We discuss a new method to compute the canonical height of an algebraic point on a hyperelliptic jacobian over a number field. The method does not require any geometrical models, neither $p$-adic nor complex analytic ones. In the case of genus 2 we also present a version that requires no factorisation at all. The method is based on a recurrence relation for the `division polynomials' associated to hyperelliptic jacobians, and a diophantine approximation result due to Faltings.
In this article we prove some interesting results on field generated by division points of several formal groups of same height, already implicit in the treatment in appendix-A of my M.Sc thesis (Points of Small Height in Certain Nonabelian Extensions). More precisely, under suitable hypothesis we show that the field generated by the division points of several formal groups is equal to the field generated by division points of an individual formal group among them, after a fixed finite unramified extension of base field.
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 canonical height associated to a polarized endomporhism of a projective variety, constructed by Call and Silverman and generalizing the Néron-Tate height on a polarized Abelian variety, plays an important role in the arithmetic theory of dynamical systems. We generalize this construction to polarized correspondences, prove various fundamental properties, and show how the global canonical height decomposes as an integral of a local height over the space of absolute values on the algebraic closure of the field of definition.
Let K be a number field, X/K a curve, and f/X a family of endomorphisms of projective N-space. It follows from a result of Call and Silverman that the canonical height associated to the family f, evaluated along a section, differs from a Weil height on the base by little-o of a Weil height. In the case where f is a family with an invariant hyperplane, whose restriction to this invariant hyperplane is isotrivial, we improve this by showing that the canonical height along a section differs from a Weil height on the base by a bounded amount.
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 above lower bound, we show that the explicit rational point is primitive as an element of the Mordell--Weil group.
Let $d\ge 2$ be an integer, let $c(t)$ be any rational map, and let $f_t(z) := (z^d+t)/z$ be a family of rational maps indexed by t. For each algebraic number $t$, we let $h_{f_t}(c(t))$ be the canonical height of $c(t)$ with respect to the rational map $f_t$. We prove that the map $H(t):=h_{f_t}(c(t))$ (as $t$ varies among the algebraic numbers) is a Weil height.
Given a family of abelian varieties over a quasiprojective smooth curve $T^0$ over a global field and a point $P$ on the generic fiber, we show that the Néron-Tate canonical height $h_{X_t}(P_t)$ of $P_t$ along each fiber is exactly equal to a Weil height $h_{\overline M}(t)$ given by an adelic metrized line bundle $\overline M$ on the unique smooth projective curve $T$ containing $T^0$. As a consequence, we show that a conjecture of Zhang on the finiteness of small-height specializations of $P$ is equivalent to $\overline M$ being big.
We adopt a statistical point of view on the conjecture of Lang which predicts a lower bound for the canonical height of non-torsion rational points on elliptic curves defined over $\mathbb{Q}$. More specifically, we prove that among the family of all elliptic curves defined over $\mathbb{Q}$ and having positive rank, there is a density one subfamily of curves which satisfy a strong form of Lang's conjecture.
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 naive height into an archimedean and a non-archimedean term. Our main contribution is an algorithm for the computation of the non-archimedean term that requires no integer factorization and runs in quasi-linear time.
In a recent breakthrough, Dimitrov solved the Schinzel-Zassenhaus Conjecture. We follow his approach and adapt it to certain dynamical systems arising from polynomials of the form $T^p+c$ where $p$ is a prime number and where the orbit of $0$ is finite. For example, if $p=2$, and $0$ is periodic under $T^2+c$ with $c\in\mathbb{R}\smallsetminus\{-2\}$, we prove a lower bound for the local canonical height of a wandering algebraic integer that is inversely proportional to the field degree. From this we are able to deduce a lower bound for the canonical height of a wandering point that decays lik…
We show that the canonical height function defined by Silverman does not have the Northcott finiteness property in general. We develop a new canonical height function for monomial maps. In certain cases, this new canonical height function has nice properties.
Denis associated to each Drinfeld module M over a global function function field L a canonical height function, which plays a role analogous to that of the Neron-Tate height in the context of elliptic curves. We prove that there exist constants ε>0 and C, depending only on the number of places at which M has bad reduction, such that either x in M is a torsion point of bounded order, or else the canonical height of x is bound below by εmax{h(j_M), deg(D_M)}, where j_M is a certain invariant of the isomorphism class of M, and D_M is the minimal discriminant of M. As an application, we make some …
Let X be a smooth projective curve of positive genus defined over a number field K. Assume given a Galois covering map x from X to the projective line over K and a place v of K. We introduce a local canonical height on the set of K_v-valued points of X associated to x as an integral with logarithmic integrand, generalizing Tate's local Neron function on an elliptic curve. The resulting global height can be viewed as a 'Mahler measure' associated to x. We prove that the local canonical height can be obtained by averaging, and taking a limit, over divisors of higher order Weierstrass points on X…
Assuming Lang's conjectured lower bound on the heights of non-torsion points on an elliptic curve, we show that there exists an absolute constant C such that for any elliptic curve E/Q and non-torsion point P in E(Q), there is at most one integral multiple [n]P such that n > C. The proof is a modification of a proof of Ingram giving an unconditional but not uniform bound. The new ingredient is a collection of explicit formulae for the sequence of valuations of the division polynomials. For P of non-singular reduction, such sequences are already well described in most cases, but for P of singul…
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 height and for which the fibre entropy is accounted for by local heights at primes of bad reduction. The proofs use transcendence theory and a strong form of Siegel's theorem. We go on to extend these ideas to the morphic heights of Call and Goldstine.
