Dragos Ghioca, Thomas J. Tucker
We prove a dynamical version of the Bogomolov conjecture in the special case of lines in affine space A^m under the action of a map (f_1,...,f_m) where each f_i is a polynomial in Q-bar[X] of the same degree.
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We prove the geometric Bogomolov conjecture over a function field of characteristic zero.
We propose a formulation of the relative Bogomolov conjecture and show that it gives an affirmative answer to a question of Mazur's concerning the uniformity of the Mordell-Lang conjecture for curves. In particular we show that the relative Bogomolov conjecture implies the uniform Manin-Mumford conjecture for curves. The proof is built up on our previous work "Uniformity in Mordell-Lang for curves".
We formulate a conjecture for semiabelian varieties A over number fields that includes both the Mordell-Lang conjecture (now proven) and the Bogomolov conjecture. We prove the "Mordellic" (finitely generated) part of the conjecture when A is isogenous to the product of an abelian variety and a torus. The proof makes use of the Mordell-Lang conjecture, the Bogomolov conjecture, and an equidistribution theorem.
In "Mordell-Lang plus Bogomolov", we formulated a conjecture combining the Mordell-Lang conjecture (now proven) and the Bogomolov conjecture, and proved the Mordellic part when A is almost split (i.e., isogenous to the product of an abelian variety and a torus). Here we prove that the full conjecture would follow from the Bogomolov conjecture and an equidistribution result for general semiabelian varieties. These assumptions have been proved when A is almost split, so the conjecture is completely proven in that case.
Geometric Bogomolov conjecture for semiabelian varieties
本文证明了函数域上半阿贝尔簇的几何Bogomolov猜想，刻画了闭子簇包含Zariski稠密小点集的充要条件，并揭示了特殊子簇可能缺乏高度为0点的Zariski稠密集这一新现象。
We prove the Effective Bogomolov Conjecture, and so the Bogomolov Conjecture, over a function field of characteristic 0 by proving Zhang's Conjecture about certain invariants of metrized graphs. In the function field case, these conjectures were previously known to be true only for curves of genus at most 4 and a few other special cases. We also either verify or improve the previous results. We relate the invariants involved in Zhang's Conjecture to the tau constant of metrized graphs. Then we use and extend our previous results on the tau constant. By proving another Conjecture of Zhang, we o…
The Bogomolov Conjecture is a finiteness statement about algebraic points of small height on a smooth complete curve defined over a global field. We verify an effective form of the Bogomolov Conjecture for all curves of genus at most 4 defined over a function field of characteristic zero. We recover the known result for genus 2 curves and in many cases improve upon the known bound for genus 3 curves. For many curves of genus 4 with bad reduction, the conjecture was previously unproved.
We prove the Bogomolov conjecture for an abelian variety A over a function field which is totally degenerate at a place v. We adapt Zhang's proof of the number field case replacing the complex analytic tools by tropical analytic geometry. A key step is the tropical equidistribution theorem for A at the totally degenerate place. As an application, we obtain finiteness of torsion points with coordinates in the maximal unramified algebraic extension over v.
Uniform Bogomolov Conjecture for Tori
本文为代数环面的统一 Bogomolov 猜想提供了新证明，引入非退化子簇定义并建立等分布定理，方法基于统一 Mordell-Lang 猜想的最新进展。
The Bogomolov conjecture claims that a closed subvariety containing a dense subset of small points is a special kind of subvarieties. In the arithmetic setting over number fields, the Bogomolov conjecture for abelian varieties has already been established as a theorem of Ullmo and Zhang, but in the geometric setting over function field, it has not yet been solved completely. There are only some partial results known such as the totally degenerate case due to Gubler and our recent work generalizing Gubler's result. The key in establishing the previous results on the Bogomolov conjecture is the …
We prove Zhang's Dynamical Manin-Mumford Conjecture and Dynamical Bogomolov Conjecture for dominant endomorphisms of (P^1)^n. We use the equidistribution theorem for points of small height with respect to an algebraic dynamical system, combined with an analysis of the symmetries of the Julia set for a rational function.
Mordell--Lang plus Geometric Bogomolov
本文证明了特征零半阿贝尔簇上 Poonen 定理的几何版本，推广了 Mordell--Lang 与几何 Bogomolov 猜想。
We prove the Dynamical Bogomolov Conjecture for endomorphisms of P^1\times P^1 defined over a number field. We use the equidistribution theorem for points of small height with respect to an algebraic dynamical system, combined with a theorem of Levin regarding symmetries of the Julia set. Using a specialization theorem of Yuan and Zhang, we prove the Dynamical Manin-Mumford Conjecture for endomorhisms of P^1\times P^1 defined over the complex numbers.
