关于结式簇的一个猜想的明确陈述

广义最大公因数、整除序列和Blowup的Vojta猜想

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

我们应用Vojta的猜想爆破和推导出一些深的声明（广义）最大公因数的品种，特别是在射影空间和阿贝尔簇。这些陈述的特殊情况推广了以前的结果和定理。我们还讨论了广义最大公因数与代数群上的整除序列之间的关系，并应用Vojta猜想得到了维数至少为2的阿贝尔簇上的整除序列的一个强界.

On the Zilber-Pink conjecture for complex abelian varieties

In this article, we prove that the Zilber-Pink conjecture for abelian varieties over an arbitrary field of characteristic $0$ is implied by the same statement for abelian varieties over the algebraic numbers. More precisely, the conjecture holds for subvarieties of dimension at most $m$ in the abelian variety $A$ if it holds for subvarieties of dimension at most $m$ in the largest abelian subvariety of $A$ that is isomorphic to an abelian variety defined over $\bar{ \mathbb{Q}}$.

阿贝尔簇上的丢番图逼近；M. Waldschmidt的一个猜想

Diophantine approximation on abelian varieties; a conjecture of M. Waldschmidt

本文研究阿贝尔簇上的丢番图逼近，将Waldschmidt猜想等价转化为矩阵条件，提出较弱猜想并证明其上界，对秩1椭圆曲线获得弱型Dirichlet定理并证明猜想。

阿贝尔簇中超曲面的Shafarevich猜想

The Shafarevich conjecture for hypersurfaces in abelian varieties

本文证明：在维数至少为4的固定阿贝尔簇中，表示给定丰富类且在固定素数集外具有良好约化的光滑超曲面，在平移意义下仅有有限多个，从而验证了该情形的Shafarevich猜想。

Unlikely Intersections in families of abelian varieties and the polynomial Pell equation

Let S be a smooth irreducible curve defined over a number field k and consider an abelian scheme A over S and a curve C inside A, both defined over k. In previous works, we proved that when A is a fibered product of elliptic schemes, if C is not contained in a proper subgroup scheme of A, then it contains at most finitely many points that belong to a flat subgroup scheme of codimension at least 2. In this article, we continue our investigation and settle the crucial case of powers of simple abelian schemes of relative dimension g bigger or equal than 2. This, combined with the above mentioned …

Weak Modular Zilber-Pink with Derivatives

In unpublished notes Pila proposed a Modular Zilber-Pink with Derivatives (MZPD) conjecture, which is a Zilber-Pink type statement for the modular $j$-function and its derivatives. In this article we define D-special varieties, then state and prove two functional (differential) analogues of the MZPD conjecture for those varieties. In particular, we prove a weak version of MZPD. As a special case of our results, we obtain a functional Modular André-Oort with Derivatives statement. The main tools used in the paper come from (model theoretic) differential algebra and complex analytic geometry, an…

半阿贝尔变异体上的小高点

Points de petite hauteur sur les varietes semi-abeliennes

我们延长的情况下，半阿贝尔簇的陈述的各种变体的猜想阿拉Bogomolov关于非密度的小点小高度的阿贝尔簇。受Ullmo，Zhang和Bilu最近工作的启发，我们证明了当半阿贝尔簇几乎分裂时的这些定理。证明使用Arakelov几何和equirepartition参数。

阿贝尔型 Shimura 簇的广义 André-Pink-Zannier 猜想

Generalised André-Pink-Zannier Conjecture for Shimura varieties of abelian type

本文证明了所有阿贝尔型 Shimura 簇的广义 André-Pink-Zannier 猜想，提出了适用于所有 Shimura 簇的通用命题，并证明了关于 p-adic 整数上齐次空间良好约化与 Mumford 稳定性关系的 p-adic Kempf-Ness 定理。

关于1-模的Mumford-Tate猜想

On the Mumford-Tate conjecture for 1-motives

我们表明，声明类似于Mumford-Tate猜想的阿贝尔簇持有1-动机的单幂部分。这是通过比较关联的霍奇群的幂幺部分和绝对伽罗瓦群的像的幂幺部分与motivic基本群的幂幺部分来完成的。包含P. Deligne的附录。在这个版本中：这里和那里的小更正。关于伽罗瓦表示的象的主要结果现在被一个阿德利克陈述所丰富。

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.

球面簇上的球面函数

Spherical functions on spherical varieties

设X=H \G是p进域k的整数o上可裂约化群G的齐次球面簇，K=G（o）是G=G（k）的超特殊极大紧子群.我们在G的非分歧（或球形）Hecke代数的作用下计算X=X（k）上的本征函数（“球形函数”），推广了“Casselman-Shalika”类型的许多经典结果。在X上的一些附加假设下，我们还证明了公式的一个变体，它涉及L值的某个商，并且我们提出了几个应用，例如：（1）关于无重数情况下“好测试向量”的说明（即，不可约非分歧表示π上的H不变泛函在π^K上非零），（2）X的非分歧Plancherel公式，包括X（o）的“Tamagawa测度”公式;（3）主Eisenstein级数H-周期积分最连续部分的计算。