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 the classical Neron-Tate height and analytic p-adic heights. Our main result is that under some general hypotheses this pairing is non-degenerate.
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 over Q. We also prove that the canonical height of a subvariety is zero if and only if it is a translation of an abelian subvariety by a torsion point.
We construct canonical heights of subvarieties for dynamical system of several morphisms associated with line bundles defined over a number field, and study some of their properties. We also construct invariant currents for such systems over $\mathbb{C}$.
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 the number of places of bad reduction.
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 $ϕ: \mathbb{P}_{\mathbb{Q}}^1 \rightarrow \mathbb{P}_{\mathbb{Q}}^1$ of degree $d \geq 2$.
We construct height functions defined stochastically on projective varieties equipped with endomorphisms, and we prove that these functions satisfy analogs of the usual properties of canonical heights. Moreover, we give a dynamical interpretation of the kernel of these stochastic height functions, and in the case of the projective line, we relate the size of this kernel to the Julia sets of the original maps. Finally, as an application, we establish the finiteness of some generalized Zsigmondy sets over global fields.
We study the interplay between canonical heights and endomorphisms of an abelian variety $A$ over a number field $k$. In particular we show that whenever the ring of endomorphisms defined over $k$ is strictly larger than $\Z$ there will be $\Q$-linear relations among the values of a canonical height-pairing evaluated at a basis modulo torsion of $A(k)$.
For $E_{b}: y^{2}=x^{3}+b$, we establish Lang's conjecture on a lower bound for the canonical height of non-torsion points along with upper and lower bounds for the difference between the canonical and logarithmic height. In many cases, our results are actually best-possible.
An expository lecture on the analogy between the subjects of the title. Delivered at the International Conference on Number Theory at the Korea Institute for Advanced Study in December 1997.
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 than the first dynamical degree. The proof uses the existence, proved by Favre and the first author, of certain compactifications of the plane adapted to the dynamics.
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 canonical height of non-torsion points on A defined over the maximal abelian extension K^ab of K.
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 using arithmetic intersection theory on a regular model of the curve in practice. In the case of hyperelliptic curves we present a complete algorithm that has been implemented in Magma. Several examples are computed and the behavior of the running time is discussed.
Let $f: \mathbb{A}^2 \to \mathbb{A}^2$ be a polynomial automorphism of dynamical degree $δ\geq 2$ over a number field $K$. (This is equivalent to say that $f$ is a polynomial automorphism that is not triangularizable.) Then we construct canonical height functions defined on $\mathbb{A}^2(\bar{K})$ associated with $f$. These functions satisfy the Northcott finiteness property, and an $\bar{K}$-valued point on $\mathbb{A}^2(\bar{K})$ is $f$-periodic if and only if its height is zero. As an application of canonical height functions, we give an estimate on the number of points with bounded height …
Call and Silverman introduced the canonical height associated to a polarized dynamical system, that is, an endomorphism of a projective variety and an ample line bundle which pulls back to a tensor power of itself. They also presented an asymptotic for the variation of this height in a family over a one-dimensional base in terms of the height on the generic fibre and the height of the parameter. Here we improve this asymptotic, saving a power in the error term. As a corollary, we give an explicit bound on the height of parameters at which the dynamical system specialized to a finite orbit, in …
For elliptic curves given by the equation $E_{a}: y^{2}=x^{3}+ax$, we establish the best-possible version of Lang's conjecture on the lower bound of the canonical height of non-torsion points along with best-possible upper and lower bounds for the difference between the canonical and logarithmic height.
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 weighted homogeneous. We prove a lower bound on the canonical height, of the form \[\hat{h}_{f_t}(z)\geq ε\max\{h_{\mathsf{M}_d}(f_t), \log|\operatorname{Norm}\mathfrak{R}_{f_t}|\},\] for values $z\in K$ which are not preperiodic for $f_t$. Here $ε$ depends only on the number of places at which $f_t$ has bad reduction. For suitably generic morphisms $φ:\math…
Let E/K be an ellptic curve defined over a number field, let h be the canonical height on E, and let K^ab be the maximal abelian extension of K. Extending work of M. Baker, we prove that there is a positive constant C(E/K) so that every nontorsion point P in E(K^ab) satisfies h(P) > C(E/K).
We prove, for the canonical height defined by Silverman [15] on monomial maps, the existence of effective lower bounds for heights of points with Zariski dense orbit, for cases with endomorphisms induced by matrices with real Jordan form.
