Canonical heights for random iterations in certain varieties

Quadratic Chabauty and rational points II: Generalised height functions on Selmer varieties

We give new instances where Chabauty--Kim sets can be proved to be finite, by developing a notion of "generalised height functions" on Selmer varieties. We also explain how to compute these generalised heights in terms of iterated integrals and give the first explicit nonabelian Chabauty result for a curve $X/\mathbb{Q}$ whose Jacobian has Mordell-Weil rank larger than its genus.

Points of small height on varieties defined over a function field

We obtain a Bogomolov type of result for the additive group scheme in characteristic $p$. Our result is equivalent with a Bogomolov theorem for Drinfeld modules defined over a finite field.

Distribution of rational points on toric varieties: All the heights

In this paper we prove a formula for the number of rational points of bounded height relative to all the generators of the cone of effective divisor for a toric variety over a number field.

Bounding heights uniformly in families of hyperbolic varieties

We show that, assuming Vojta's height conjecture, the height of a rational point on an algebraically hyperbolic variety can be bounded "uniformly" in families. This generalizes a result of Su-Ion Ih for curves of genus at least two to higher-dimensional varieties. As an application, we show that, assuming Vojta's height conjecture, the height of a rational point on a curve of general type is uniformly bounded. Finally, we prove a similar result for smooth hyperbolic surfaces with $c_1^2 > c_2$.

环面簇上有界高度的半整点

Semi-integral points of bounded height on toric varieties

本文证明了环面簇上有界高度半整点的渐近公式，并在修正首项常数后验证了光滑及部分奇异环面轨形上的Manin型猜想。

Torsion points and height jumping in higher-dimensional families of abelian varieties

In 1983 Silverman and Tate showed that the set of points in a 1-dimensional family of abelian varieties where a section of infinite order has `small height' is finite. We conjecture a generalisation to higher-dimensional families, where we replace `finite' by `not Zariski dense'. We show that this conjecture would imply the Uniform Boundedness Conjecture for torsion points on abelian varieties. We then prove a few special cases of this new conjecture.

Degree and height estimates for modular equations on PEL Shimura varieties

We define modular equations in the setting of PEL Shimura varieties as equations describing Hecke correspondences, and prove upper bounds on their degrees and heights. This extends known results about elliptic modular polynomials, and implies complexity bounds for number-theoretic algorithms using these modular equations. In particular, we obtain tight degree bounds for modular equations of Siegel and Hilbert type for abelian surfaces.

余维分析代数簇的有理点分布

Multi-height analysis of rational points of toric varieties

研究光滑、射影且分裂的托里克簇上有理点的多高度分布，通过将点数提升到通用 torsors 中。

极值论中极核种的极值理论：全局理论

Heights on toric varieties for singular metrics: Global theory

本文提出了一种极核种的极值理论，通过凹函数积分计算极核算术品种的极值。

toric varieties 的高度：对于奇异度量的局部理论

Heights on toric varieties for singular metrics: Local theory

本文研究了toric varieties在奇异度量下的局部高度计算，通过凹函数积分推广了先前关于连续度量的结果。

Counting integral points of bounded height on varieties with large fundamental group

The present note is devoted to an amendment to a recent paper of Ellenberg, Lawrence and Venkatesh. Roughly speaking, the main result here states the subpolynomial growth of the number of integral points with bounded height of a variety over a number field whose fundamental group is large. Such an improvement, i.e. requiring large fundamental group as opposed to the existence of a geometric variation of pure Hodge structures, was already asked in op.cit..

A Lower Bound for the Canonical Height on Abelian Varieties over Abelian Extensions

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.

Shimura 簇上的典范高度与 André-Oort 猜想

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。

环面簇中完全交的高度

Heights of complete intersections in toric varieties

研究环面簇中完全交的高度问题，通过给出极限公式将环面簇及其超曲面高度的相关结果扩展到二维余维情况，证明了特定交周期高度收敛到相关混合积分的阿戴尔和，部分证实先前猜想。

$\mathcal{M}$-点有界高度在分裂紧致态射上的渐近公式

$\mathcal{M}$-points of bounded height on toric varieties

本文研究了紧致态射上$\mathcal{M}$-点有界高度的渐近公式，并扩展了曼因猜想，同时得到了弱坎帕纳点的新结果。

计算全局函数域上Hirzebruch-Kleinschmidt簇的有理点数

Counting rational points on Hirzebruch-Kleinschmidt varieties over global function fields

