Points of small height on varieties defined over a function field
Dragos Ghioca
对于特征$p$中的加群方案，得到了一个Bogomolov型结果.我们的结果等价于定义在有限域上的Drinfeld模的Bogomolov定理。
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We establish asymptotic formulas for the number of integral points of bounded height on toric varieties.
Positivity of Heights of Semistable Varieties
我们完全解决了S的一个问题。Zhang关于给定次数和维数的半稳定簇的模空间上的正规化高度的正性。
Canonical heights for random iterations in certain varieties
证明了有界态射序列的子簇的典范高的存在性，并给出了一些应用。
Manin's conjecture on toric varieties with different heights
在本文中，我验证了Manin猜想的一类合理的射影环面品种的一个大类的高度以外的通常的一个来自标准度量的射影空间。
Points of bounded height on certain subvarieties of toric varieties
本文结合分裂挠子法与双曲线法，计算环面簇某些子簇上高度有界的有理点和Campana点。
p-adic height pairings on abelian varieties with semistable ordinary reduction
证明了对于具有半稳定普通约化的阿贝尔簇，第一deRham上同调的Hodge滤子的单位根分裂诱导p-adic Mazur-Tate高度配对.
We show that the height of a variety over a finitely generated field of characteristic zero can be written as an integral of local heights over the set of places of the field. This allows us to apply our previous work on toric varieties and extend our combinatorial formulae for the height to compute some arithmetic intersection numbers of non toric arithmetic varieties over the rational numbers.
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.
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.
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 $c_1^2 > c_2$.
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。
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.
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.
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。
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。
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。
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
设A是数域K上的一个阿贝尔簇，考虑A上的对称充线丛L的标准高度函数.本文证明了定义在K的极大阿贝尔扩张K^ab上的A上的非挠点的标准高度存在正的下界C（依赖于A，K和L）.
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。
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。
$\mathcal{M}$-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。
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。
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…
Normalized height of projective toric varieties
给出了射影复曲面簇的归一化高度的一个显式表达式。该表达式分解为局部贡献的和，每项是某个函数的积分，凹的和分段线性仿射的。更一般地说，我们得到了一个明确的表达式的正规化多高度的环面相对于几个单项嵌入。所介绍的一组功能的行为作为一个算术模拟的多面体经典与环面行动。除了公式的高度和multiheight，我们表明，这个对象的行为在一个自然的方式与几个标准的建设：分解成轨道，加入，塞格雷产品和Veronese嵌入。证明遵循一种间接的方式：而不是归一化高度的定义，我们依赖于一个适当的算术希尔伯特函数的计算。
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。
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。
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。
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。
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…
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…
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…
Generalised Hermite Constants, Voronoi Theory and Heights on Flag Varieties
本文研究了T。渡边为此，一个高度，这自然适用于旗品种，是建立和概念的完美和eutaxy特征的极值。最后，我们熟悉了一些关系式（如Korkine-Zolotareff约化），上界和与这些常数有关的计算。
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…
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.
A polarizable endomorphism on a projective variety enables us to consider given morphism as constant multiplication in the height function. In this paper, we will generalize it for arbitrary dominant endomorphism by defining the height expansion and contraction coefficients.
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…
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.
Heights on 'Hybrid orbits' in Shimura varieties
本文在 abelian 型 Shimura 簇情形下证明“混合猜想”，统一推广了 Andreé-Oort 与 André-Pink-Zannier 两个猜想。
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。
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。
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…
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 M be the Shimura variety associated with the group of spinor similitudes of a rational quadratic space over of signature (n,2). We prove a conjecture of Bruinier-Kudla-Yang, relating the arithmetic intersection multiplicities of special divisors and big CM points on M to the central derivatives of certain $L$-functions. As an application of this result, we prove an averaged version of Colmez's conjecture on the Faltings heights of CM abelian varieties.
