Minhyong Kim
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.
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We show the existence of canonical heights of subvarieties for bounded sequences of morphisms and give some applications.
Canonical local heights and Berkovich skeleta
本文从Berkovich解析空间视角研究非阿基米德域上阿贝尔簇的典范局部高度，细化了Néron关于其与相交重数关系的经典结果，并将Tate的显式公式推广到高维。
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 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 $A(k)$.
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 …
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).
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.
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…
On relative fields of definition for log pairs, Vojta's height inequalities and asymptotic coordinate size dynamics
本文基于相关文献视角，研究射影簇有理点的动力算术复杂度，在渐近坐标大小动力学问题上取得进展，提出对数典范配对的猜想性Vojta型不等式新表述，并引出对数分解存在性的新概念。
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.
Canonical Heights on Shimura Varieties and the André-Oort Conjecture
本文旨在完全一般性地证明 André-Oort 猜想，该猜想涉及 Shimura 簇上特殊子簇的分布。
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.
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.
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…
Explicit Uniform Lower Bounds for the Canonical Height on Elliptic Curves over Abelian Extensions
本文通过显式方法为椭圆曲线在阿贝尔扩张上的诺尔-塔特高度提供了一个不依赖基域判别式的下界，改进了 Baker 的定理证明。
We define an "ample canonical height" for an endomorphism on a projective variety, which is essentially a generalization of the canonical heights for polarized endomorphisms introduced by Call--Silverman. We formulate a dynamical analogue of the Northcott finiteness theorem for ample canonical heights as a conjecture, and prove it for endomorphisms on varieties of small Picard numbers, abelian varieties, and surfaces. As applications, for the endomorphisms which satisfy the conjecture, we show the non-density of the set of preperiodic points over a fixed number field, and obtain a dynamical Mo…
We prove a more general and precise version of the Noether-Fano inequalities for birational maps between Mori fiber spaces. This is applied to give descriptions of global canonical thresholds on Fano varieties of Picard number one.
We obtain all linear Chern number inequalities satisfied by any smooth complete intersection threefold with ample canonical bundle.
We extend the Faltings modular heights of abelian varieties to general arithmetic varieties and show direct relations with the Kahler-Einstein geometry, the Minimal Model Program, heights of Bost and Zhang, and give some applications. Along the way, we propose arithmetic Yau-Tian-Donaldson conjecture, an equivalence of a purely arithmetic property of variety and its metrical property, and partially confirm it.
We present a dynamical proof of the well-known fact that the Neron-Tate canonical height (and its local counterpart) takes rational values at points of an elliptic curve over a function field k of transcendence degree 1 over an algebraically closed field K of characteristic 0. More generally, we investigate the mechanism for which the local canonical height for a rational function f defined over k can take irrational values (at points in a local completion of k), providing examples in all degrees greater than 1. Building on Kiwi's classification of non-archimedean Julia sets for quadratic maps…
We give a mathematical structure on an arithmetic surface, that has algebraic meanings over finite places and can estimate the canonical norm for a relative differential form on the arithmetic surface. This will give a lower bound for the canonical norm for a relative differential form on an arithmetic surface, which proves a Height Inequality on the arithmetic surface.
We construct a quasi-canonical lifting of a $K3$ surface of finite height over a finite field of characteristic $p\geq3$. Such results are previously obtained by Nygaard-Ogus when $p\geq5$. For this purpose, we use the display-theoretic deformation theory developed by Langer, Zink, and Lau. We study the display structure of the crystalline cohomology of deformations of a $K3$ surface of finite height in terms of the Dieudonné display of the enlarged formal Brauer group.
Canonical heights, periods and the Hurwitz zeta function
本文为相对丰富的对数典范线丛引入典范高度，证明其与 K-稳定性及 Kähler-Einstein 度量高度的关系，并用周期极限公式显式计算算术对数曲面等情形，应用于志村曲线整模型和 Arakelov 度量界。
The Kawaguchi--Silverman conjecture predicts that if $f\colon X \dashrightarrow X$ 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 Zariski-dense orbit, then the dynamical and arithmetic degrees of $f$ coincide: $λ_1(f) = α_f(P)$. 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 varieties in any dimension. In the latter case, we construct a canonical…
Let S be a smooth minimal complex projective surface of maximal Albanese dimension. Under the assumption that the canonical class of S is ample and the irregularity of S, q(S), is greater or equal to 5 we show that K^2>= 4χ(S)+(10/3)q(S)-8, thus improving the well known Severi inequality K^2>=4χ(S). We also give stronger inequalities under extra assumptions on the Albanese map or on the canonical map of S.
