Some cases of Vojta's Conjecture on integral points over function fields

Some examples of exceptional loci in Vojta Conjecture

In this short note we discuss the exceptional locus for the Lang-Vojta's conjecture in the case of the complement of two completely reducible hyperplane sections in a cubic surface. Using elementary methods, we show that generically the exceptional set is the union of the remaining 21 lines in the surface. We also describe examples in which the exceptional set is strictly larger.

The Lang-Vojta conjectures on projective pseudo-hyperbolic varieties

These notes grew out of a mini-course given from May 13th to May 17th at UQAM in Montreal during a workshop on Diophantine Approximation and Value Distribution Theory. We start with an overview of Lang-Vojta's conjectures on pseudo-hyperbolic projective varieties. These conjectures relate various different notions of hyperbolicity. We begin with Brody hyperbolicity and discuss conjecturally related notions of hyperbolicity in arithmetic geometry and algebraic geometry in subsequent sections. We slowly work our way towards the most general version of Lang-Vojta's conjectures and provide a summa…

Generalized Greatest Common Divisors, Divisibility Sequences, and Vojta's Conjecture for Blowups

We apply Vojta's conjecture to blowups and deduce a number of deep statements regarding (generalized) greatest common divisors on varieties, in particular on projective space and on abelian varieties. Special cases of these statements generalize earlier results and conjectures. We also discuss the relationship between generalized greatest common divisors and the divisibility sequences attached to algebraic groups, and we apply Vojta's conjecture to obtain a strong bound on the divisibility sequences attached to abelian varieties of dimension at least two.

Vojta's conjecture for singular varieties

We formulate a generalization of Vojta's conjecture in terms of log pairs and variants of multiplier ideals. In this generalization, a variety is allowed to have singularities. It turns out that the generalized conjecture for a log pair is equivalent to the original conjecture applied to a log resolution of the pair. A special case of the generalized conjecture can be interpreted as representing a general phenomenon that there tend to exist more rational points near singular points than near smooth points. The same phenomenon is also observed in relation between greatest common divisors of int…

Lang-Vojta Conjecture over function fields for surfaces dominating $\mathbb{G}_m^2$

We prove the nonsplit case of the Lang-Vojta conjecture over function fields for surfaces of log general type that are ramified covers of $\mathbb{G}_m^2$. This extends results of Corvaja and Zannier, who proved the conjecture in the split case, and results of Corvaja and Zannier and the second author that were obtained in the case of the complement of a degree four and three component divisor in $\mathbb{P}^2$. We follow the strategy developed by Corvaja and Zannier and make explicit all the constants involved.

加权射影簇上的Vojta猜想

Vojta's conjecture on weighted projective varieties

本文针对光滑加权射影簇等三类对象给出Vojta猜想的等价表述，引入广义加权最大公因子并证明猜想成立时可对相关log高度给出界，也得到整系数加权齐次多项式的对应结果。

Campana points, Vojta's conjecture, and level structures on semistable abelian varieties

We introduce a qualitative conjecture, in the spirit of Campana, to the effect that certain subsets of rational points on a variety over a number field, or a Deligne-Mumford stack over a ring of S-integers, cannot be Zariski dense. The conjecture interpolates, in a way that we make precise, between Lang's conjecture for rational points on varieties of general type over number fields, and the conjecture of Lang and Vojta that asserts that S-integral points on a variety of logarithmic general type are not Zariski-dense. We show our conjecture follows from Vojta's conjecture. Assuming our conject…

The Vojta conjecture implies Galois rigidity in dynamical families

We show that the Vojta (or Hall-Lang) conjecture implies that the arboreal Galois representations in a 1-parameter family of quadratic polynomials are surjective if and only if they surject onto some finite and uniform quotient. As an application, we use the Vojta conjecture, our uniformity theorem over $\mathbb{Q}(t)$, and Hilbert's irreducibility theorem to prove that the prime divisors of many quadratic orbits have density zero.

On Vojta's $1+ε$ Conjecture

I gave a geometric proof of Vojta's 1 + epsilon conjecture. Some gaps in the published paper were spotted and kindly pointed out to me by Paul Vojta. These were addressed in "Erratum".

