Related papers: Canonical heights and division polynomials

We give an algorithm which requires no integer factorization for computing the canonical height of a point in $\mathbb{P}^1(\mathbb{Q})$ relative to a morphism $\phi: \mathbb{P}_{\mathbb{Q}}^1 \rightarrow \mathbb{P}_{\mathbb{Q}}^1$ of…

Let $K$ be a number field and let $C/K$ be a curve of genus 2 with Jacobian variety $J$. In this paper, we study the canonical height $\hat{h} \colon J(K) \to \mathbb R$. More specifically, we consider the following two problems, which are…

For several applications in the arithmetic of abelian varieties it is important to compute canonical heights. Following Faltings and Hriljac, we show how the canonical height on the Jacobian of a smooth projective curve can be computed…

We introduce an algorithm that can be used to compute the canonical height of a point on an elliptic curve over the rationals in quasi-linear time. As in most previous algorithms, we decompose the difference between the canonical and the…

We define a new canonical height pairing on the rational points of elliptic curves over global function fields which takes values in the multiplicative group of a completion of the function field. This height serves as an analogue of both…

We prove a lower bound on the canonical height associated to polynomials over number fields evaluated at points with infinite forward orbit. The lower bound depends only on the degree of the polynomial, the degree of the number field, and…

Canonical heights and Arakelov geometry on semi-abelian varieties. In this paper, we propose a construction of the canonical heights on an extension of an abelian variety by the multiplicative group, in the framework of Arakelov geometry.…

Let $f: \mathbb{A}^2 \to \mathbb{A}^2$ be a polynomial automorphism of dynamical degree $\delta \geq 2$ over a number field $K$. (This is equivalent to say that $f$ is a polynomial automorphism that is not triangularizable.) Then we…

This paper is the sequel of our paper "Arithmetic height functions over finitely generated fields" (cf. math.NT/9809016). In this paper, we define the canonical height of subvarieties of an abelian variety over a finitely generated field…

A practical version of the polynomial canonical formalism is developed for normal mesoscopic systems consisting of N independent electrons. Drastic simplification of calculations is attained by means of proper ordering excited states of the…

We use Arakelov theory to define a height on divisors of degree zero on a hyperelliptic curve over a global field, and show that this height has computably bounded difference from the N\'eron-Tate height of the corresponding point on the…

We introduce a new canonical height function for Jordan blocks of small eigenvalues for endomorphisms on smooth projective varieties over a number field. We prove that under an assumption on the eigenvalues of the endomorphism on the group…

Let A be an abelian variety defined over a number field K, and consider the canonical height function attached to a symmetric ample line bundle L on A. We prove that there is a positive lower bound C (depending on A, K, and L) for the…

We study canonical heights for plane polynomial mappings of small topological degree. In particular, we prove that for points of canonical height zero, the arithmetic degree is bounded by the topological degree and hence strictly smaller…

An approach to the calculation of local canonical morphic heights is described, motivated by the analogy between the classical height in Diophantine geometry and entropy in algebraic dynamics. We consider cases where the local morphic…

A system of transformations is associated to a rational point on an elliptic curve. The sequence entropy is connected to the canonical height, and in some cases there is a canonically defined quotient system whose entropy is the canonical…

A new proof is given for the explicit formulae for the non-archimedean canonical height on an elliptic curve. This arises as a direct calculation of the Haar integral in the elliptic Jensen formula.

We compute a lower bound of the canonical height on quadratic twists of certain elliptic curves. Also we show a simple method for constructing families of quadratic twists with an explicit rational point. % from cubic polynomials. Using the…

A family $f_t(z)$ of polynomials over a number field $K$ will be called \emph{weighted homogeneous} if and only if $f_t(z)=F(z^e, t)$ for some binary homogeneous form $F(X, Y)$ and some integer $e\geq 2$. For example, the family $z^d+t$ is…

Let $K$ be a field of characteristic $p \neq 2$, and let $f(x)$ be a sextic polynomial irreducible over $K$ with no repeated roots, whose Galois group is isomorphic to $\A_5$. If the jacobian $J(C)$ of the hyperelliptic curve $C:y^2=f(x)$…