We deduce an analogue of the Bogomolov conjecture for non-degenerate subvarieties in fibered products of families of elliptic curves from the author's recent theorem on equidistribution in families of abelian varieties. This generalizes results of DeMarco and Mavraki and improves certain results of Manin-Mumford type proven by Masser and Zannier to results of Bogomolov type, yielding the first results of this type for subvarieties of relative dimension $>1$ in families of abelian varieties with trivial trace.
Let K be a finitely generated field over Q, and A an abelian variety over K. Let <, > : A(K^a) x A(K^a) --> R be an arithmetic height pairing on A, where K^a is the algebric closure of K. For x_1,..., x_l \in A(K^a), we denote det(<x_i, x_j>) by d(x_1,..., x_l). Let G be a subgroup of finite rank in A(K^a), and X a subvariety of A_{K^a}. Fix a basis {g_1,..., g_n} of G_Q. In this note, we prove a generalization of Poonen's theorem: If the set {x \in X(K^a) | d(g_1,..., g_n, x) <= e} is Zariski dense in X for every positive number e, then X is a translation of an abelian subvariety by an elemen…
This is an introduction to the topics of the title, from the 2017 Grenoble Summer school on Arakelov geometry and arithmetic applications. We review Arithmetic intersection numbers, explain the definition of the height of a variety and its properties, both in the framework of classical Arakelov geometry and of Zhang's adelic formalism. We then discuss arithmetic ampleness and its application to the equidistribution theorem of Szpiro-Ullmo-Zhang-Yuan, both at complex and nonarchimedean places. We conclude with Ullmo-Zhang's proof of the Bogomolov conjecture over number fields.
Uniform Height Gaps in Arbitrary Characteristics
该研究推广了Gao-Ge-Kühne的间隙原理，通过阿代尔曲线与全局赋值域理论证明任意特征下阿贝尔簇子簇的一致高度间隙，还给出了Bogomolov猜想的新证明途径。
A uniform lower bound for the Zhang-Kawazumi invariant and applications to the Bogomolov conjecture
本文证明了紧致黎曼曲面Zhang-Kawazumi不变量的统一下界，并应用于Bogomolov猜想，得到自交数与高度界的新下界。
In this paper we study the degeneration behavior of the norm of the Riemann $θ$-function in a family of principally polarized abelian varieties over the punctured complex unit disc in terms of the associated polarized real torus. As an application, we obtain the degeneration behavior of the Zhang--Kawazumi invariant $φ(M_t)$ of a family of Riemann surfaces $M_t$ in terms of Zhang's invariant $φ(Γ)$ of the associated metrized reduction graph $Γ$. This allows us to deduce a uniform lower bound for the essential minimum of the Néron-Tate height on the tautological cycles of any Jacobian variety o…
This is an enhanced version of the author's 1998 Harvard Ph.D. thesis, as published by IMRN in 2005. We propose an extension of Bogomolov's conjecture about commutator subgroups of Galois groups to arbitrary fields. A somewhat weaker conjecture is then shown to follow, together with the famous Milnor-Bloch-Kato conjecture, from a certain hypothesis about homological properties of the Milnor K-theory algebras.
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 …
New gap principle for semiabelian varieties using globally valued fields
本文基于 Hrushovski 的观察，通过将 Bogomolov 猜想归约，证明了半阿贝尔簇在正特征下的新间隙原理，其中阿贝尔商为椭圆曲线。
We finish the proof of the conjecture of F. Bogomolov and F. Pop: Let $F_{1}$ and $F_{2}$ be fields finitely-generated and of transcendence degree $\geq 2$ over $k_{1}$ and $k_{2}$, respectively, where $k_{1}$ is either $\bar{\mathbb{Q}}$ or $\bar{\mathbb{F}}_{p}$, and $k_{2}$ is algebraically closed. We denote by $G_{F_1}$ and $G_{F_2}$ their respective absolute Galois groups. Then the canonical map $φ_{F_{1}, F_{2}}: \Isom^i(F_1, F_2)\rightarrow \Isom^{\Out}_{\cont}(G_{F_2}, G_{F_1})$ from the isomorphisms, up to Frobenius twists, of the inseparable closures of $F_1$ and $F_2$ to continuous …
On the dynamical Bogomolov conjecture for families of split rational maps
本文证明张寿武动力学Bogomolov猜想在有理分裂映射单参数族上一致成立，并建立分裂映射与簇的高度不等式，从而首次将Baker-DeMarco猜想推广到高维并得到任意维几何Bogomolov定理。
Dynamique analytique sur $\mathbf{Z}$. II : Écart uniforme entre Lattès et conjecture de Bogomolov-Fu-Tschinkel
本文证明了 Lattès 动力系统相互作用能的一致下界，进而证明了 Bogomolov-Fu-Tschinkel 猜想，核心依赖于 $\mathbf{Z}$ 上的 Berkovich 空间理论与从中心估计推导全局估计的原创论证。
On the Bogomolov-Positselski Conjecture
本文提出了一个新准则，用于判断定向pro-p群是否具有博戈莫洛夫-波西茨尔基性质，并展示了初等类型pro-p群满足更强条件
We prove a sharp lower bound for the essential minimum of a non-translate variety in certain abelian varieties. This uses and generalises a result of Galateau. Our bound is a new step in direction of an abelian analogue by David and Philippon of a toric conjecture of Amoroso and David and has applications in the framework of anomalous intersections.
