Weak approximation over function fields

Campana有理连通性与Del Pezzo轨形族的弱逼近

Campana Rational Connectedness and Weak Approximation of Del Pezzo Orbifolds

本文证明低次及具不可约边界的高次del Pezzo轨形族是强Campana无理性覆盖的，从而推出Campana有理连通性，为弱逼近定理提供二维非环面例子，并在附录中分类了次数6、7、8的情形。

Campana有理连通性与弱逼近

Campana rational connectedness and weak approximation

本文证明了复曲线上Campana纤维化在良好约化位置处Campana截面的弱逼近性质，并验证了环状Campana轨道体的更强有理连通性，关键工具为对数几何与稳定对数映射模栈。

伴随群中弱逼近的失效

Failure of Weak Approximation in Adjoint Groups

本文研究伴随群的弱逼近性质，证明其在任意无限域上不成立，反驳了Platonov的猜想。

Weak approximation on Châtelet surfaces

We study weak approximation for Châtelet surfaces over number fields when all singular fibers are defined over rational points. We consider Châtelet surfaces which satisfy weak approximation over every finite extension of the ground field. We prove many of these results by showing that the Brauer-Manin obstruction vanishes, then apply results of Colliot-Thélène, Sansuc, and Swinnerton-Dyer.

低次数del Pezzo曲面的弱逼近

Weak approximation for del Pezzo surfaces of low degree

本研究受Denef工作启发，采用算术满射性方法，证明满足一般性假设的带两个圆锥纤维化的曲面成立弱弱逼近，且次数为1或2的带圆锥纤维化的一般del Pezzo曲面也满足该性质。

某些二次曲面丛的弱弱逼近

Weak weak approximation for certain quadric surface bundles

研究 $\mathbb{P}^3 \times \mathbb{P}^2$ 中一类双二次四维簇在远离有限位集合时的弱逼近问题，部分情形涉及 Hassett--Pirutka--Tschinkel 的近期工作。

Campana点的弱逼近与希尔伯特性

Weak approximation and the Hilbert property for Campana points

该研究探讨Campana点的弱逼近与希尔伯特性，证明弱弱逼近蕴含后者，并借此构造出Campana点集非薄的Campana轨形，为相关Manin型猜想研究提供支撑。

Obstructions to weak approximation for reductive groups over $p$-adic function fields

We establish arithmetic duality theorems for short complexes associated to reductive groups over $p$-adic function fields. Using dualities, we deduce obstructions to weak approximation for certain reductive groups (especially quasi-split ones) and relate this obstruction to an unramified Galois cohomology group.

对称积与范数簇的弱逼近

Weak approximation of symmetric products and norm varieties

本文研究数域上簇的对称积及其光滑射影模型是否保持带Brauer-Manin障碍的弱逼近性质，并将同一方法推广到范数簇。

Compatibility of weak approximation for zero-cycles on products of varieties

Zero-cycles are conjectured to satisfy weak approximation with Brauer-Manin obstruction for proper smooth varieties defined over number fields. Roughly speaking, we prove that the conjecture is compatible for products of rationally connected varieties, K3 surfaces, Kummer varieties, and one curve.

Weak Approximation for Cubic Hypersurfaces and Degree 4 del Pezzo Surfaces

In this article we prove the following theorems about weak approximation of smooth cubic hypersurfaces and del Pezzo surfaces of degree 4 defined over global fields. (1) For cubic hypersurfaces defined over global function fields, if there is a rational point, then weak approximation holds at places of good reduction whose residual field has at least 11 elements. (2) For del Pezzo surfaces of degree 4 defined over global function fields, if there is a rational point, then weak approximation holds at places of good reduction whose residual field has at least 13 elements. (3) Weak approximation …

An example of a Brauer-Manin obstruction to weak approximation at a prime with good reduction

Following Bright and Newton, we construct an explicit K$3$ surface over the rational numbers having good reduction at 2, and for which 2 is the only prime at which weak approximation is obstructed.

Hasse principle and weak approximation for multinorm equations

In this note, we are interested in local-global principles for multinorm equations of the form $\prod_{i=1}^n N_{L_i /k}(z_i) = a$ where $k$ is a global field, $L_i/k$ are finite separable field extensions and $a \in k^*$. In particular, we prove a result relating weak approximation for this equation to weak approximation for some classical norm equation $N_{F/k}(w) = a$ where $F := \bigcap_{i=1}^n L_i$. It provides a proof of a "weak approximation" analogue of a recent conjecture by Pollio and Rapinchuk about multinorm principle. We also provide a counterexample to the original conjecture con…

A Comparison of Cohomological Obstructions to the Hasse Principle and to Weak Approximation

We show that certain Tate--Shafarevich groups are unramified which enables us to give an obstruction to the Hasse principle for torsors under tori over p-adic function fields.