受Bourqui工作启发，本文研究了全局函数域上Picard秩为2的分裂环面簇的高度zeta函数，将其分解为子簇并给出精确解析性质，进而得到各子簇上大高度有理点计数的渐近公式。

Local heights of toric varieties over non-archimedean fields

We generalize results about local heights previously proved in the case of discrete absolute values to arbitrary non-archimedean absolute values of rank 1. First, this is done for the induction formula of Chambert-Loir and Thuillier. Then we prove the formula of Burgos--Philippon--Sombra for the toric local height of a proper normal toric variety in this more general setting. We apply the corresponding formula for Moriwaki's global heights over a finitely generated field to a fibration which is generically toric. We illustrate the last result in a natural example where non-discrete non-archime…

On the height of Calabi-Yau varieties in positive characteristic

We study invariants of Calabi-Yau varieties in positive characteristic, especially the height of the Artin-Mazur formal group. We illustrate these results by Calabi-Yau varieties of Fermat and Kummer type.

Normalized height of projective toric varieties

We present an explicit expression for the normalized height of a projective toric variety. This expression decomposes as a sum of local contributions, each term being the integral of a certain function, concave and piecewise linear-affine. More generally, we obtain an explicit expression for the normalized multiheight of a torus with respect to several monomial embeddings. The set of functions introduced behaves as an arithmetic analog of the polytope classically associated with the torus action. Besides the formulae for the height and multiheight, we show that this object behaves in a natural…

拟射影簇上的阿代尔线丛

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。

Height zeta functions of toric bundles over flag varieties

We investigate analytic properties of height zeta functions of toric bundles over flag varieties.

Kuga-Sato簇上CM闭链的模性与高度

Modularity and Heights of CM cycles on Kuga-Sato varieties

本文在任意级模曲线的Kuga-Sato簇上证明了CM闭链的高权一般Gross-Zagier公式，并证明CM闭链生成的Hecke模是半单模，为Beilinson-Bloch猜想提供证据。

数域上 Hirzebruch-Kleinschmidt 簇的有理点计数

Counting rational points on Hirzebruch-Kleinschmidt varieties over number fields

本文研究了数域上 Picard 秩为 2 的光滑射影分裂环面簇在 Arakelov 高度下的有理点渐近增长，证明其可分解为有限个不相交子簇并给出显式渐近公式，同时提供了包含 Hirzebruch 曲面的实例。

族中高度的连续性与环面簇中的完全交

Continuity of heights in families and complete intersections in toric varieties

本文研究全局赋值域上平坦族中闭链高度的变化，通过引入 GVF 解析化证明纤维高度的连续性，并以此证实了 Roberto Gualdi 关于环面簇中完全交极限高度的猜想。

Good height functions on quasi-projective varieties: equidistribution and applications in dynamics

In the present article, we define a notion of good height functions on quasi-projective varieties $V$ defined over number fields and prove an equidistribution theorem of small points for such height functions. Those good height functions are defined as limits of height functions associated with semi-positive adelic metrization on big and nef $\mathbb{Q}$-line bundles on projective models of $V$ satisfying mild assumptions. Building on a recent work of the author and Vigny as well as on a classical estimate of Call and Silverman, and inspiring from recent works of Kühne and Yuan and Zhang, we d…

Ample canonical heights for endomorphisms on projective varieties

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…

Heights of hypersurfaces in toric varieties

For a cycle of codimension 1 in a toric variety, its degree with respect to a nef toric divisor can be understood in terms of the mixed volume of the polytopes associated to the divisor and to the cycle. We prove here that an analogous combinatorial formula holds in the arithmetic setting: the global height of a 1-codimensional cycle with respect to a toric divisor equipped with a semipositive toric metric can be expressed in terms of mixed integrals of the $v$-adic roof functions associated to the metric and the Legendre-Fenchel dual of the $v$-adic Ronkin function of the Laurent polynomial o…

Height Zeta Functions of Toric Varieties

We investigate analytic properties of height zeta functions of toric varieties. Using the height zeta functions, we prove an asymptotic formula for the number of rational points of bounded height with respect to an arbitrary line bundle whose first Chern class is contained in the interior of the cone of effective divisors

The motivic anabelian geometry of local heights on abelian varieties

We study the problem of describing local components of height functions on abelian varieties over characteristic $0$ local fields as functions on spaces of torsors under various realisations of a $2$-step unipotent motivic fundamental group naturally associated to the defining line bundle. To this end, we present three main theorems giving such a description in terms of the $\mathbb Q_\ell$- and $\mathbb Q_p$-pro-unipotent étale realisations when the base field is $p$-adic, and in terms of the $\mathbb R$-pro-unipotent Betti--de Rham realisation when the base field is archimedean. In the cours…

