scholarly journals The Weil height in terms of an auxiliary polynomial

2007 ◽  
Vol 128 (3) ◽  
pp. 209-221
Author(s):  
Charles L. Samuels
Keyword(s):  
Weil Height

scholarly journals Multiplicative approximation by the Weil height

2020 ◽  
Vol 373 (5) ◽  
pp. 3235-3259
Author(s):  
Robert Grizzard ◽  
Jeffrey D. Vaaler
Keyword(s):  
Weil Height

scholarly journals A Banach space determined by the Weil height

2009 ◽  
Vol 136 (3) ◽  
pp. 279-298 ◽  
Author(s):  
Daniel Allcock ◽  
Jeffrey D. Vaaler
Keyword(s):  
Banach Space ◽  
Weil Height

scholarly journals Decreasing height along continued fractions

2018 ◽  
Vol 40 (3) ◽  
pp. 763-788
Author(s):  
GIOVANNI PANTI
Keyword(s):  
Continued Fractions ◽  
Number Fields ◽  
Gauss Map ◽  
Projective Line ◽  
Triangle Groups ◽  
Quadratic Invariant ◽  
Weil Height ◽  
Symbolic Sequences ◽  
Commensurability Class ◽  
Quadratic Integer

The fact that the euclidean algorithm eventually terminates is pervasive in mathematics. In the language of continued fractions, it can be stated by saying that the orbits of rational points under the Gauss map$x\mapsto x^{-1}-\lfloor x^{-1}\rfloor$eventually reach zero. Analogues of this fact for Gauss maps defined over quadratic number fields have relevance in the theory of flows on translation surfaces, and have been established via powerful machinery, ultimately relying on the Veech dichotomy. In this paper, for each commensurability class of non-cocompact triangle groups of quadratic invariant trace field, we construct a Gauss map whose defining matrices generate a group in the class; we then provide a direct and self-contained proof of termination. As a byproduct, we provide a new proof of the fact that non-cocompact triangle groups of quadratic invariant trace field have the projective line over that field as the set of cross-ratios of cusps. Our proof is based on an analysis of the action of non-negative matrices with quadratic integer entries on the Weil height of points. As a consequence of the analysis, we show that long symbolic sequences in the alphabet of our maps can be effectively split into blocks of predetermined shape having the property that the height of points which obey the sequence and belong to the base field decreases strictly at each block end. Since the height cannot decrease infinitely, the termination property follows.

POINTS OF SMALL HEIGHT ON AFFINE VARIETIES DEFINED OVER FUNCTION FIELDS OF FINITE TRANSCENDENCE DEGREE

2020 ◽  
pp. 1-10
Author(s):  
DRAGOS GHIOCA ◽  
DAC-NHAN-TAM NGUYEN
Keyword(s):  
Lower Bounds ◽  
Function Fields ◽  
Direct Proof ◽  
Transcendence Degree ◽  
Arbitrary Field ◽  
Constant Field ◽  
Weil Height ◽  
Small Height ◽  
Type Statement ◽  
Special Case

Abstract We provide a direct proof of a Bogomolov-type statement for affine varieties V defined over function fields K of finite transcendence degree over an arbitrary field k, generalising a previous result (obtained through a different approach) of the first author in the special case when K is a function field of transcendence degree $1$ . Furthermore, we obtain sharp lower bounds for the Weil height of the points in $V(\overline {K})$ , which are not contained in the largest subvariety $W\subseteq V$ defined over the constant field $\overline {k}$ .

scholarly journals Dynamical Uniform Bounds for Fibers and a Gap Conjecture

2019 ◽  
Author(s):  
Jason Bell ◽  
Dragos Ghioca ◽  
Matthew Satriano
Keyword(s):  
Growth Rate ◽  
Positive Integer ◽  
Number Field ◽  
Projective Variety ◽  
Rational Map ◽  
Uniform Bounds ◽  
Weil Height ◽  
Forward Orbit

