On different notions of tameness in arithmetic geometry

Tame coverings of arithmetic schemes

We extend the notion of a tame covering of a pair (X,D) where X is a regular scheme and D is a normal crossing divisor (cf. SGA1), to pairs (X,Y) where X is an arbitrary scheme and Y is a closed subset in X. We show that the abelianized tame fundamental group of a regular scheme which is flat and of finite type over Spec(Z) is finite and does not depend on the choice of a particular compactification.

Good Sections of Arithmetic Fundamental Groups

In this paper we exhibit the notion of (uniformly) good sections of arithmetic fundamental groups. We introduce and investigate the problem of cuspidalisation of sections of arithmetic fundamental groups, its ultimate aim is to reduce the solution of the Grothendieck anabelian section conjecture to the solution of its birational version. We show that (uniformly) good sections of arithmetic fundamental groups of smooth, proper, and geometrically connected hyperbolic curves over slim (and regular) fields can be lifted to sections of cuspidally abelian absolute Galois groups. As an application we…

等分布与算术$Λ$-分布

Equidistribution and arithmetic $Λ$-distributions

本文提出了容许$\mathbb{Z}$-集参数化概率空间族的抽象等分布概念，证明其矩母函数可表为动机Euler乘积，并结合Poonen筛法推广计算了函数域$L$-函数和zeta函数的渐近$Λ$-分布。

局部域上双曲曲线的算术

Arithmetic of hyperelliptic curves over local fields

本文提出一种描述局部域上双曲曲线性质的聚类图方法，用于编码其伽罗瓦表示、判别式及其他不变量。

Higgs Bundles in Geometry and Arithmetic

We introduce the notions of deformation Higgs bundle and Riemann-Finsler metric on the moduli space of polarized varieties. We also use the Higgs-de Rham flow in the p-adic setting. These are the key novelties in our program.

Self-Similarity in Geometry, Algebra and Arithmetic

We define the concept of self-similarity of an object by considering endomorphisms of the object as `similarity' maps. A variety of interesting examples of self-similar objects in geometry, algebra and arithmetic are introduced. Self-similar objects provide a framework in which, one can unite some results and conjectures in different mathematical frameworks. In some general situations, one can define a well-behaved notion of dimension for self-similar objects. Morphisms between self-similar objects are also defined and a categorical treatment of this concept is provided. We conclude by some ph…

Supergeometry and Arithmetic Geometry

We define a superspace over a ring $R$ as a functor on a subcategory of the category of supercommutative $R$-algebras. As an application the notion of a $p$-adic superspace is introduced and used to give a transparent construction of the Frobenius map on $p$-adic cohomology of a smooth projective variety over the ring of $p$-adic integers.

On the arithmetic of weighted complete intersections of low degree

A variety is rationally connected if two general points can be joined by a rational curve. A higher version of this notion is rational simple connectedness, which requires suitable spaces of rational curves through two points to be rationally connected themselves. We prove that smooth, complex, weighted complete intersections of low enough degree are rationally simply connected. This result has strong arithmetic implications for weighted complete intersections defined over the function field of a smooth, complex curve. Namely, it implies that these varieties satisfy weak approximation at all p…