Fibered Threefolds and Lang-Vojta's Conjecture over Function Fields

Using the techniques introduced by Corvaja and Zannier we solve the non-split case of the geometric Lang-Vojta Conjecture for affine surfaces isomorphic to the complement of a conic and two lines in the projective plane. In this situation we deal with sections of an affine threefold fibered over a curve, whose boundary, in the natural projective completion, is a quartic bundle over the base whose fibers have three irreducible components. We prove that the image of each section has bounded degree in terms of the Euler Characteristic of the base curve.

Vojta's conjecture, heights associated with subschemes, and primitive prime divisors in arithmetic dynamics

Assuming Vojta's conjecture, we give a sufficient condition for the limit \[ \lim_{n \to \infty} \frac{h_{Y}(f^{n}(x))}{h_{H}(f^{n}(x))} \] is equal to zero, where $f \colon X \longrightarrow X$ is a surjective self-morphism on a smooth projective variety $X$, $h_{H}$ is an ample height function on $X$, and $h_{Y}$ is a global height function associated with a closed subscheme $Y \subset X$ of codimension at least two. Based on this, we propose a conjecture on a sufficient condition for the limit to be zero. We point out that our conjecture implies Dynamical Mordell-Lang conjecture for endomor…

Divisorial instability and Vojta's Main Conjecture for $\mathbb{Q}$-Fano varieties

We study Diophantine arithmetic properties of birational divisors in conjunction with concepts that surround $\mathrm{K}$-stability for Fano varieties. There is also an interpretation in terms of the barycentres of Newton-Okounkov bodies. Our main results show how the notion of divisorial instability, in the sense of K. Fujita, implies instances of Vojta's Main Conjecture for Fano varieties. A main tool in the proof of these results is an arithmetic form of Cartan's Second Main Theorem that has been obtained by M. Ru and P. Vojta.

单幂零秩交换代数群的非阿基米德Green--Griffiths--Lang--Vojta猜想

The Green--Griffiths--Lang--Vojta conjecture for commutative algebraic groups with unipotent rank 1

针对线性部分同构于$\bbG_a$的交换代数群的闭子簇，证明其Kawamata轨迹等于类Lang例外轨迹并确定真子概形条件，还证明了线性部分为$\bbG_a \times \bbG_m^t$情形的非阿基米德Green--Griffiths--Lang--Vojta猜想强形式。

走向Lang-Vojta猜想的退化研究

Towards Lang--Vojta via Degeneration

本文通过研究曲线模空间的算术和几何性质，证明了关于数字段上变种S-积分点有限性和Zariski退化的结果，并提供了几何上不可约除子的例子，回答了关于正常射影品种的积分点有限性问题。

Nonspecial varieties and Generalized Lang-Vojta conjectures

We construct a family of fibered threefolds $X_m \to (S , Δ)$ such that $X_m$ has no étale cover that dominates a variety of general type but it dominates the orbifold $(S,Δ)$ of general type. Following Campana, the threefolds $X_m$ are called \emph{weakly special} but not \emph{special}. The Weak Specialness Conjecture predicts that a weakly special variety defined over a number field has a potentially dense set of rational points. We prove that if $m$ is big enough the threefolds $X_m$ present behaviours that contradict the function field and analytic analogue of the Weak Specialness Conject…

函数域上非常一般对数射影空间的Lang-Vojta猜想

Lang-Vojta conjecture over function fields for very general log projective spaces

研究对数对$(\mathbb{P}^{n}\times C, D)$中曲线度数由其亏格界定的问题，其中$C$为光滑复射影曲线，$D$为$\mathbb{P}^{n}\times C$上非常一般光滑除子，受Lang-Vojta猜想的函数域类比驱动。

关于Vojta的更难蕴含、可容许对的高度不等式、有界次数点与Deligne-Mumford栈

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。

Greatest common divisors of integral points of numerically equivalent divisors

We generalize the G.C.D. results of Corvaja--Zannier and Levin on $\mathbb G_m^n$ to more general settings. More specifically, we analyze the height of a closed subscheme of codimension at least $2$ inside an $n$-dimensional Cohen-Macaulay projective variety, and show that this height is small when evaluated at integral points with respect to a divisor $D$ when $D$ is a sum of $n+1$ effective divisors which are all numerically equivalent to some multiples of a fixed ample divisor. Our method is inspired by Silverman's G.C.D. estimate as an application of Vojta's conjecture, which is substitute…

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 $c_1^2 > c_2$.

