Shu Kawaguchi
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 $\mathbb{C}$.
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We describe the action of the different Frobenius morphisms on the cohomology ring of the moduli stack of algebraic vector bundles of fixed rank and determinant on an algebraic curve over a finite field in characteristic p and analyse special situations like vector bundles on the projective line and relations with infinite Grassmannians.
Certain lower bounds are obtained on the canonical height associated to the morphism $ϕ(z)=z^d+c$.
Cyclotomic preperiodic points for morphisms in affine spaces and preperiodic points with bounded house and height
本文研究了仿射空间中模形的周期点集在循环闭包上的稠密性问题，并探讨了具有有界house和高度的周期点的性质。
Points of bounded height in images of morphisms of weighted projective stacks with applications to counting elliptic curves
研究有界高度点在加权投射堆映射中的渐进行为，并应用于计数具有指定水平结构的椭圆曲线，包括Γ(N)、Γ₁(N)和Γ₀(N)等情形。
We show the existence of canonical heights of subvarieties for bounded sequences of morphisms and give some applications.
In this paper, we consider the limit $ \lim_{n \to \infty} \sum_{v\in S} λ_{Y,v}(f^{n}(x))/h_{H}(f^{n}(x)) $ where $f \colon X \longrightarrow X$ is a surjective self-morphism on a smooth projective variety $X$ over a number field, $S$ is a finite set of places, $ λ_{Y,v}$ is a local height function associated with a proper closed subscheme $Y \subset X$, and $h_{H}$ is an ample height function on $X$. We give a geometric condition which ensures that the limit is zero, unconditionally when $\dim Y=0$ and assuming Vojta's conjecture when $\dim Y\geq1$. In particular, we prove (one is unconditio…
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).
Let F and G be morphisms of degree at least 2 from P^N to P^N that are defined over the algebraic closure of Q. We define the arithmetic distance d(F,G) between F and G to be the supremum over all algebraic points P of |h_F(P)-h_G(P)|, where h_F and h_G are the canonical heights associated to the morphisms F and G, respectively. We prove comparison theorems relating d(F,G) to more naive height functions and show that for a fixed G, the set of F satisfying d(F,G) < B is a set of bounded height. In particular, there are only finitely many such F defined over any given number field.
Heights and morphisms in number fields
本文针对数域上射影空间之间的非常值态射，给出了满足高度条件的有理点计数公式，并由此导出关于韦伊高度和 Call-Silverman 典范高度的计数函数结果。
We prove lower bound and finiteness properties for arakelovian heights with respect to pre-log-log hermitian ample line bundles. These heights were introduced by Burgos, Kramer and Kühn, in their extension of the arithmetic intersection theory of Gillet and Soulé, aimed to deal with hermitian vector bundles equipped with metrics admitting suitable logarithmic singularities. Our results generalize the corresponding properties for the heights of Bost-Gillet-Soulé, as well as the properties established by Faltings for heights of points attached to hermitian line bundles whose metrics have logarit…
A conjecture of Manin predicts the asymptotic distribution of rational points of bounded height on Fano varieties. In this paper we use conic bundles to obtain correct lower bounds or a wide class of surfaces over number fields for which the conjecture is still far from being proved. For example, we obtain the conjectured lower bound of Manin's conjecture for any del Pezzo surface whose Picard rank is sufficiently large, or for arbitrary del Pezzo surfaces after possibly an extension of the ground field of small degree.
Duality invariance of Faltings heights, Hodge line bundles and global periods
本文证明整体域上 Abel 簇与其对偶的 Faltings 高度、带自然度量的 Hodge 线丛，以及 BSD 猜想中的实周期和整体周期均具有对偶不变性。
Explicit Kodaira-Spencer maps over PEL Shimura varieties
本文通过Kodaira-Spencer映射构造两个canonical线丛间的映射，并显式计算其对线丛canonical度量的影响，为比较两个canonical线丛及其算术交数提供了具体方法。
Let K be an algebraic number field. For a degree d rational morphism of projective n-space defined over K let R denote its minimal resultant ideal. For a fixed height function on the moduli space of dynamical systems this paper shows that all such morphisms of bounded height and resultant are contained in finitely many PGL(K) equivalence classes. This answers a question of Silverman in the affirmative.
Parametrization of geometric Beilinson--Bloch heights via adelic line bundles
本文针对光滑射影态射构造取值于几何 adelic 线丛群的新高程配对，用以参数化 Brosnan–Pearlstein 渐近配对，并证明其在条件满足时等于 Beilinson–Bloch 配对。
Adelic line bundles on quasi-projective varieties
本书构建有限生成域上拟射影簇的阿代尔线丛理论，探讨其相交、体积与高度理论，并应用于拟射影簇代数点的高度研究。
Adelic line bundles over a Néron model
受Néron-Tate高度配对变分问题的启发，在曲线上的阿贝尔方案的Néron模型上构造了adelic Poincaré丛，推广了Yuan-Zhang的adelic Poincaré丛。
For a given conic bundle X over a curve C defined over F_q, we count irreducible branch covers of C in X of degree d and height e>>1. As a special case, we get the number of algebraic numbers of degree d and height e over the function field F_q (C).
We introduce a new approach to study height zeta functions of projective spaces and projective bundles. To study height zeta functions of projective spaces $Z(\mathbb{P}^n, H_{\mathcal{O}(1)}; s)$, we apply the Riemann-Roch theorem of Arakelov vector bundles by van der Geer and Schoof to the integrand of an integral expression of $Z(\mathbb{P}^n, H_{\mathcal{O}(1)}; s)$. We give a proof of the analytic continuation and functional equations of height zeta functions of projective spaces with respect to various height functions. Motivic analogues of these results are also proved. We also study he…
A conjecture of Batyrev and Manin predicts the asymptotic behaviour of rational points of bounded height on smooth projective varieties over number fields. We prove some new cases of this conjecture for conic bundle surfaces equipped with some non-anticanonical height functions. As a special case, we verify these conjectures for the first time for some smooth cubic surfaces for height functions associated to certain ample line bundles.
Absolute minimum of a height associated to a semipositive adelic line bundle
研究有理数域上射影簇半正定阿戴尔线丛绝对最小值问题，基于\(\mathbb{P}^1\)动力系统算术性质构造，指出其绝对最小值不由代数点达到。
The holomorphic endomorphism f of projective space is called post-critically finite (PCF) if the forward image of the critical locus, under iteration of f, has algebraic support (i.e., is a finite union of hypersurfaces). In the case of dimension 1, a deep result of Thurston implies that there are no algebraic families of PCF morphisms, other than a well-understood exceptional class known as the flexible Lattes maps. This note proves a corresponding result in arbitrary dimension, for a certain subclass of morphism. Specifically, we restrict attention to morphisms f of degree at least two, with…
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 show that the height of a toric variety with respect to a toric metrized line bundle can be expressed as the integral over a polytope of a certain adelic family of concave functions. To state and prove this result, we study the Arakelov geometry of toric varieties. In particular, we consider models over a discrete valuation ring, metrized line bundles, and their associated measures and heights. We show that these notions can be translated in terms of convex analysis, and are closely related to objects like polyhedral complexes, concave functions, real Monge-Ampère measures, and Legendre-Fen…
Geometric Height on Flag Varieties in Positive Characteristic
本文在正特征下研究旗流形上几何高度函数的高度滤波与连续最小值计算方法。
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 height is expressed as an integral average of the logarithmic distance to the closure of the periodic points of the underlying morphism. The results may be thought of as a kind of morphic Jensen formula.
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