Let phi(z) be a polynomial of degree at least 2 with coefficients in a number field K. Iterating phi gives rise to a dynamical system and a corresponding canonical height function, as defined by Call and Silverman. We prove a simple product formula relating the transfinite diameters of the filled Julia sets of phi over various completions of K, and we apply this formula to give a generalization of Bilu's equidistribution theorem for sequences of points whose canonical heights tend to zero.
Let $k$ be a number field, suppose that $B$ is a central simple division algebra over $k$, and choose any maximal order $\mathcal{D}$ of $B$. The object of this paper is to show that the group $\mathcal{D}_S^*$ of $S$-units of $B$ is generated by elements of small height once $S$ contains an explicit finite set of places of $k$. This generalizes a theorem of H.\ W.\ Lenstra Jr., who proved such a result when $B = k$. Our height bound is an explicit function of the number field and the discriminant of a maximal order in $B$ used to define its $S$-units.
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…
Given an elliptic curve E over a function field K=Q(T_1,...,T_n), we study the behavior of the canonical height ^h_(E_w) of the specialized elliptic curve E_w with respect to the height of w in Q^n. In this paper, we prove that there exists a uniform non-zero lower bound for the average of the quotient ^h_(E_w)(P_w)/h(w) for all non-torsion P in E(K).
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 consider the arithmetic of Henon maps f(x, y)=(ay, x+f(y)) defined over number fields and function fields, usually with the restriction that a=1. We prove a result on the variation of Kawaguchi's canonical height in families of Henon maps, and derive from this a specialization theorem, showing that the set of parameters above which a given non-periodic point becomes periodic is a set of bounded height. Proving this involves showing that the only points of canonical height zero for a Henon map over a function field are those which are periodic (in the non-isotrivial case). In the case of qua…
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.
In an earlier paper by the first author, an argument for the nonexistence of canonical vector heights on K3 surfaces of Picard number three was given, based on an explicit surface that was not proved to have Picard number three. In this paper, we fill the gap in the argument by redoing the computations for another explicit surface for which we prove that the Picard number equals three. The conclusion remains unchanged.
We prove that if $f$ is a polynomial over a number field $K$ with a finite superattracting periodic point and a non-archimedean place of bad reduction, then there is an $ε>0$ such that only finitely many $P\in K^{\text{ab}}$ have canonical height less than $ε$ with respect to $f$. The key ingredient is the geometry of the filled Julia set at a place of bad reduction. We also prove a conditional uniform boundedness result for the $K$-rational preperiodic points of such polynomials, as well as a uniform lower bound on the canonical height of non-preperiodic points in $K$. We further prove uncond…
Let f: A^N \to A^N be a regular polynomial automorphism defined over a number field K. For each place v of K, we construct the v-adic Green functions G_{f,v} and G_{f^{-1},v} (i.e., the v-adic canonical height functions) for f and f^{-1}. Next we introduce for f the notion of good reduction at v, and using this notion, we show that the sum of v-adic Green functions over all v gives rise to a canonical height function for f that satisfies the Northcott-type finiteness property. Using previous results, we recover results on arithmetic properties of f-periodic points and non f-periodic points. We…
Let K be a number field and let E/K be an elliptic curve. If E has complex multiplication, we show that there is a positive lower bound for the canonical height of non-torsion points on E defined over the maximal abelian extension K^ab of K. This is analogous to results of Amoroso-Dvornicich and Amoroso-Zannier for the multiplicative group. We also show that if E has non-integral j-invariant (so that in particular E does not have complex multiplication), then there exists C > 0 such that there are only finitely many points P in E(K^ab) of canonical height less than C. This strengthens a result…
Let p be a rational prime and K/Q_p be an extension of complete discrete valuation fields. Let G be a truncated Barsotti-Tate group of level n, height h and dimension d over O_K with 0<d<h. In this paper, we prove the existence of higher canonical subgroups with expected properties for G if the Hodge height of G is less than 1/(p^{n-2}(p+1)), including the case of p=2.
We prove that a positive proportion of squarefree integers are congruent numbers such that the canonical height of the lowest non-torsion rational point on the corresponding elliptic curve satisfies a strong lower bound.
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 of divisors modulo numerical equivalence, the arithmetic degree at a rational point is equal to one if and only if it is preperiodic under the endomorphism.
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. These canonical heights are the sum of some height coming from the abelian variety and something we call a relative height. We finally give some complements about the points whose relative height is zero.
We study families of varieties endowed with polarized canonical eigensystems of several maps, inducing canonical heights on the dominating variety as well as on the "good" fibers of the family. We show explicitely the dependence on the parameter for global and local canonical heights defined by Kawaguchi when the fibers change, extending previous works of J. Silverman and others. Finally, fixing an absolute value $v \in K$ and a variety $V/K$, we descript the Kawaguchi`s canonical local height $\hatλ_{V,E,\mathcal{Q},}(.,v)$ as an intersection number, provided that the polarized system $(V,\ma…
