduce the problem to the positivity of the height of Y with respect to certain metrized line bundle. To do this, we will ¯rst extend Gillet- Soul¶e'sintersection theory of hermitian line bundles to certain limits of line bundles which are called integrable metrized line bundles, then for a dynamic system, construct certain special integrable metrized line bundles which are called admissible metrized line bundles, and ¯nally prove the positivity of heights. Integrable metrized line bundles. Consider a projective variety X over Spec Q. For a e e e line bundle L on X and an© arithmeticª model (X; L) of (X; L ) over Spec Z, one can de¯ne an adelic metric k ¢ kL~ = k ¢ kp; p 2 S on L, where e is a positive integer, S is the set of places of Q, and k ¢ kp is a metric on L ­Q Qp on X(Qp).
    [Show full text]
  • Arxiv:Cond-Mat/9606220V2 28 Jul 1997 .INTRODUCTION 1
    To appear J. Stat. Phys., February, 1997. NEW TWO-COLOR DIMER MODELS WITH CRITICAL GROUND STATES R. Raghavan, Christopher L. Henley,* Scott L. Arouh Laboratory of Atomic and Solid State Physics † Ithaca, NY 14853-2501 Received ABSTRACT: We define two new models on the square lattice, in which each allowed configuration is a superposition of a covering by “white” dimers and one by “black” dimers. Each model maps to a solid-on-solid (SOS) model in which the “height” field is two dimensional. Measuring the stiffness of the SOS fluctuations in the rough phase provides critical exponents of the dimer models. Using this “height” representation, we have performed Monte Carlo simulations. They confirm that each dimer model has critical correlations and belongs to a new universality class. In the “dimer-loop” model (which maps to a loop model) one height component is smooth but has unusual correlated fluctuations; the other height component is rough. In the “noncrossing-dimer” model, the heights are rough having two different elastic constants; an unusual form of its elastic theory implies anisotropic critical correlations. 1. INTRODUCTION Certain discrete spin models are defined by a constraint on the configurations, with all allowed config- urations receiving equal statistical weights. Among these are dimer models and ice models, as well as the (highly degenerate) ground state ensembles of antiferromagnetic Potts models and frustrated Ising models. Many of the models on two-dimensional lattices are critical, in that correlation functions decay with distance as power laws [1]; and every one of these critical models has been found to have a “height representation” [2,3], that is, a 1-to-1 mapping of the microstates to those of a sort of interface model.
    [Show full text]
  • Generalized Greatest Common Divisors for Orbits Under Rational
    Generalized Greatest Common Divisors for Orbits under Rational Functions Keping Huang ∗∗ Abstract Assume Vojta’s Conjecture (Conjecture 2.6). Suppose a, b, α, β ∈ Z, and f(x), g(x) ∈ Z[x] are polynomials of degree d ≥ 2. Assume that the sequence ◦n ◦n (f (a), g (b))n is generic and α, β are not exceptional for f, g respectively. We prove that for each given ε> 0, there exists a constant C = C(ε, a, b, α, β, f, g) > 0, such that for all n ≥ 1, we have gcd(f ◦n(a) − α, g◦n(b) − β) ≤ C · exp(ε · dn). We prove an estimate for rational functions and for a more general gcd and then obtain the above inequality as a consequence. 1 Introduction In [BCZ03], Bugeaud, Corvaja, and Zannier proved the following theorem. Theorem 1.1. Let a, b be multiplicatively independent integers ≥ 2, and let ε > 0. Then, provided n is sufficiently large, we have gcd(an − 1, bn − 1) < exp(εn). The authors of that paper obtained the result by contradiction. They began by constructing a family of vectors in terms of n, a, and b. Then they showed that if the arXiv:1702.03881v2 [math.NT] 8 Aug 2018 bound is not satisfied, then the vectors must lie in a lower-dimensional linear subspace by the Schmidt Subspace Theorem. Using this result they are able to derive algebraic relations on powers of a and b, which guarantee that a, b are multiplicatively dependent. One may ask whether a similar inequality holds for iterations of polynomials, as iterations are dynamical analogues of power maps.
    [Show full text]
  • Math 249A Fall 2010: Transcendental Number Theory
    Math 249A Fall 2010: Transcendental Number Theory A course by Kannan Soundararajan LATEXed by Ian Petrow September 19, 2011 Contents 1 Introduction; Transcendence of e and π α is algebraic if there exists p 2 Z[x], p 6= 0 with p(α) = 0, otherwise α is called transcendental . Cantor: Algebraic numbers are countable, so transcendental numbers exist, and are a measure 1 set in [0; 1], but it is hard to prove transcendence for any particular number. p p 2 Examples of (proported) transcendental numbers: e, π, γ, eπ, 2 , ζ(3), ζ(5) ::: p p 2 Know: e, π, eπ, 2 are transcendental. We don't even know if γ and ζ(5), ζ(7);::: are irrational or rational, and we know that ζ(3) is irrational, but not whether or not it is transcendental! Lioville showed that the number 1 X 10−n! n=1 is transcendental, and this was one of the first numbers proven to be transcen- dental. Theorem 1 (Lioville). If 0 6= p 2 Z[x] is of degree n, and α is a root of p, α 62 , then Q a C(α) α − ≥ : q qn Proof. Assume without loss of generality that α < a=q, and that p is irreducible. Then by the mean value theorem, p(α) − p(a=q) = (α − a=q)p0(ξ) 1 for some point ξ 2 (α; a=q). But p(α) = 0 of course, and p(a=q) is a rational number with denominator qn. Thus 1=qn ≤ jα − a=qj sup jp0(x)j: x2(α−1,α+1) This simple theorem immediately shows that Lioville's number is transcen- dental because it is approximated by a rational number far too well to be al- gebraic.
    [Show full text]