The height of a polynomial with integer coefficients is the largest coefficient in absolute value. Many papers have been written on the subject of bounding heights of cyclotomic polynomials. One result, due to H. Maier, gives a best possible upper bound of n^{ψ(n)} for almost all n, where ψ(n) is any function that approaches infinity as n tends to infinity. We will discuss the related problem of bounding the maximal height over all polynomial divisors of x^n - 1 and give an analogue of Maier's result in this scenario.
Given an endomorphism f of projective space, we exhibit explicit bounds on the difference between the naive height of a divisor and its canonical height relative to f.
The height of a polynomial $f(x)$ is the largest coefficient of $f(x)$ in absolute value. Let B(n) be the largest height of a polynomial in $\mathbb{Z}[x]$ dividing $x^n-1$. In this paper we investigate the maximal height of divisors of $x^{pq^b}-1$ and prove that some conjectures on the maximal height of divisors of $x^{pq^b}-1$ are true.
We introduce the weighted greatest common divisor of a tuple of integers and explore some of it basic properties. Furthermore, for a set of heights $\mathfrak w=(q_0, \ldots , q_n)$, we use the concept of the weighted greatest common divisor to define a height $\mathfrak{h} (\mathfrak p)$ on weighted projective spaces $\mathbb{WP}_{\mathfrak w}^n (k)$. We prove some of the basic properties of this weighted height, including an analogue of the Northcott's theorem for heights on projective spaces.
Define B(n) to be the largest height of a polynomial in $\mathbb{Z}[x]$ dividing $x^n-1$. We formulate a number of conjectures related to the value of B(n) when $n$ is of a prescribed form. Additionally, we prove a lower bound for $B(p^aq^b)$ where $p,q$ are distinct primes.
We prove a formula, which, given a principally polarized abelian variety $(A,λ)$ over the field of algebraic numbers, relates the stable Faltings height of $A$ with the Néron--Tate height of a symmetric theta divisor on $A$. Our formula completes earlier results due to Bost, Hindry, Autissier and Wagener. The local non-archimedean terms in our formula can be expressed as the tropical moments of the tropicalizations of $(A,λ)$.
In this paper, we study properties of Weil height functions associated with numerically trivial divisors. It helps us to define the fractional limit of $h_E$ with respect to $h_D$ on $U$, with $D$ ample: \[ \Flim_D(E,U) := \liminf_{\substack{P \in U h_D(P) \rightarrow \infty}}\dfrac{h_E(P)}{h_D(P)}. \] The value of $\Flim_D(E,U)$ contains numerical information about a divisor $E$, enough to determine whether $E$ is ample, numerically effective or pseudo-effective.
We study special cycles on integral models of Shimura varieties associated with unitary similitude groups of signature (n-1,1). We construct an arithmetic theta lift from harmonic Maass forms of weight 2-n to the arithmetic Chow group of the integral model of a unitary Shimura variety, by associating to a harmonic Maass form f a linear combination of Kudla-Rapoport divisors, equipped with the Green function given by the regularized theta lift of f. Our main result is an equality of two complex numbers: (1) the height pairing of the arithmetic theta lift of f against a CM cycle, and (2) the cen…
We define an algebraic analogue, in the case of jacobians of curves, of the height jump divisor introduced recently by R. Hain. We give explicit combinatorial formulae for the height jump for families of semistable curves using labelled reduction graphs. With these techniques we prove a conjecture of Hain on the effectivity of the height jump, and also give a new proof of a theorem of Tate, Silverman and Green on the variation of heights in families of abelian varieties.
When we have a morphism f : P^n -> P^n, then we have an inequality \frac{1}{°f} h(f(P)) +C > h(P) which provides a good upper bound of $h(P)$. However, if $f$ is a rational map, then \frac{1}{°f} h(f(P))+C cannot be an upper bound of h(P). In this paper, we will define the $D$-ratio of a rational map $f$ which will replace the degree of a morphism in the height inequality of h(P).
We introduce the notion of an adelically summable normalized weight $g$, which is a family of normalized weights on the Berkovich projective lines satisfying a summability condition. We then establish an adelic equidistribution of effective $k$-divisors on the projective line over the separable closure $k_s$ in $\overline{k}$ of a product formula field $k$ having small $g$-heights and small diagonals. This equidistribution result generalizes Ye's for the Galois conjugacy classes of algebraic numbers with respect to quasi-adelic measures.
For a function field $K$ and fixed polynomial $F\in K[x]$ and varying $f\in F$ (under certain restrictions) we give a lower bound for the degree of the greatest prime divisor of $F(f)$ in terms of the height of $f$, establishing a strong result for the function field analogue of a classical problem in number theory.