A New Northcott Property for Faltings Height

In this work we prove a new Northcott property for the Faltings height. Namely we show, assuming the Colmez Conjecture and the Artin Conjecture, that there are finitely many CM abelian varieties over the complex numbers of a fixed dimension which have bounded Faltings height. The technique developed uses new tools from integral p-adic Hodge theory to study the variation of Faltings height within an isogeny class of CM abelian varieties. In special cases, we are moreover able to use the technique to develop new Colmez-type formulas for the Faltings height.

Arithmetic geometry of toric varieties. Metrics, measures and heights

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…

函数域上同源类中高度的变化

Variation of height in an isogeny class over a function field

本文研究任意特征下曲线函数域上 Abel 簇同源类中的高度变化，给出微分高度与模高度变化的最优估计，并证明平行四边形不等式及相关推论。

关于主导有理自映射在射影簇上周期点和前周期点高度有界的性质

On the height boundedness of periodic and preperiodic points of dominant rational self-maps on projective varieties

本文推翻了关于周期点高度有界的猜想，并证明了在射影簇上某些主导有理自映射的周期点集高度有界，同时指出前周期点可能不满足这一性质。

Gorensteinness and iteration of Cox rings for Fano type varieties

We show that finitely generated Cox rings are Gorenstein. This leads to a refined characterization of varieties of Fano type: they are exactly those projective varieties with Gorenstein canonical quasicone Cox ring. We then show that for varieties of Fano type and Kawamata log terminal quasicones, iteration of Cox rings is finite with factorial master Cox ring. Moreover, we prove a relative version of Cox ring iteration for almost principal solvable $G$-bundles.

Heights on stacks and a generalized Batyrev-Manin-Malle conjecture

We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties. We explain how to compute this height for various stacks of interest (for instance: classifying stacks of finite groups, symmetric products of varieties, moduli stacks of abelian varieties, weighted projective spaces). In many cases our uniform definition reproduces ways already in use for measuring the complexity of rational points, while in others it is something n…

Canonical heights on hyper-Kähler varieties and the Kawaguchi-Silverman conjecture

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…

Lower bounds for Galois orbits of special points on Shimura varieties: a point-counting approach

Let $S$ be a Shimura variety and let $h$ be a Weil height function on $S$. We conjecture that the heights of special points in $S$ are discriminant negligible. Assuming this conjecture to be true, we prove that the sizes of the Galois orbits of special points grow as a fixed power of their discriminant (an invariant we will define in the text). In the case of Shimura varieties of abelian type, the height bound holds by the recently proved averaged Colmez formula, and our theorem gives a new proof of Tsimerman's Galois lower bound in this case. The main novelty is that our approach avoids the u…

Diophantine Approximation on varieties III: Approximation of non-algebraic points by algebraic points

For $θ$ a non-algebraic point on a quasi projective variety over a number field, I prove that $θ$ has an approximation by a series of algebraic points of bounded height and degree which is essentially best possible. Applications of this result will include a proof of a slightly strengthened version of the Philippon criterion, some new algebraic independence criteria, statements concerning metric transcendence theory on varieties of arbitrary dimension, and a rather accurate estimate for the number of algebraic points of bounded height and degree on quasi projective varieties over number fields…

环面簇上整点的Manin猜想

Manin's conjecture for integral points on toric varieties

本研究类比有理点的Manin猜想，提出对数Fano簇上有界高度整点数量的猜想及首项常数预测，并借助万有挠子上的调和分析在环面簇上证明该猜想，为其提供证据。

簇与非阿基米德空间的局部一致化的高度约化

Height reduction for local uniformization of varieties and non-archimedean spaces

本文证明概形与非阿基米德解析空间的局部一致化可严格归结为高度为一的赋值情形，从而将非阿基米德空间的问题转化为研究光滑Berkovich空间的局部结构。

部分高度、全纯曲线与几何 Bombieri-Lang 猜想

Partial Heights, Entire Curves, and the Geometric Bombieri-Lang Conjecture

本文提出利用部分高度与 Brody 引理从无穷有理点序列构造全纯曲线的新方法，并在部分高度非退化猜想下证明了特征 0 函数域上具有到阿贝尔簇有限态射的双曲射影簇的几何 Bombieri-Lang 猜想。