关于对数配对的相对定义域、Vojta高度不等式与渐近坐标大小动力学

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。

阿贝尔簇上的广义整点与几何 Lang-Vojta 猜想

Generalized integral points on abelian varieties and the Geometric Lang-Vojta conjecture

本文研究紧黎曼曲面函数域上阿贝尔簇的广义整点，发展 Parshin 思想提出双曲同伦高度替代相交理论，证明了整点大并集的有限性与多项式增长，为几何 Lang-Vojta 猜想提供了新证据。

特征零函数域上的显式一致 Mordell 猜想

An Explicit Uniform Mordell Conjecture over Function Fields of Characteristic Zero

本文基于Vojta方法、Zhang的容许adelic线丛及Bogomolov猜想的定量证明，证明了特征零函数域上非同位曲线的显式一致Mordell猜想。

函数域上环面簇覆叠的Campana猜想

Campana conjecture for coverings of toric varieties over function fields

本文在函数域上证明了沿边界具有高重数的射影环面簇上Campana点的Vojta abc猜想，并以此推导出函数域上射影环面簇有限覆叠的Campana猜想版本。

A conjecture on rational approximations to rational points

In this paper, we examine how well a rational point P on an algebraic variety X can be approximated by other rational points. We conjecture that if P lies on a rational curve, then the best approximations to P on X can be chosen to lie along a rational curve. We prove this conjecture for a wide range of examples, and for a great many more examples we deduce our conjecture from Vojta's Main Conjecture.

A variant of the Mordell-Lang conjecture

The Mordell-Lang conjecture (proven by Faltings, Vojta and McQuillan) states that the intersection of a subvariety $V$ of a semiabelian variety $G$ defined over an algebraically closed field $\mathbb{k}$ of characteristic $0$ with a finite rank subgroup $Γ\le G(\mathbb{k})$ is a finite union of cosets of subgroups of $Γ$. We explore a variant of this conjecture when $G$ is a product of an abelian variety $A$ defined over $\mathbb{k}$ with the additive group $\mathbb{G}_a$.

Mordell猜想中的定量性

Quantitativity in the Mordell Conjecture

本文综述了Mordell猜想的定量版本，即通过Vojta、Dimitrov-Habegger-Gao和Kuhne的工作解决的一致性问题，并介绍了Yu-Yuan-Zhou最新证明的定量结果。

Approximating rational points on toric varieties

Given a smooth projective variety $X$ over a number field $k$ and $P\in X(k)$, the first author conjectured that in a precise sense, any sequence that approximates $P$ sufficiently well must lie on a rational curve. We prove this conjecture for smooth split toric surfaces conditional on Vojta's conjecture. More generally, we show that if $X$ is a $\mathbb{Q}$-factorial terminal split toric variety of arbitrary dimension, then $P$ is better approximated by points on a rational curve than by any Zariski dense sequence.

Divisibility of polynomials and degeneracy of integral points

We prove several statements about arithmetic hyperbolicity of certain blow-up varieties. As a corollary we obtain multiple examples of simply connected quasi-projective varieties that are pseudo-arithmetically hyperbolic. This generalizes results of Corvaja and Zannier obtained in dimension 2 to arbitrary dimension. The key input is an application of the Ru-Vojta's strategy. We also obtain the analogue results for function fields and Nevanlinna theory with the goal to apply them in a future paper in the context of Campana's conjectures.

逼近曲面上的有理点

Approximating rational points on surfaces

本文研究代数簇上有理点的最佳逼近问题，提出由 Vojta 猜想推出相关较弱猜想的策略，并在分裂曲面上完成完整猜想的论证。

双有理平面上的二次点

Quadratic points on double planes

本文研究高维代数簇上Zariski稠密二次点的几何刻画，证明在双覆盖情形下Vojta猜想可排除典范丛充分正时的其他稠密来源，并给出反例说明该条件不可省略。

重访沙法列维奇猜想：带基点的极化代数簇族的有限性

The Shafarevich conjecture revisited: Finiteness of pointed families of polarized varieties