We establish a quantitative adelic equidistribution theorem for a sequence of algebraic zeros divisors on the projective line over the separable closure of a product formula field having small diagonals and small $g$-heights with respect to an adelic normalized weight $g$ in arbitrary characteristic and in possibly non-separable setting, and obtain local proximity estimates between the iterations of a rational function $f\in k(z)$ of degree $>1$ and a rational function $a\in k(z)$ of degree $>0$ over a product formula field $k$ of characteristic $0$, applying this quantitative adelic equidistr…
Assuming Vojta's conjecture, we give a sufficient condition for the limit \[ \lim_{n \to \infty} \frac{h_{Y}(f^{n}(x))}{h_{H}(f^{n}(x))} \] is equal to zero, where $f \colon X \longrightarrow X$ is a surjective self-morphism on a smooth projective variety $X$, $h_{H}$ is an ample height function on $X$, and $h_{Y}$ is a global height function associated with a closed subscheme $Y \subset X$ of codimension at least two. Based on this, we propose a conjecture on a sufficient condition for the limit to be zero. We point out that our conjecture implies Dynamical Mordell-Lang conjecture for endomor…
We address the distribution properties of points of small height on proper toric varieties and applications to the related Bogomolov property. We introduce the notion of monocritical toric metrized divisor and we prove that equidistribution occurs for every generic, small sequence with respect to a toric metrized divisor, for every place if and only if the divisor is monocritical. Furthermore, when this is the case, the limit measure is a translate of the natural measure on the compact torus sitting in the principal orbit of the ambient toric variety. We also study the $v$-adic modulus distrib…
Let $C$ be a genus $2$ hyperelliptic curve over a number field $K$, with a Weierstrass point $\infty$ at infinity, let $J$ be its Jacobian, let $Θ$ be the theta divisor with respect to $\infty$, and let $p$ be any prime number. We give an explicit construction of a $p$-adic height $h_p\colon J(\overline{\mathbb{Q}})\to \mathbb{Q}_p$ by means of $p$-adic analogues of Néron functions of divisor $2Θ$. We define such Néron functions using division polynomials and a generalisation of Blakestad's $p$-adic sigma function on the formal group of $J$. We prove that our $p$-adic Néron function $λ_v$ at a…
We generalize the G.C.D. results of Corvaja--Zannier and Levin on $\mathbb G_m^n$ to more general settings. More specifically, we analyze the height of a closed subscheme of codimension at least $2$ inside an $n$-dimensional Cohen-Macaulay projective variety, and show that this height is small when evaluated at integral points with respect to a divisor $D$ when $D$ is a sum of $n+1$ effective divisors which are all numerically equivalent to some multiples of a fixed ample divisor. Our method is inspired by Silverman's G.C.D. estimate as an application of Vojta's conjecture, which is substitute…
For a cycle of codimension 1 in a toric variety, its degree with respect to a nef toric divisor can be understood in terms of the mixed volume of the polytopes associated to the divisor and to the cycle. We prove here that an analogous combinatorial formula holds in the arithmetic setting: the global height of a 1-codimensional cycle with respect to a toric divisor equipped with a semipositive toric metric can be expressed in terms of mixed integrals of the $v$-adic roof functions associated to the metric and the Legendre-Fenchel dual of the $v$-adic Ronkin function of the Laurent polynomial o…
Let $A_n$ denote the height of cyclotomic polynomial $Φ_n$, where $n$ is a product of $k$ distinct odd primes. We prove that $A_n \le ε_kϕ(n)^{k^{-1}2^{k-1}-1}$ with $-\logε_k\sim c2^k$, $c>0$. The same statement is true for the height $C_n$ of the inverse cyclotomic polynomial $Ψ_n$. Additionally, we improve on a bound of Kaplan for the maximal height of divisors of $x^n-1$, denoted by $B_n$. We show that $B_n<η_k n^{(3^k-1)/(2k)-1}$, with $-\log η_k \sim c3^k$ and the same $c$.
