Fundamental groups and Diophantine geometry

Adelic 线丛、算术正性与丢番图几何

Adelic Line Bundles, Arithmetic Positivity and Diophantine Geometry

本文综述拟射影簇上 adelic 线丛的基础、算术正性及其在均匀分布定理和一致 Bogomolov 猜想中的应用。

Théorèmes d'algébrisation en géométrie diophantienne (Algebricity theorems in diophantine geometry)

In a recent paper, J.-B. Bost establishes a criterion for certain ``formal subvarieties'' of algebraic varieties to be algebraic. His theorem unifies and generalizes results of Chudnovsky's and Y. André, motivated by an arithmetic conjecture of Grothendieck that predicts that the solutions of certain differential equations are algebraic functions. The proof makes use of the diophantine approximation techniques introduced by these authors but with a systematic geometric point of view, notably via Arakelov geometry and the formalism of ``slopes''.

Diophantine geometry and non-abelian reciprocity laws I

We use non-abelian fundamental groups to define a sequence of higher reciprocity maps on the adelic points of a variety over a number field satisfying certain conditions in Galois cohomology. The non-abelian reciprocity law then states that the global points are contained in the kernel of all the reciprocity maps.

On Profinite Hyperbolicity and Diophantine Geometry

In this note, we explore the notion of hyperbolicity of topologically finitely generated profinite groups. Some applications to diophantine geometry are suggested and we try to reformulate certain problems in diophantine geometry in terms of hyperbolic profinite groups. Then, we introduce many occasions in which Galois groups are free profinite and try to explore implications of this condition in the world of diophantine geometry. In particular, we prove that, Grothendieck's "section conjecture" plus Shafarevich's "freeness conjecture" imply that hyperbolic curves have infinitely many solution…

On the Complexity of Diophantine Geometry in Low Dimensions

We consider the average-case complexity of some otherwise undecidable or open Diophantine problems. More precisely, we show that the following two problems can be solved in the complexity class PSPACE: (I) Given polynomials f_1,...,f_m in Z[x_1,...,x_n] defining a variety of dimension <=0 in C^n, find all solutions in Z^n of f_1=...=f_m=0. (II) For a given polynomial f in Z[v,x,y] defining an irreducible nonsingular non-ruled surface in C^3, decide the sentence ``\exists v \forall x \exists y such that f(v,x,y)=0?'' quantified over N. Better still, we show that the truth of the Generalized Rie…

Finiteness Problems in Diophantine Geometry

This survey contains an exposition of ideas and results related to Faltings' proof of the conjectures of Shafarevich, Tate and Mordell. This paper originally appeared in 1986 as an Appendix to the Russian translation of Serge Lang, "Fundamentals of Diophantine Geometry" (Springer Verlag, 1983) published by "Mir", Moscow (MR0854670, 88a:11054). A history of the publication of the Appendix is briefly described by Lang in Section 4 of his paper "Mordell's review, Siegel's letter to Mordell, Diophantine geometry, and 20th century mathematics" that was published (in 1995) simultaneously in Notices …

Hensel 极小性 II：混合特征与丢番图应用

Hensel minimality II: Mixed characteristic and a diophantine application

本文为特征零 Hensel 赋值域建立 Hensel 极小性框架，处理混合特征情形，证明 Jacobian 性质、Taylor 逼近、胞腔分解等几何结果，并应用于计数曲线上有界高度有理点。

Invitation to Integral and Rational points on curves and surfaces

This survey article is an introduction to Diophantine Geometry at a basic undergraduate level. It focuses on Diophantine Equations and the qualitative description of their solutions rather than detailed proofs.

The motivic fundamental group of the projective line minus three points and the theorem of Siegel

In this paper, we establish a link between the structure theory of the pro-unipotent motivic fundamental group of the projective line minus three points and Diophantine geometry. In particular, we give a p-adic proof of Siegel's theorem.