受Lang-Vojta双曲簇猜想启发，研究者证明了带基点极化代数簇族有限性的新版Shafarevich猜想，并将其应用于极化代数簇模空间整点有限性的算术问题。

$abcd$ 猜想、一致有界性与动力系统

The $abcd$ conjecture, uniform boundedness, and dynamical systems

本文综述了Vojta对$abc$猜想和Szpiro猜想的高维推广及其在算术动力学中的应用，特别指出$abcd$猜想蕴含了挠点一致有界性及典范高度下界的动力学类比猜想。

关于Mordell猜想中理性点数量的量化

Quantitativity on the number of rational points in the Mordell conjecture

本文针对Mordell猜想中的理性点数量问题，通过算术和分析估计方法，给出了数域上光滑射影曲线有理点的显式上界。

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…

前向轨道中$(D,S)$-整点的非Zariski密度与子空间定理

On non-Zariski density of $(D,S)$-integral points in forward orbits and the Subspace Theorem

本文在数域上研究射影簇自映射前向轨道中$(D,S)$-整点的非Zariski密度，基于Schmidt子空间定理的推广形式，给出由除子动力学表述的充分条件，拓展了已有相关研究结果。

Uniformity of stably integral points on principally polarized abelian surfaces

We prove, assuming that the conjecture of Lang and Vojta holds true, that there is a uniform bound on the number of stably integral points in the complement of the theta divisor on a principally polarized abelian surface defined over a number field. This gives a uniform version, in the spirit of a result of Caporaso-Harris-Mazur, of an unconditional theorem of Faltings. We utilize recent results of Alexeev and Nakamura on complete moduli for quasi-abelian varieties. We expect that a thorough understanding of current work of Alexeev should give a more general result for abelian varieties of an …

ABC inequalities for some moduli spaces of log-general type

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.

Greatest Common Divisors on the Complement of Numerically Parallel Divisors

We prove inequalities involving greatest common divisors of functions at integral points with respect to numerically parallel divisors, generalizing a result of Wang and Yasufuku (after work of Bugeaud-Corvaja-Zannier, Corvaja-Zannier, and the second author). After applying a result of Vojta on integral points on subvarieties of semiabelian varieties, we use geometry and the theory of heights to reduce to the (known) case of $\mathbb{G}_m^n$. In addition to proving results in a broader context than previously considered, we also study the exceptional set in this setting, for both the counting …

The Hilbert property for arithmetic schemes

We extend the usual Hilbert property for varieties over fields to arithmetic schemes over integral domains by demanding the set of near-integral points (as defined by Vojta) to be non-thin. We then generalize results of Bary-Soroker-Fehm-Petersen and Corvaja-Zannier by proving several structure results related to products and finite étale covers of arithmetic schemes with the Hilbert property.

Uniform bound for the number of rational points on a pencil of curves

Consider a one-parameter family of smooth, irreducible, projective curves of genus $g\ge 2$ defined over a number field. Each fiber contains at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of the family and the Mordell--Weil rank of the fiber's Jacobian. Our proof uses Vojta's approach to the Mordell Conjecture furnished with a height inequality due to the second- and third-named authors. In addition we obtain uniform bounds for the number of torsion in the Jacobian that lie each fiber o…

Uniformity of stably integral points on elliptic curves

A common practice in arithmetic geometry is that of generalizing rational points on projective varieties to integral points on quasi-projective varieties. Following this practice, we demonstrate an analogue of a result of L. Caporaso, J. Harris and B. Mazur, showing that the Lang - Vojta conjecture implies a uniform bound on the number of stably integral points on an elliptic curve over a number field, as well as the uniform boundedness conjecture (Merel's theorem).

Uniformity in Mordell-Lang for curves

Consider a smooth, geometrically irreducible, projective curve of genus $g \ge 2$ defined over a number field of degree $d \ge 1$. It has at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of $g$, $d$, and the Mordell-Weil rank of the curve's Jacobian, thereby answering in the affirmative a question of Mazur. In addition we obtain uniform bounded, in $g$ and $d$, for the number of geometric torsion points of the Jacobian which lie in the image of an Abel-Jacobi map. Both estimates generaliz…

阿贝尔簇上有理挠点与小点位的一致性

Uniformity in rational torsion and small points on abelian varieties

本文提出一种源于沃伊塔证明思路的方法，针对整体域上的阿贝尔簇，明确证明了一致有界性猜想与朗-西尔弗曼猜想，还给出特征p>0函数域上椭圆曲线的相关明确界限，并在提出的弱于Hindry版本的高维斯皮罗猜想下对数域证明了这两个猜想。