Let X be a projective integral normal scheme over a number field F; let L be a ample line bundle on X together with a semi-positive adelic metric in the sense of Zhang. The main results of this article are 1) A formula which computes the height (relative to L) of a Cartier divisor D on X as generalized "Mahler measures", i.e. by integrating Green functions for D against measures attached to L. 2) a theorem of equidistribution of points of "small" height valid for functions with logarithmic singularities along a divisor D, provided the height of D is "minimal". In the context of algebraic dynam…
Let f : X --> X be an endomorphism of a normal projective variety defined over a global field K, and let D_0,D_1,D_2,... be divisor classes that form a Jordan block with eigenvalue b for the action of f^* on Pic(X) tensored with C. We construct appropriately normalized canonical heights h_0,h_1,h_2,... associated to D_0,D_1,D_2,... and satisfying Jordan transformation formulas h_k(f(x)) = b h_k(x) + h_{k-1}(x). As an application, we prove that for every x in X, the arithmetic degree a_f(x) exists, is an algebraic integer, and takes on only finitely many values as x varies over X. Further, if X…
We initiate a systematic quantitative study of subsets of rational points that are integral with respect to a weighted boundary divisor on Fano orbifolds. We call the points in these sets Campana points. Earlier work of Campana and subsequently Abramovich shows that there are several reasonable competing definitions for Campana points. We use a version that delineates well different types of behaviour of points as the weights on the boundary divisor vary. This prompts a Manin-type conjecture on Fano orbifolds for sets of Campana points that satisfy a klt (Kawamata log terminal) condition. By i…
In this paper we consider the problem of counting algebraic numbers $α$ of fixed degree $n$ and bounded height $Q$ such that the derivative of the minimal polynomial $P_α(x)$ of $α$ is bounded, $|P_α'(α)| < Q^{1-v}$. This problem has many applications to the problems of the metric theory of Diophantine approximation. We prove that the number of $α$ defined above on the interval $\left(-\frac12, \frac12\right)$ doesn't exceed $c_1(n)Q^{n+1-\frac{1}{7}v}$ for $Q>Q_0(n)$ and $1.4 \le v \le \frac{7}{16}(n+1)$. Our result is based on an improvement to the lemma on the order of zero approximation by…
We study the Faltings height pairing of arithmetic Heegner divisors and CM cycles on Shimura varieties associated to orthogonal groups. We compute the Archimedian contribution to the height pairing and derive a conjecture relating the total pairing to the central derivative of a Rankin L-function. We prove the conjecture in certain cases where the Shimura variety has dimension 0, 1, or 2. In particular, we obtain a new proof of the Gross-Zagier formula.
In this note we compute a constant $N$ that bounds the number of non--primitive divisors in elliptic divisibility sequences over function fields of any characteristic. We improve a result of Ingram--Mah{é}--Silverman--Stange--Streng, 2012, and we show that the constant can be chosen independently of the specific point and to some extent of the specific curve, as predicted in loc. cit.
Given a toric metrized R-divisor on a toric variety over a global field, we give a formula for the essential minimum of the associated height function. Under suitable positivity conditions, we also give formulae for all the successive minima. We apply these results to the study, in the toric setting, of the relation between the successive minima and other arithmetic invariants like the height and the arithmetic volume. We also apply our formulae to compute the successive minima for several families of examples, including weighted projective spaces, toric bundles and translates of subtori.
We give a direct proof that the Mazur-Tate and Coleman-Gross heights on elliptic curves coincide. The main ingredient is to extend the Coleman-Gross height to the case of divisors with non-disjoint support and, doing some $p$-adic analysis, show that, in particular, its component above $p$ gives, in the special case of an ordinary elliptic curve, the $p$-adic sigma function. We use this result to give a short proof of a theorem of Kim characterizing integral points on elliptic curves in some cases under weaker assumptions. As a further application, we give new formulas to compute double Colema…
We give a formula for the component at p of the p-adic height pairing of a divisor of degree 0 on a hyperelliptic curve. We use this to give a Chabauty-like method for finding p-adic approximations to p-integral points on such curves when the Mordell-Weil rank of the Jacobian equals the genus. In this case we get an explicit bound for the number of such p-integral points, and we are able to use the method in explicit computation. An important aspect of the method is that it only requires a basis of the Mordell-Weil group tensored with the rationals.
Let f : X --> X be a dominant rational map of a projective variety defined over a global field, let d_f be the dynamical degree of f, and let h_X be a Weil height on X relative to an ample divisor. We prove that h_X(f^n(P)) << (d_f + e)^n h_X(P), where the implied constant depends only on X, h_X, f, and e. As applications, we prove a fundamental inequality a_f(P) \le d_f for the upper arithmetic degree and we construct canonical heights for (nef) divisors. We conjecture that a_f(P) = d_f whenever the orbit of P is Zariski dense, and we describe some cases for which we can prove our conjecture.
A theorem of Tate asserts that, for an elliptic surface E/X defined over a number field k, and a section P of E, there exists a divisor D on X such that the canonical height of the specialization of P to the fibre above t differs from the height of t relative to D by at most a bounded amount. We prove the analogous statement for a one-parameter family of polynomial dynamical systems. Moreover, we compare, at each place of k, the local canonical height with the local contribution to the height relative to D, and show that the difference is analytic near the support of D, a result which is analo…
We give a proof for a fact that for any Weil height $h_X$ with respect to an ample divisor on a projective variety $X$, any dynamical system $\mathcal{F}$ of rational self-maps on $X$, and any $ε>0$, there is a positive constant $C=C(X, h_X, f, ε)$ such that $\sum_{f \in \mathcal{F}_n} h^+_X(f(P)) \leq C. k^n.(δ_{\mathcal{F}} + ε)^n . h^+_X(P)$ for all points $P$ whose $\mathcal{F}$-orbit is well defined, with $δ_{\mathcal{F}}$ being a dynamical degree associated with a system of several maps, defined by the author in the previous paper mentioned above.