The distribution of Galois orbits of points of small height in toric varieties

We address the distribution properties of points of small height on proper toric varieties and applications to the related Bogomolov property. We introduce the notion of monocritical toric metrized divisor and we prove that equidistribution occurs for every generic, small sequence with respect to a toric metrized divisor, for every place if and only if the divisor is monocritical. Furthermore, when this is the case, the limit measure is a translate of the natural measure on the compact torus sitting in the principal orbit of the ambient toric variety. We also study the $v$-adic modulus distrib…

Heights of varieties in multiprojective spaces and arithmetic Nullstellensatze

We present bounds for the degree and the height of the polynomials arising in some central problems in effective algebraic geometry including the implicitation of rational maps and the effective Nullstellensatz over a variety. Our treatment is based on arithmetic intersection theory in products of projective spaces and extends to the arithmetic setting constructions and results due to Jelonek. A key role is played by the notion of canonical mixed height of a multiprojective variety. We study this notion from the point of view of resultant theory and establish some of its basic properties, incl…

Counting Rational Points on Ruled Varieties

In this paper, we prove a general result computing the number of rational points of bounded height on a projective variety $V$ which is covered by lines. The main technical result used to achieve this is an upper bound on the number of rational points of bounded height on a line. This upper bound is such that it can be easily controlled as the line varies, and hence is used to sum the counting functions of the lines which cover the original variety $V$.

A fixed point formula of Lefschetz type in Arakelov geometry III: representations of Chevalley schemes and heights of flag varieties

We give a new proof of the Jantzen sum formula for integral representations of Chevalley schemes over Spec Z. This is done by applying the fixed point formula of Lefschetz type in Arakelov geometry to generalized flag varieties. Our proof involves the computation of the equivariant Ray-Singer torsion for all equivariant bundles over complex homogeneous spaces. Furthermore, we find several explicit formulae for the global height of any generalized flag variety.

非阿基米德局部高度的热带公式

A tropical formula for non-Archimedean local heights

本文提出了一种基于热带几何的非阿基米德局部高度计算方法，通过定义delta形式和星积来实现。

正特征下旗流形上的几何高度

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。

A Note on the Specialization Theorem for Families of Abelian Varieties

In this note, we apply Moriwaki's arithmetic height functions to obtain an analogue of Silverman's Specialization Theorem for families of Abelian varieties over $K$, where $K$ is any field finitely generated over ${\Q}$.

Growth of local height functions along orbits of self-morphisms on projective varieties

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…

Canonical Kahler metrics and Arithmetics -- Generalising Faltings heights

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.

Successive minima of projective toric varieties

We compute the successive minima of the projective toric variety $X_\cA$ associated to a finite set $ \cA \subset \Z^n$. As a consequence of this computation and of the results of S.-W. Zhang on the distribution of small points, we derive estimates for the height of the subvariety $X_\cA$ and of the $\cA$-resultant. These estimates allow us to obtain an arithmetic analogue of the Bezout-Kushnirenko's theorem concerning the number of solutions of a system of polynomial equations. As an application of this result, we improve the known estimates for the height of the polynomials in the sparse Nul…

Height Estimates for Equidimensional Dominant Rational Maps

Let F : W --> V be a dominant rational map between quasi-projective varieties of the same dimension. We give two proofs that h_V(F(P)) >> h_W(P) for all points P in a nonempty Zariski open subset of W. For dominant rational maps F : P^n --> P^n, we give a uniform estimate in which the implied constant depends only on n and the degree of F. As an application, we prove a specialization theorem for equidimensional dominant rational maps to semiabelian varieties, providing a complement to Habegger's recent theorem on unlikely intersections.

Motivic Euler products and motivic height zeta functions

A motivic height zeta function associated to a family of varieties parametrised by a curve is the generating series of the classes, in the Grothendieck ring of varieties, of moduli spaces of sections of this family with varying degrees. This text is devoted to the study of the motivic height zeta function associated to a family of varieties with generic fiber having the structure of an equivariant compactification of a vector group. Our main theorem describes the convergence of this motivic height zeta function with respect to a topology on the Grothendieck ring of varieties coming from the th…

K-半稳定Fano簇高度的确界I，环面情形

Sharp bounds on the height of K-semistable Fano varieties I, the toric case

受Fujita结果启发，本文猜想整数射影空间在K-半稳定度量算术Fano簇中高度最大，并在n≤6时证明了环面Fano簇的该猜想，给出了Donaldson环面不变量与Odaka模高度的尖锐下界。