On Vojta's harder implication, height inequalities for admissible pairs, points of bounded degree and Deligne-Mumford stacks
本研究扩展可容许对概念，证明Vojta的更难蕴含，并针对一般型Deligne-Mumford栈推导出有界次数整点的Bombieri-Lang猜想形式，结合切片定理与Silverman不等式。
Canonical heights for abelian group actions of maximal dynamical rank
针对正熵自由阿贝尔群作用，构造满足Northcott性质的典范高度函数，证明Kawaguchi-Silverman猜想并确定非周期点高度计数。
Variation of canonical heights of subvarieties for polarized endomorphisms
本文将Call和Silverman关于极化自同态下点的典范高度在族中变化的收敛结果，推广到了任意相对维数的子簇族的情形，证明了对应的等价命题。
p-adic adelic metrics and Quadratic Chabauty I
本文提出了一种基于p-进阿雷克洛夫理论的新方法，用于构造abelian簇上的p-进高度，并简化了二次查巴乌方法用于计算有理点。
Gap for geometric canonical height functions
该研究证明复函数域上射影空间自同态的典范高度函数在零附近存在间隙，且零高度有理点Zariski稠密时自同态双有理等平凡，进而推出射影平面的几何Northcott性质。
GCD and ABC inequalities for abelian varieties over function fields
该研究证明了特征0函数域上阿贝尔簇有理点的GCD不等式与几何abc猜想的特殊情形，采用三步证明模式，借助Xie-Yuan高度等工具解决了Buium的相关猜想。
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 important in applications: (1) for a given $P \in J(K)$, compute $\hat{h}(P)$ efficiently; (2) for a given bound $B > 0$, find all $P \in J(K)$ with $\hat{h}(P) \le B$. We develop an algorithm running in polynomial time (and fast in practice) to deal with the first problem. Regarding the second problem, we show how one can tweak the naive height $h$ that is …
Canonical currents and heights for K3 surfaces
本文在K3曲面丰沛锥的扩大边界上构造了自同构群等变的典范正流与高度连续族，并为椭圆纤维化曲面上的高度函数和流构造了优选代表元。
We prove that a conjecture of Fomin, Fulton, Li, and Poon, associated to ordered pairs of partitions, holds for many infinite families of such pairs. We also show that the bounded height case can be reduced to checking that the conjecture holds for a finite number of pairs, for any given height. Moreover, we propose a natural generalization of the conjecture to the case of skew shapes.
We prove a new bound for the Arakelov-Faltings height of an abelian variety over a function field of characteristic zero and relate it to inequalities of ABC-type as conjectured by Buium and Vojta.
Unlikely intersections with CM abelian varieties in a family and explicit bounds for canonical heights under endomorphisms
研究\(\overline{\mathbb{Q}}\)上阿贝尔概型中曲线与CM纤维代数子群的交集问题，推广前人成果。核心方法是通过控制自同态下典范高度，确定相关除子充裕时\(\lambda\)值来显式界定高度，给出了关于曲线与特定子群交集及高度界的结论。
We give a proof of generalizations of the classical Arakelov inequality valid for the degree $d$ of the relative canoincal bundle of a family of curves of genus $g$ over a complete curve of genus $p$ under the assumption that the monodromy around the singular fibers is unipotent. This relative canonical bundle is the (canonical extension of) the Hodge bundle and the inequality is generalized to the degrees of the Hodge bundles of a complex variation of Hodge structures.
The cohomology algebra of the canonical bundle of a compact Kähler manifold is naturally viewed as a module over an exterior algebra. We use the Bernstein-Gel'fand-Gel'fand correspondence, together with Generic Vanishing theory, in order to understand the regularity properties of this module. We also relate it to the infinitesimal theory of the canonical linear series inside paracanonical space. Finally, we apply vector bundle methods on the polynomial ring side to obtain inequalities for the holomorphic Euler characteristic and the Hodge numbers of compact Kähler manifolds without irregular f…
Let $X$ be a nonsingular projective $n$-fold $(n\ge 2)$ of Fano or of general type with ample canonical bundle $K_X$ over an algebraic closed field $κ$ of any characteristic. We produce a new method to give a bunch of inequalities in terms of all the Chern classes $c_1, c_2, \cdots, c_n$ by pulling back Schubert classes in the Chow group of Grassmannian under the Gauss map. Moreover, we show that if the characteristic of $κ$ is $0$, then the Chern ratios $(\frac{c_{2,1^{n-2}}}{c_{1^n}}, \frac{c_{2,2,1^{n-4}}}{c_{1^n}}, \cdots, \frac{c_{n}}{c_{1^n}})$ are contained in a convex polyhedron for al…
If $X$ is a smooth complex projective 3-fold with ample canonical divisor $K$, then the inequality $K^3\ge {2/3}(2p_g-7)$ holds, where $p_g$ denotes the geometric genus. This inequality is nearly sharp. We also give similar, but more complicated, inequalities for general minimal 3-folds of general type. (A noether type of inequality lies, by all means, on the other side of the Miyaoka-Yau inequality from the geographical point of view.)
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