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
对于数域上拟射影簇上的非代数点$θ$，证明了$θ$具有一系列高度和次数有界的代数点的逼近，这是本质上最好的逼近.这一结果的应用将包括证明一个稍微加强版本的Philippon标准，一些新的代数独立性标准，报表有关度量超越理论的品种任意尺寸，和一个相当准确的估计代数点的数量有界高度和程度的拟射影簇在数域。
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。
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。
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…
Local and Global Heights on Weighted Projective Varieties
本文通过Cartier与Weil除子将加权高度推广至加权簇，证明线丛存在局部有界加权$M$-度量，并定义局部与全局高度，展示其基本性质。
The Equidistribution Conjecture is proved for general semiabelian varieties over number fields. Previously, this conjecture was only known in the special case of almost split semiabelian varieties through work of Chambert-Loir. The general case has remained intractable so far because the height of a semiabelian variety is negative unless it is almost split. In fact, this places the conjecture outside the scope of Yuan's equidistribution theorem on algebraic dynamical systems. To overcome this, an asymptotic adaption of the equidistribution technique invented by Szpiro, Ullmo, and Zhang is used…
Heights of Special Points on Quaternionic Shimura Varieties
本文用 CM 阿贝尔簇的 Faltings 高度给出四元数志村簇上特殊点高度的显式公式，并证明其与部分 CM 型的典范高度相容，从而重新证明部分 CM 型高度关于判别式具有次多项式上界。
We investigate Fano varieties defined over a number field that contain subvarieties whose number of rational points of bounded height is comparable to the total number on the variety.
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…
We lower bound the Faltings height of an abelian variety over a number field by the sum of its injectivity diameter and the norm of its bad reduction primes. It leads to an unconditional bound on the rank of Mordell-Weil groups. Assuming the height conjecture of Lang and Silverman, we then obtain a Northcott property for the regulator on the set of simple abelian varieties defined over a fixed number field, of fixed dimension $g$, bounded rank and with dense rational points over a number field. We remove the simplicity assumption in the principally polarized case by giving a refined version of…
Counting Rational Points on Ruled Varieties
本文证明了一个计算被直线覆盖的射影簇$V$上有界高的有理点个数的一般结果。用于实现这一点的主要技术结果是一个上限的有理点的数量有界的高度线。该上界是这样的，即当线变化时，它可以容易地控制，并因此用于对覆盖原始品种$V$的线的计数函数求和。
Lehmer-Type bounds and counting rational points of bounded heights on Abelian varieties
本文研究了数域K上阿贝尔簇A的$\bar{K}$-点Néron-Tate高度的Lehmer型界，估计了高度$\leq \log B$的K-有理点数量，并针对椭圆曲线及其乘积将估计中的常数C变为显式可计算。
In the first section of his seminal paper on height pairings, Beilinson constructed an $\ell$-adic height pairing for rational Chow groups of homologically trivial cycles of complementary codimension on smooth projective varieties over the function field of a curve over an algebraically closed field, and asked about an generalization to higher dimensional bases. In this paper we answer Beilinson's question by constructing a pairing for varieties defined over the function field of a smooth variety $B$ over an algebraically closed field, with values in the second $\ell$-adic cohomology group of …
The Bounded Height Conjecture of Bombieri, Masser, and Zannier states that for any sufficiently generic algebraic subvariety of a semiabelian $\overline{\mathbb{Q}}$-variety $G$ there is an upper bound on the Weil height of the points contained in its intersection with the union of all algebraic subgroups having (at most) complementary dimension in $G$. This conjecture has been shown by Habegger in the case where $G$ is either a multiplicative torus or an abelian variety. However, there are new obstructions to his approach if $G$ is a general semiabelian variety. In particular, the lack of Poi…
On an abelian scheme over a smooth curve over $\overline{\mathbb Q}$ a symmetric relatively ample line bundle defines a fiberwise Néon-Tate height. If the base curve is inside a projective space, we also have a height on its $\overline{\mathbb Q}$-points that serves as a measure of each fiber, an abelian variety. Silverman proved an asymptotic equality between these two heights on a curve in the abelian scheme. In this paper we prove an inequality between these heights on a subvariety of any dimension of the abelian scheme. As an application we prove the Geometric Bogomolov Conjecture for the …
Modular Heights of Unitary Shimura Varieties II: Arithmetic Generating Series of Divisors
本文通过定义并计算酉 Shimura 簇上除子的算术生成级数及其算术相交数，推导了酉 Shimura 曲线的模高公式及其上 CM 点的高度公式。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。