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

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 $\overline {k}$ .

The bounded height conjecture for semiabelian varieties

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.

Dynamical Uniform Bounds for Fibers and a Gap Conjecture

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.

Small totally p-adic algebraic numbers

The purpose of this note is to give a short and elementary proof of the fact that the absolute logarithmic Weil-height is bounded from below by a positive constant for all totally [Formula: see text]-adic numbers which are neither zero nor a root of unity. The proof is based on an idea of C. Petsche and gives the best known lower bounds in this setting. These bounds differ from the truth by a term of less than [Formula: see text].

Decreasing height along continued fractions

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.

ALGEBRAIC NUMBERS WITH BOUNDED DEGREE AND WEIL HEIGHT

For a positive integer $d$ and a nonnegative number $\unicode[STIX]{x1D709}$, let $N(d,\unicode[STIX]{x1D709})$ be the number of $\unicode[STIX]{x1D6FC}\in \overline{\mathbb{Q}}$ of degree at most $d$ and Weil height at most $\unicode[STIX]{x1D709}$. We prove upper and lower bounds on $N(d,\unicode[STIX]{x1D709})$. For each fixed $\unicode[STIX]{x1D709}>0$, these imply the asymptotic formula $\log N(d,\unicode[STIX]{x1D709})\sim \unicode[STIX]{x1D709}d^{2}$ as $d\rightarrow \infty$, which was conjectured in a question at Mathoverflow [https://mathoverflow.net/questions/177206/].

EQUIDISTRIBUTION OF ALGEBRAIC NUMBERS OF NORM ONE IN QUADRATIC NUMBER FIELDS

Given a fixed quadratic extension K of ℚ, we consider the distribution of elements in K of norm one (denoted [Formula: see text]). When K is an imaginary quadratic extension, [Formula: see text] is naturally embedded in the unit circle in ℂ and we show that it is equidistributed with respect to inclusion as ordered by the absolute Weil height. By Hilbert's Theorem 90, an element in [Formula: see text] can be written as [Formula: see text] for some [Formula: see text], which yields another ordering of [Formula: see text] given by the minimal norm of the associated algebraic integers. When K is imaginary we also show that [Formula: see text] is equidistributed in the unit circle under this norm ordering. When K is a real quadratic extension, we show that [Formula: see text] is equidistributed with respect to norm, under the map β ↦ log |β|( mod log |ϵ2|) where ϵ is a fundamental unit of [Formula: see text].