A quantitative dynamical Zhang fundamental inequality and Bogomolov-type problems
本文通过证明张基本不等式的定量版本，扩展了Néron-Tate高度的间隙原理，并在算术和几何情形中提供了动力学波戈莫洛夫猜想的有效实例。
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。
We give a short proof of the "prime-to-$p$ version" of the Manin-Mumford conjecture for an abelian variety over a number field, when it has supersingular reduction at a prime dividing $p$, by combining the methods of Bogomolov, Hrushovski, and Pink-Roessler. Our proof here is quite simple and short, and neither $p$-adic Hodge theory nor model theory is used. The observation is that a power of a lift of the Frobenius element at a supersingular prime acts on the prime-to-$p$ torsion points via nontrivial homothety.
We extend to the case of semi-abelian varieties the statements of various variants of the conjecture alla Bogomolov about the non-density of small points of small height in abelian varieties. Inspired by recent work of Ullmo, Zhang and Bilu, we then prove these conjectures when the semi-abelian variety is almost split. The proof uses Arakelov geometry and equirepartition arguments.
Intersecting the torsion of elliptic curves
本文结合Nevanlinna理论与Uniform Manin-Mumford猜想，证明了Bogomolov-Fu-Tschinkel猜想的推广，允许使用任意有界次数且分支轨迹不同的有理函数代替偶次度-2映射，并给出了数域上椭圆曲线秩的下界结果。
The Geometric Dynamical Northcott and Bogomolov Properties
本研究在特征零函数域上建立极化自同态的动力Northcott性质，关联复动力系统稳定性，推广已有一维与高维结果，还提出几何Bogomolov猜想的动力版本。
Consider a fibered power of an elliptic surface. We characterize its subvarieties that contain a Zariski dense set of points that are torsion points in fibers with complex multiplication. This result can be viewed as a mix of the Manin-Mumford and André-Oort Conjecture and is related to a conjecture of Pink. The main technical tool is a new height inequality. We also use it to give another proof of a case of Gubler's result on the Bogomolov Conjecture over function fields.
A Bogomolov property for moduli spaces of polynomials over abelian extensions
本文证明了在阿贝尔扩张上，多项式模空间中临界高度小于某正数的点仅有有限多个，即后临界有限点有限，方法涉及通用临界除子族、adelic线丛、交织关系、局部分歧估计及Ji–Song–Xie对应方法。
Weakly-special threefolds and non-density of rational points
本文验证了Campana猜想的一部分，通过建立轨道映射模空间性质并利用Kobayashi-Ochiai定理的推广，证明了Bogomolov-Tschinkel构造的弱特殊三维簇上有理点不稠密。
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…
DAO for curves
本文证明了 Baker–DeMarco 关于代数曲线参数化有理映射族的 DAO 猜想，并进一步给出其 Bogomolov 型推广。
We consider hypersurfaces in $(\mathbb{P}^1)^n$ that contain a generic sequence of small dynamical height with respect to a split map and project onto $n-1$ coordinates. We show that these hypersurfaces satisfy strong coincidence relations between their points with zero height coordinates. More precisely, it holds that in a Zariski-open dense subset of such a hypersurface $n-1$ coordinates have height zero if and only if all coordinates have height zero. This is a key step in the resolution of the dynamical Bogomolov conjecture for split maps.
The conjecture of Lehmer is proved to be true. The proof mainly relies upon: (i) the properties of the Parry Upper functions $f_{house(α)}(z)$ associated with the dynamical zeta functions $ζ_{house(α)}(z)$ of the Rényi--Parry arithmetical dynamical systems, for $α$ an algebraic integer $α$ of house "$house(α)$" greater than 1, (ii) the discovery of lenticuli of poles of $ζ_{house(α)}(z)$ which uniformly equidistribute at the limit on a limit "lenticular" arc of the unit circle, when $house(α)$ tends to $1^+$, giving rise to a continuous lenticular minorant ${\rm M}_{r}(house(α))$ of the Mahler…
Dobbertin, Mills, Müller, Pott and Willems conjecture that two families of power mapping are families of APN functions. Here we prove those two conjectures.
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