Weil Height Research Articles

Multiplicative dependence of rational values modulo approximate finitely generated groups

Abstract In this paper, we establish some finiteness results about the multiplicative dependence of rational values modulo sets which are ‘close’ (with respect to the Weil height) to division groups of finitely generated multiplicative groups of a number field K. For example, we show that under some conditions on rational functions $f_1, \ldots, f_n\in K(X)$ , there are only finitely many elements $\alpha \in K$ such that $f_1(\alpha),\ldots,f_n(\alpha)$ are multiplicatively dependent modulo such sets.

On Pisot units and the fundamental domain of Galois extensions of [formula omitted]

Let K be a number field that is Galois over Q with unit group of rank d. We obtain an upper bound on the number of facets of the reduction domain (and therefore the fundamental domain) of K of O(((d+1)ρ∞(ΛK)+log⁡(d2))d(d+1)!), where ρ∞(ΛK) is the covering radius of the log-unit lattice in the infinity norm. As a side-result, we also show that the Weil height of the shortest Pisot unit in the number field can be no greater than γ[K:Q]((d+1)ρ∞(ΛK)+dϵ) where γ=1 if K is totally real or 2 otherwise and ϵ>0 is some arbitrarily small constant.

Large algebraic integers

An algebraic integer is said large if all its real or complex embeddings have absolute value larger than [Formula: see text]. An integral ideal is said large if it admits a large generator. We investigate the notion of largeness, relating it to some arithmetic invariants of the field involved, such as the regulator and the covering radius of the lattice of units. We also study its connection with the Weil height and the Bogomolov property. We provide an algorithm for testing largeness and give some applications to the construction of floor functions arising in the theory of continued fractions.

Northcott numbers for the weighted Weil heights

We answer the question of Vidaux and Videla about the distribution of the Northcott numbers for the Weil height. We solve the same problem for the weighted Weil heights. These heights generalize both the absolute and relative Weil height. Our results also refine those of Pazuki, Technau, and Widmer.

D-finiteness, rationality, and height II: Lower bounds over a set of positive density

We consider D-finite power series f(z)=∑n≥0anzn with coefficients in a number field K. We show that there is a dichotomy governing the behaviour of h(an) as a function of n, where h is the absolute logarithmic Weil height. As an immediate consequence of our results, we have that either f(z) is rational or h(an)>[K:Q]−1⋅log⁡(n)+O(1) for n in a set of positive upper density and this is best possible when K=Q.

A height gap theorem for coefficients of Mahler functions

We study the asymptotic growth of coefficients of Mahler power series with algebraic coefficients, as measured by their logarithmic Weil height. We show that there are five different growth behaviors, all of which being reached. Thus, there are gaps in the possible growths. In proving this height gap theorem, we find that a k -Mahler function is k -regular if and only if its coefficients have height in O(\log n) . Moreover, we deduce that, over an arbitrary ground field of characteristic 0, a k -Mahler function is k -automatic if and only if its coefficients belong to a finite set. As a by-product of our results, we also recover a conjecture of Becker which was recently settled by Bell, Chyzak, Coons, and Dumas.