Modularity and integral points on moduli schemes

The purpose of this paper is to give some new Diophantine applications of modularity results. We use the Shimura-Taniyama conjecture to prove effective finiteness results for integral points on moduli schemes of elliptic curves. For several fundamental Diophantine problems (e.g. $S$-unit and Mordell equations), this gives an effective method which does not rely on Diophantine approximation or transcendence techniques. We also combine Faltings' method with Serre's modularity conjecture, isogeny estimates and results from Arakelov theory, to establish the effective Shafarevich conjecture for abe…

Hensel minimality I

We present a framework for tame geometry on Henselian valued fields which we call Hensel minimality. In the spirit of o-minimality, which is key to real geometry and several diophantine applications, we develop geometric results and applications for Hensel minimal structures that were previously known only under stronger, less axiomatic assumptions. We show existence of t-stratifications in Hensel minimal structures and Taylor approximation results which are key to non-archimedean versions of Pila-Wilkie point counting, Yomdin's parameterization results and to motivic integration. In this firs…

Equivariant geometry of odd-dimensional complete intersections of two quadrics

Fix a finite group $G$. We seek to classify varieties with $G$-action equivariantly birational to a representation of $G$ on affine or projective space. Our focus is odd-dimensional smooth complete intersections of two quadrics, relating the equivariant rationality problem with analogous Diophantine questions over nonclosed fields. We explore how invariants -- both classical cohomological invariants and recent symbol constructions -- control rationality in some cases.

An Introduction to $\mathbf{A}^1$-Enumerative Geometry

We provide an expository introduction to $\mathbb{A}^1$-enumerative geometry, which uses the machinery of $\mathbb{A}^1$-homotopy theory to enrich classical enumerative geometry questions over a broader range of fields. Included is a discussion of enriched local degrees of morphisms of smooth schemes, following Morel, $\mathbb{A}^1$-Milnor numbers, as well as various computational tools and recent examples. Based off lectures delivered by Kirsten Wickelgren at the 2018 LMS-CMI summer school ''Homotopy Theory and Arithmetic Geometry, Motivic and Diophantine Aspects.''

Algebraization, transcendence, and D-group schemes

We present a conjecture in Diophantine geometry concerning the construction of line bundles over smooth projective varieties over $\bar{\mathbb Q}}$. This conjecture, closely related to the Grothendieck Period Conjecture for cycles of codimension 1, is also motivated by classical algebraization results in analytic and formal geometry and in transcendence theory. Its formulation involves the consideration of $D$-group schemes attached to abelian schemes over algebraic curves over $\bar{\mathbb Q}}$. We also derive the Grothendieck Period Conjecture for cycles of codimension 1 in abelian varieti…

Non-abelian fundamental groups in arithmetic geometry

This is a brief exposition of the mathematical themes that motivate the special programme at the Newton Institute in 2009. It is mostly intended for the general public having mathematical training up to the level of secondary school.

计算算术几何 I：接近多项式层次的句子

Computational Arithmetic Geometry I: Sentences Nearly in the Polynomial Hierarchy

本文研究丢番图问题的平均情况复杂性，证明问题(I)在coNP内，问题(II)在广义黎曼猜想下属于多项式层次第三层，并建议避免在密码系统中使用此类问题。

Some fundamental groups in arithmetic geometry

Those are the notes for the 2015 Summer Research Institute on Algebraic Geometry. We report on Deligne's finiteness theorem for $\ell$-adic representations on smooth varieties defined over a finite field, on its crystalline version, and on how the geometric étale fundamental group of a smooth projective variety defined over a characteristic $p>0$ field controls crystals on the infinitesimal site and should control those on the crystalline site. v2: last results added to the report, and some typos corrected.

Fundamental Theorem of Projective Geometry over Semirings

We state the fundamental theorem of projective geometry for semimodules over semirings, which is facilitated by recent work in the study of bases in semimodules defined over semirings. In the process we explore in detail the linear algebra setup over semirings. We also provide more explicit results to understand the implications of our main theorem on maps between tropical lines in the tropical plane. Along with this we also look at geometrical connections to the rich theory of tropical geometry

折线凸几何基础

Fundamentals of Broken Line Convex Geometry

本文提出了一种基于折线段的凸几何理论，将环境空间定义为聚类簇的有理热带化，并证明了多项标准凸几何结果的折线凸几何版本。

Étale 基本群——代数几何中基本群的几何与拓扑方法

Étale Fundamental Groups: a geometric and topological approach to fundamental groups in algebraic geometry

本文通过拓扑覆盖空间、Étale拓扑和Tannakian对偶三种框架，系统构建并比较了代数几何中的基本群概念。