逼近球对称簇上的有理点

Approximating rational points on horospherical varieties

该研究在Vojta猜想成立的前提下，构造过球对称簇上有理点的曲线以实现更优的有理点逼近，证明了McKinnon猜想在球对称情形的弱化版本，结果可推广至带末端奇点的Q-阶乘球对称簇。

Finiteness properties of pseudo-hyperbolic varieties

Motivated by Lang-Vojta's conjecture, we show that the set of dominant rational self-maps of an algebraic variety over a number field with only finitely many rational points in any given number field is finite by combining Amerik's theorem for dynamical systems of infinite order with properties of Prokhorov-Shramov's notion of quasi-minimal models. We also prove a similar result in the geometric setting by using again Amerik's theorem and Prokhorov-Shramov's notion of quasi-minimal model, but also Weil's regularization theorem for birational self-maps and properties of dynamical degrees. Furth…

Kodaira维数为零的三维簇的积分Hodge猜想

The integral Hodge conjecture for 3-folds of Kodaira dimension zero

本文证明了Kodaira维数为零且具有非零正则典范形式的三维簇满足积分Hodge猜想，并推广至积分Tate猜想，涵盖任意特征的阿贝尔三维簇。

2-adic integral canonical models and the Tate conjecture in characteristic 2

We use E. Lau's classification of 2-divisible groups using Dieudonné displays to construct integral canonical models for Shimura varieties of abelian type at 2-adic places where the level is hyperspecial. We apply this to prove the Tate conjecture for K3 surfaces in characteristic 2.

Kudla's modularity conjecture on integral models of orthogonal Shimura varieties

We construct a family of special cycle classes on the regular integral model of an orthogonal Shimura variety, and show that these cycle classes appear as Fourier coefficients of a Siegel modular form. Passing to the generic fiber of the Shimura variety recovers a result of Bruinier and Raum, originally conjectured by Kudla.

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。

On some differences between number fields and function fields

The analogy between the arithmetic of varieties over number fields and the arithmetic of varieties over function fields is a leading theme in arithmetic geometry. This analogy is very powerful but there are some gaps. In this note we will show how the presence of isotrivial varieties over function fields (the analogous of which do not seems to exist over number fields) breaks this analogy. Some counterexamples to a statement similar to Northcott Theorem are proposed. In positive characteristic, some explicit counterexamples to statements similar to Lang and Vojta conjectures are given.

Integral and adelic aspects of the Mumford-Tate conjecture

Let $Y$ be an abelian variety over a subfield $k \subset \mathbb{C}$ that is of finite type over $\mathbb{Q}$. We prove that if the Mumford-Tate conjecture for $Y$ is true, then also some refined integral and adelic conjectures due to Serre are true for $Y$. In particular, if a certain Hodge-maximality condition is satisfied, we obtain an adelic open image theorem for the Galois representation on the (full) Tate module of $Y$. Our second main result is an (unconditional) adelic open image theorem for K3 surfaces. The proofs of these results rely on the study of a natural representation of the …

The canonicity of the integral models of RSZ Shimura varieties

We show that the integral models of Shimura varieties of Rapoport, Smithling and Zhang in relation to variants of the arithmetic Gan-Gross-Prasad conjecture, the arithmetic fundamental lemma conjecture and the arithmetic transfer conjecture are canonical in the sense prescribed by Pappas. In particular, we prove that they are isomorphic to the models constructed by Kisin and Pappas.

Manin猜想的半整曲线和$\mathbb A^1$-连通性

Manin's conjecture for semi-integral curves and $\mathbb A^1$-connectedness

本文研究了积分点的Log Manin猜想及其与$\mathbb A^1$-连通性的联系，证明了Campana有理曲线和分裂托里除数上的Log Manin猜想，并结合Cox环描述和Batyrev的启发式计数方法给出几何解释。

关于帕帕斯和拉波波特猜想的证明

On a conjecture of Pappas and Rapoport

本文证明了帕帕斯和拉波波特关于Shimura型变种的 canonical 整体模型存在性的猜想，并展示了通过整体局部Shimura变种统一同源类，以及证明了Kisin和帕帕斯关于局部模型图的猜想。