Northcott numbers for the house and the Weil height

For an algebraic number α $\alpha$ and γ ∈ R $\gamma \in \mathbb {R}$ , let be the house, h ( α ) $h(\alpha )$ be the (logarithmic) Weil height, and h γ ( α ) = ( deg α ) γ h ( α ) $h_\gamma (\alpha )=(\deg \alpha )^\gamma h(\alpha )$ be the γ $\gamma$ -weighted (logarithmic) Weil height of α $\alpha$ . Let f : Q ¯ → [ 0 , ∞ ) $f:\overline{\mathbb {Q}}\rightarrow [0,\infty )$ be a function on the algebraic numbers Q ¯ $\overline{\mathbb {Q}}$ , and let S ⊂ Q ¯ $S\subset \overline{\mathbb {Q}}$ . The Northcott number N f ( S ) $\mathcal {N}_f(S)$ of S $S$ , with respect to f $f$ , is the infimum of all X ⩾ 0 $X\geqslant 0$ such that { α ∈ S ; f ( α ) < X } $\lbrace \alpha \in S; f(\alpha )< X\rbrace$ is infinite. This paper studies the set of Northcott numbers N f ( O ) $\mathcal {N}_f({\mathcal {O}})$ for subrings of Q ¯ $\overline{\mathbb {Q}}$ for the house, the Weil height, and the γ $\gamma$ -weighted Weil height. We show: (1) Every t ⩾ 1 $t\geqslant 1$ is the Northcott number of a ring of integers of a field w.r.t. the house . (2) For each t ⩾ 0 $t\geqslant 0$ , there exists a field with Northcott number in [ t , 2 t ] $ [t,2t]$ w.r.t. the Weil height h ( · ) $h(\cdot )$ . (3) For all 0 ⩽ γ ⩽ 1 $0\leqslant \gamma \leqslant 1$ and γ ′ < γ $\gamma ^{\prime }<\gamma$ , there exists a field K $K$ with N h γ ′ ( K ) = 0 $\mathcal {N}_{h_{\gamma ^{\prime }}}(K)=0$ and N h γ ( K ) = ∞ $\mathcal {N}_{h_\gamma }(K)=\infty$ . For (1) we provide examples that satisfy an analogue of Julia Robinson's property (JR), examples that satisfy an analogue of Vidaux and Videla's isolation property, and examples that satisfy neither of those. Item (2) concerns a question raised by Vidaux and Videla due to its direct link with decidability theory via the Julia Robinson number. Item (3) is a strong generalisation of the known fact that there are fields that satisfy the Lehmer conjecture but which are not Bogomolov in the sense of Bombieri and Zannier.

Lower bounds for Galois orbits of special points on Shimura varieties: a point-counting approach

Let S be a Shimura variety. We conjecture that the heights of special points in \(S(\overline{\mathbb {Q}})\) are discriminant negligible with respect to some Weil height function \(h:S(\overline{\mathbb {Q}})\rightarrow \mathbb {R}\). Assuming this conjecture to be true, we prove that the sizes of the Galois orbits of special points grow as a fixed power of their discriminant (an invariant we will define in the text). In particular, we give a new proof of a theorem of Tsimerman on lower bounds for Galois degrees of special points in Shimura varieties of abelian type. This gives a new proof of the André–Oort conjecture for such varieties that avoids the use of Masser–Wüstholz isogeny estimates, replacing them by a point-counting argument.

Upper bounds on the heights of polynomials and rational fractions from their values

Let $F$ be a univariate polynomial or rational fraction of degree $d$ defined over a number field. We give bounds from above on the absolute logarithmic Weil height of $F$ in terms of the heights of its values at small integers: we review well-known bounds obtained from interpolation algorithms given values at $d+1$ (resp. $2d+1$) points, and obtain tighter results when considering a larger number of evaluation points.

Counting rational points on elliptic curves with a rational 2-torsion point

Let E/\mathbb{Q} be an elliptic curve over the rational numbers. It is known, by the work of Bombieri and Zannier, that if E has full rational 2-torsion, the number N_E(B) of rational points with Weil height bounded by B is \operatorname{exp}\big(O \big(\frac{\operatorname{log}B}{\sqrt{\operatorname{log}\operatorname{log} B}}\big)\big) . In this paper we exploit the method of descent via 2-isogeny to extend this result to elliptic curves with just one nontrivial rational 2-torsion point. Moreover, we make use of a result of Petsche to derive the stronger upper bound N_E(B) = \operatorname{exp}\big(O \big(\frac{\operatorname{log}B}{\sqrt{\operatorname{log}\operatorname{log} B}}\big)\big) for these curves and to remove a deep transcendence theory ingredient from the proof.

On the frequency of height values

We count algebraic numbers of fixed degree d and fixed (absolute multiplicative Weil) height {mathcal {H}} with precisely k conjugates that lie inside the open unit disk. We also count the number of values up to {mathcal {H}} that the height assumes on algebraic numbers of degree d with precisely k conjugates that lie inside the open unit disk. For both counts, we do not obtain an asymptotic, but only a rough order of growth, which arises from an asymptotic for the logarithm of the counting function; for the first count, even this rough order of growth exists only if k in {0,d} or gcd (k,d) = 1. We therefore study the behaviour in the case where 0< k < d and gcd (k,d) > 1 in more detail. We also count integer polynomials of fixed degree and fixed Mahler measure with a fixed number of complex zeroes inside the open unit disk (counted with multiplicities) and study the dynamical behaviour of the height function.

