Canonical Height Pairings via Biextensions

1983 ◽  
pp. 195-237 ◽  
Author(s):  
Barry Mazur ◽  
John Tate
Keyword(s):  
Canonical Height

scholarly journals Entropy and the Canonical Height

2001 ◽  
Vol 91 (2) ◽  
pp. 256-273 ◽  
Author(s):  
M Einsiedler ◽  
G Everest ◽  
T Ward
Keyword(s):  
Canonical Height

scholarly journals On the complex dynamics of birational surface maps defined over number fields

2016 ◽  
Vol 0 (0) ◽  
Author(s):  
Mattias Jonsson ◽  
Paul Reschke
Keyword(s):  
Number Field ◽  
Complex Dynamics ◽  
Energy Condition ◽  
Height Function ◽  
Number Fields ◽  
Projective Surface ◽  
Complex Projective Surface ◽  
Canonical Height ◽  
Dynamical Degree ◽  
Complex Projective

AbstractWe show that any birational selfmap of a complex projective surface that has dynamical degree greater than one and is defined over a number field automatically satisfies the Bedford–Diller energy condition after a suitable birational conjugacy. As a consequence, the complex dynamics of the map is well behaved. We also show that there is a well-defined canonical height function.

scholarly journals Points of low height on elliptic surfaces with torsion

2010 ◽  
Vol 13 ◽  
pp. 370-387
Author(s):  
Sonal Jain
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Open Subset ◽  
Function Field ◽  
Rational Point ◽  
Torsion Point ◽  
Integral Points ◽  
Elliptic Surfaces ◽  
Canonical Height

AbstractWe determine the smallest possible canonical height$\hat {h}(P)$for a non-torsion pointPof an elliptic curveEover a function field(t) of discriminant degree 12nwith a 2-torsion point forn=1,2,3, and with a 3-torsion point forn=1,2. For eachm=2,3, we parametrize the set of triples (E,P,T) of an elliptic curveE/with a rational pointPandm-torsion pointTthat satisfy certain integrality conditions by an open subset of2. We recover explicit equations for all elliptic surfaces (E,P,T) attaining each minimum by locating them as curves in our projective models. We also prove that forn=1,2 , these heights are minimal for elliptic curves over a function field of any genus. In each case, the optimal (E,P,T) are characterized by their patterns of integral points.

scholarly journals Canonical Heights on Hyper-Kähler Varieties and the Kawaguchi–Silverman Conjecture

2019 ◽  
Author(s):  
John Lesieutre ◽  
Matthew Satriano
Keyword(s):  
Projective Variety ◽  
Kähler Manifold ◽  
Height Function ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Dense Orbit ◽  
Canonical Height ◽  
Projective Threefolds ◽  
Higher Dimensional ◽  
Dynamical Degrees

Abstract The Kawaguchi–Silverman conjecture predicts that if $f: X \dashrightarrow X$ is a dominant rational-self map of a projective variety over $\overline{{\mathbb{Q}}}$, and $P$ is a $\overline{{\mathbb{Q}}}$-point of $X$ with a Zariski dense orbit, then the dynamical and arithmetic degrees of $f$ coincide: $\lambda _1(f) = \alpha _f(P)$. We prove this conjecture in several higher-dimensional settings, including all endomorphisms of non-uniruled smooth projective threefolds with degree larger than $1$, and all endomorphisms of hyper-Kähler manifolds in any dimension. In the latter case, we construct a canonical height function associated with any automorphism $f: X \to X$ of a hyper-Kähler manifold defined over $\overline{{\mathbb{Q}}}$. We additionally obtain results on the periodic subvarieties of automorphisms for which the dynamical degrees are as large as possible subject to log concavity.

scholarly journals Rationality of dynamical canonical height

2018 ◽  
Vol 39 (9) ◽  
pp. 2507-2540
Author(s):  
LAURA DE MARCO ◽  
DRAGOS GHIOCA
Keyword(s):  
Elliptic Curve ◽  
Function Field ◽  
Rational Maps ◽  
Puiseux Series ◽  
Quadratic Maps ◽  
Canonical Height ◽  
Complete Answer ◽  
Irrational Values ◽  
Invent Math

We present a dynamical proof of the well-known fact that the Néron–Tate canonical height (and its local counterpart) takes rational values at points of an elliptic curve over a function field $k=\mathbb{C}(X)$, where $X$ is a curve. More generally, we investigate the mechanism by which the local canonical height for a map $f:\mathbb{P}^{1}\rightarrow \mathbb{P}^{1}$ defined over a function field $k$ can take irrational values (at points in a local completion of $k$), providing examples in all degrees $\deg f\geq 2$. Building on Kiwi’s classification of non-archimedean Julia sets for quadratic maps [Puiseux series dynamics of quadratic rational maps. Israel J. Math.201 (2014), 631–700], we give a complete answer in degree 2 characterizing the existence of points with irrational local canonical heights. As an application we prove that if the heights $\widehat{h}_{f}(a),\widehat{h}_{g}(b)$ are rational and positive, for maps $f$ and $g$ of multiplicatively independent degrees and points $a,b\in \mathbb{P}^{1}(\bar{k})$, then the orbits $\{f^{n}(a)\}_{n\geq 0}$ and $\{g^{m}(b)\}_{m\geq 0}$ intersect in at most finitely many points, complementing the results of Ghioca et al [Intersections of polynomials orbits, and a dynamical Mordell–Lang conjecture. Invent. Math.171 (2) (2008), 463–483].

scholarly journals Canonical heights and division polynomials

2014 ◽  
Vol 157 (2) ◽  
pp. 357-373 ◽  
Author(s):  
ROBIN de JONG ◽  
J. STEFFEN MÜLLER
Keyword(s):  
Recurrence Relation ◽  
Number Field ◽  
Diophantine Approximation ◽  
New Method ◽  
Approximation Result ◽  
Algebraic Point ◽  
Canonical Height ◽  
Division Polynomials ◽  
Genus 2 ◽  
Hyperelliptic Jacobians

AbstractWe 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, neitherp-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.

scholarly journals The canonical height and elliptic surfaces

1990 ◽  
Vol 36 (2) ◽  
pp. 201-211 ◽  
Author(s):  
Masato Kuwata
Keyword(s):  
Elliptic Surfaces ◽  
Canonical Height
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