Equidistribution of Dynamically Small Subvarieties over the Function Field of a Curve

Galois orbits and equidistribution: towards the Andre-Oort conjecture

In this paper we develop a strategy and some technical tools for proving the Andre-Oort conjecture. We give lower bounds for the degrees of Galois orbits of geometric components of special subvarieties of Shimura varieties, assuming the Generalised Riemann Hypothesis. We proceed to show that sequences of special subvarieties whose Galois orbits have bounded degrees are equidistributed in a suitable sense.

Equidistribution des sous-varietes de petite hauteur

In this paper, the equidistribution theorem of Szpiro-Ullmo-Zhang about sequences of small points in an abelian variety is extended to the case of sequences of higher dimensional subvarieties. A quantitative version of this result is also given.

On Special Subvarieties of Kuga Varieties

In this paper we prove the equidistribution of certain families of special subvarieties in Kuga varieties, which is a special case of the general Andre-Oort conjecture formulated for mixed Shimura varieties proposed by R.Pink. Our approach is parallel to the pure case treated in the works of L.Clozel, E.Ullmo, and A.Yafaev using tools from ergodic theory.

Equidistribution of expanding translates of curves and Diophantine approximation on matrices

We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space $G/Γ$ of a semisimple algebraic group $G$. We define two families of algebraic subvarieties of the associated partial flag variety $G/P$, which give the obstructions to non-divergence and equidistribution. We apply this to prove that for Lebesgue almost every point on an analytic curve in the space of $m\times n$ real matrices whose image is not contained in any subvariety coming from these two families, the Dirichlet's theorem on simultaneous Dio…

Bounded eqidistribution of special subvarieties in mixed Shimura varieties

In this paper we prove the equidistribution of bounded sequences of special subvarieties in a general mixed Shimura varieties, a notion adapted from the pure case treated by Clozel, Ullmo, and Yafaev in the study of the Andre-Oort conjecture. We also discuss the relation of bounded sequences with bounded Galois orbits of special subvarieties.

Equidistribution theorem over an adelic curve

In this article, we introduce the notion of global adelic space of an arithmetic variety over an adelic curve and prove an equidistribution theorem for a generic sequence of subvarieties. As an application, we prove a Bogomolov type theorem for Abelian varieties over an adelic curve of characteristic 0.

Equidistribution de sous-variétés spéciales et o-minimalité: André-Oort géométrique

A characterization of subvarieties of Shimura varieties which contain a Zariski dense subset of weakly special subvarieties has been proved by the second author, by combining o-minimality results and functional transcendence results. In this paper, we obtain a new proof of this statement by dynamics techniques on homogeneous spaces in the spirit of the earlier work of Clozel and the second author. The proof combines ergodic theory à la Ratner, with a statement on the dimension of a Hausdorff limit of a sequence of definable subsets (in an o-minimal theory) extracted from a definable family. On…

Equidistribution over function fields

We prove equidistribution of a generic net of small points in a projective variety X over a function field K. For an algebraic dynamical system over K, we generalize this equidistribution theorem to a small generic net of subvarieties. For number fields, these results were proved by Yuan and we transfer here his methods to function fields. If X is a closed subvariety of an abelian variety, then we can describe the equidistribution measure explicitly in terms of convex geometry.

Some problems of arithmetic origin in rational dynamics

These are lecture notes from a course in arithmetic dynamics given in Grenoble in June 2017. The main purpose of this text is to explain how arithmetic equidistribution theory can be used in the dynamics of rational maps on P^1. We first briefly introduce the basics of the iteration theory of rational maps on the projective line over C, as well as some elements of iteration theory over an arbitrary complete valued field and the construction of dynamically defined height functions for rational functions defined over \overline Q. The equidistribution of small points gives some original informati…

高维本质最小值与闭链的等分布

Higher dimensional essential minima and equidistribution of cycles

针对高维子簇缺乏本质最小值概念及其等分布研究不足的问题，本文提出高维本质最小值，并据此证明通有小有效闭链的等分布，推广了既有定理。

The Relative Bogomolov Conjecture for Fibered Products of Elliptic Curves

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.

Equidistribution quantitative des points de petite hauteur sur la droite projective

We introduce a new class of adelic heights on the projective line. We estimate their essential minimum and prove a result of equidistribution (at every place) for points of small height with estimates on the speed of convergence. To each rational function R in one variable and defined over a number field K, is associated a normalized height on the algebraic closure of K. We show that these dynamically defined heights are adelic in our sense, and deduce from this equidistribution results for preimages of points under R at every place of K. Our approach follows that of Bilu, and relies on potent…

模曲线乘积中特殊轨迹的p进等分布性

$p$-adic Equidistribution of Special Loci in a Product of Modular Curves

该论文研究定义在ℂₚ上的光滑曲线C与模子簇Y₀(n)交轨迹的p进等分布性，给出等分布成立的条件，证明对应交集合非离散，并给出等分布不成立的典型例子。

相对马宁-芒福德猜想

The Relative Manin-Mumford Conjecture

本文在特征0下证明阿贝尔簇族的相对马宁-芒福德猜想，采用皮拉-赞尼尔方法，结合新的高度不等式与退化轨迹研究，还强化结果给出复数域上阿贝尔概法子簇挠点稠密判据，并给出一致马宁-芒福德猜想的无等分布新证明。

Strict supports of canonical measures and applications to the geometric Bogomolov conjecture

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 …