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

AbstractWe 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.

Multiplicative approximation by the Weil height

Let $K/\mathbb{Q}$ be an algebraic extension of fields, and let $\alpha \not= 0$ be contained in an algebraic closure of $K$. If $\alpha$ can be approximated by roots of numbers in $K^{\times}$ with respect to the Weil height, we prove that some nonzero integer power of $\alpha$ must belong to $K^{\times}$. More generally, let $K_1, K_2, \dots , K_N$, be algebraic extensions of $\mathb{Q}$ such that each pair of extensions includes one which is a (possibly infinite) Galois extension of a common subfield. If $\alpha \not= 0$ can be approximated by a product of roots of numbers from each $K_n$ with respect to the Weil height, we prove that some nonzero integer power of $\alpha$ must belong to the multiplicative group $K_1^{\times} K_2^{\times} \cdots K_N^{\times}$. Our proof of the more general result uses methods from functional analysis.

Positive semigroups and generalized Frobenius numbers over totally real number fields

Frobenius problem and its many generalizations have been extensively studied in several areas of mathematics. We study semigroups of totally positive algebraic integers in totally real number fields, defining analogues of the Frobenius numbers in this context. We use a geometric framework recently introduced by Aliev, De Loera and Louveaux to produce upper bounds on these Frobenius numbers in terms of a certain height function. We discuss some properties of this function, relating it to absolute Weil height and obtaining a lower bound in the spirit of Lehmer's conjecture for algebraic vectors satisfying some special conditions. We also use a result of Borosh and Treybig to obtain bounds on the size of representations and number of elements of bounded height in such positive semigroups of totally real algebraic integers.

On upper bounds of arithmetic degrees

Let $X$ be a smooth projective variety over $ \overline{\mathbb Q}$, and $f:X -rightarrow X$ be a dominant rational map. Let $\delta_{f}$ be the first dynamical degree of $f$ and $h_{X}:X( \overline{\mathbb Q})\to [1,\infty)$ be a Weil height function on $X$ associated with an ample divisor on $X$. We prove several inequalities which give upper bounds of the sequence $(h_X (f^n(P)))_{n\geq0}$ where $P$ is a point of $X( \overline{\mathbb Q})$ whose forward orbit by $f$ is well-defined. As a corollary, we prove that the upper arithmetic degree is less than or equal to the first dynamical degree; $ \overline{\alpha}_{f}(P) \leq \delta_{f}$. Furthermore, if the Picard number of $X$ is one, $f$ is algebraically stable and $\delta_{f}>1$, we prove that the limit defining canonical height $\lim_{n\to \infty} h_{X} (f^{n}(P)) \big/ \delta_f^n$ converges.

Dynamical Uniform Bounds for Fibers and a Gap Conjecture

AbstractWe 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 $\limsup _{n\to \infty } h(f(\Phi ^n(x)))/\log (n)&amp;gt;0$, where $h(\cdot )$ represents the usual logarithmic Weil height for algebraic points.

A Weil Banach Algebra for Multiplicative Algebraic Numbers

We present a generalization of the construction of Allcock and Vaaler in [AV09] which established that the Weil height provides non-zero algebraic numbers modulo torsion with the structure of a normed vector space and construct its completion as L1 of a locally compact measure space. In the present work we show that this space is naturally a Banach algebra and describe its structure and a method for computing the Banach multiplication explicitly using the theory of harmonic analysis on compact groups.

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.

The critical height is a moduli height

Silverman defined the critical height of a rational function f(z) of degree d≥2 in terms of the asymptotic rate of growth of the Weil height along the critical orbits of f. He also conjectured that this quantity was commensurate to an ample Weil height on the moduli space of rational functions degree d. We prove that conjecture.