Abstract We prove a uniform version of the Dynamical Mordell–Lang Conjecture for étale maps; also, we obtain a gap result for the growth rate of heights of points in an orbit along an arbitrary endomorphism of a quasiprojective variety defined over a number field. More precisely, for our 1st result, we assume $X$ is a quasi-projective variety defined over a field $K$ of characteristic $0$, endowed with the action of an étale endomorphism $\Phi $, and $f\colon X\longrightarrow Y$ is a morphism with $Y$ a quasi-projective variety defined over $K$. Then for any $x\in X(K)$, if for each $y\in Y(K)$, the set $S_{x,y}:=\{n\in{\mathbb{N}}\colon f(\Phi ^n(x))=y\}$ is finite, then there exists a positive integer $N_x$ such that $\sharp S_{x,y}\le N_x$ for each $y\in Y(K)$. For our 2nd result, we let $K$ be a number field, $f:X\dashrightarrow{\mathbb{P}}^1$ is a rational map, and $\Phi $ is an arbitrary endomorphism of $X$. If ${\mathcal{O}}_{\Phi }(x)$ denotes the forward orbit of $x$ under the action of $\Phi $, then either $f({\mathcal{O}}_{\Phi }(x))$ is finite, or $\limsup _{n\to \infty } h(f(\Phi ^n(x)))/\log (n)>0$, where $h(\cdot )$ represents the usual logarithmic Weil height for algebraic points.

On the difference of the Weil height and the N�ron-Tate height

1976 ◽  
Vol 147 (1) ◽  
pp. 35-51 ◽  
Author(s):  
Horst G�nter Zimmer
Keyword(s):  
Weil Height ◽  
The Difference

scholarly journals The bounded height conjecture for semiabelian varieties

2020 ◽  
Vol 156 (7) ◽  
pp. 1405-1456
Author(s):  
Lars Kühne
Keyword(s):  
Abelian Variety ◽  
Upper Bound ◽  
Line Bundles ◽  
Weil Height

The bounded height conjecture of Bombieri, Masser, and Zannier states that for any sufficiently generic algebraic subvariety of a semiabelian $\overline{\mathbb{Q}}$-variety $G$ there is an upper bound on the Weil height of the points contained in its intersection with the union of all algebraic subgroups having (at most) complementary dimension in $G$. This conjecture has been shown by Habegger in the case where $G$ is either a multiplicative torus or an abelian variety. However, there are new obstructions to his approach if $G$ is a general semiabelian variety. In particular, the lack of Poincaré reducibility means that quotients of a given semiabelian variety are intricate to describe. To overcome this, we study directly certain families of line bundles on $G$. This allows us to demonstrate the conjecture for general semiabelian varieties.

scholarly journals On algebraic numbers close to 1

1998 ◽  
Vol 58 (3) ◽  
pp. 423-434 ◽  
Author(s):  
Artūras Dubickas
Keyword(s):  
Explicit Construction ◽  
Mahler Measure ◽  
Algebraic Numbers ◽  
Weil Height ◽  
Small Height

We prove that there exists a polynomial of small height with a root close to 1. This implies that there are algebraic numbers close to 1 with relatively small Mahler measure. We also give an explicit construction of such numbers with small Weil height.

Canonical Vector Heights on Algebraic K3 Surfaces with Picard Number Two

2003 ◽  
Vol 46 (4) ◽  
pp. 495-508 ◽  
Author(s):  
Arthur Baragar
Keyword(s):  
Positive Integer ◽  
Number Field ◽  
Nonnegative Integer ◽  
Bounded Function ◽  
K3 Surface ◽  
Picard Number ◽  
Weil Height ◽  
Logarithmic Height ◽  
Real Quadratic ◽  
Canonical Vector

AbstractLet V be an algebraic K3 surface defined over a number field K. Suppose V has Picard number two and an infinite group of automorphisms A = Aut(V/K). In this paper, we introduce the notion of a vector height h: V → Pic(V) ⊗ and show the existence of a canonical vector height with the following properties:where σ ∈ A, σ* is the pushforward of σ (the pullback of σ−1), and hD is a Weil height associated to the divisor D. The bounded function implied by the O(1) does not depend on P. This allows us to attack some arithmetic problems. For example, we show that the number of rational points with bounded logarithmic height in an A-orbit satisfiesHere, μ(P) is a nonnegative integer, s is a positive integer, and ω is a real quadratic fundamental unit.

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