弱逼近与R-等价的Manin群

Weak approximation and Manin group of R-equivalences

我们扩展了一个确切的序列Colliot-Thelene和Santiago连接算术和双有理不变量的环面在数域上的任意连接的线性代数群。

高维函数域上约化群的弱逼近研究

On Weak Approximation of Reductive Groups over Higher Dimensional Function Fields

该研究针对特征0的d进局部域上光滑曲线的函数域，借助算术对偶性给出了域上约化群的弱逼近偏差，推进了高维函数域算术几何相关研究。

Non-invariance of weak approximation with Brauer-Manin obstruction for surfaces

In this paper, we study the property of weak approximation with Brauer-Manin obstruction for surfaces with respect to field extensions of number fields. For any nontrivial extension of number fields L/K, assuming a conjecture of M. Stoll, we construct a smooth, projective, and geometrically connected surface over K such that it satisfies weak approximation with Brauer-Manin obstruction off all archimedean places, while its base change to L fails. Then we illustrate this construction with an explicit unconditional example.

Weak approximation for homogeneous spaces over some two-dimensional geometric global fields

In this article, we study obstructions to weak approximation for connected linear groups and homogeneous spaces with connected or abelian stabilizers over finite extensions of $\mathbb C((x,y))$ or function fields of curves over $\mathbb C((t))$. We show that for connected linear groups, the usual Brauer-Manin obstruction works as in the case of tori. However, this Brauer-Manin obstruction is not enough for homogeneous spaces, as shown by the examples we give. We then construct an obstruction using torsors under quasi-trivial tori that explains the failure of weak-approximation.

Failures of weak approximation in families

Given a family of varieties $X\to \mathbb{P}^n$ over a number field $k$, we determine conditions under which there is a Brauer-Manin obstruction to weak approximation for $100\%$ of the fibres which are everywhere locally soluble.

Weak approximation for tori over $p$-adic function fields

This is the companion piece to "Local-global questions for tori over p-adic function fields" by the first and third authors. We study local-global questions for Galois cohomology over the function field of a curve defined over a p-adic field, the main focus here being weak approximation of rational points. We construct a 9-term Poitou--Tate type exact sequence for tori over a field as above (and also a 12-term sequence for finite modules). Like in the number field case, part of the sequence can then be used to analyze the defect of weak approximation for a torus. We also show that the defect o…

Transcendental obstructions to weak approximation on general K3 surfaces

We construct an explicit K3 surface over the field of rational numbers that has geometric Picard rank one, and for which there is a transcendental Brauer-Manin obstruction to weak approximation. To do so, we exploit the relationship between polarized K3 surfaces endowed with particular kinds of Brauer classes and cubic fourfolds.

弱弱逼近与二次 del Pezzo 曲面的 Hilbert 性质

Weak weak approximation and the Hilbert property for degree-two del Pezzo surfaces

本文证明含特定有理点的二次 del Pezzo 曲面在数域上满足弱弱逼近，在 Hilbert 域上满足 Hilbert 性质，基于单有理性和圆锥纤维结果。

Weak approximation for homogeneous spaces: reduction to the case with finite stabilizer

We reduce the question about whether the Brauer-Manin obstruction to weak approximation for homogeneous spaces is the only obstruction to the "simpler" question of the particular case of homogeneous spaces of $\mathrm{SL}_n$ with finite stabilizer. We also give unconditional results based on the few known results on finite stabilizers.

A cohomological obstruction to weak approximation for homogeneous spaces

Let X be a homogeneous space, X = G/H, where G is a connected linear algebraic group over a number field k, and H is a k-subgroup of G (not necessarily connected). Let S be a finite set of places of k. We compute the Brauer-Manin obstruction to weak approximation for X in S in terms of Galois cohomology.

Weak approximation, Brauer and R-equivalence in algebraic groups over arithmetical fields

We prove some new relations between weak approximation and some rational equivalence relations (Brauer and R-equivalence) in algebraic groups over arithmetical fields. By using weak approximation and local - global approach, we compute completely the group of Brauer equivalence classes of connected linear algebraic groups over number fields, and also completely compute the group of R-equivalence classes of connected linear algebraic groups $G$, which either are defined over a totally imaginary number field, or contains no anisotropic almost simple factors of exceptional type $^{3,6}\D_4$, nor …

Campana点在射影空间上的弱逼近

Weak approximation for Campana points on projective spaces

本文证明了在射影直线（带log Fano边界）和射影$n$-空间（边界为$n+3$个一般位置超平面或两个系数$1/2$的横截光滑二次曲面）上，Campana点具有弱逼近性质。

Separable rational connectedness and weak approximation in positive characteristic

In this short note we give a characterization of smooth projective varieties of Picard number one that are separably uniruled but not separably rationally connected. We also give a sufficient condition involving the torsion order and the uniruling index for a smooth Fano variety of Picard number one to be separably rationally connected. As an application, we prove some weak approximation results for Fano complete intersections in positive charactersitic. For example, we show that weak approximation holds at place of strong potentially good reduction for a Fano complete intersection in $\mathbb…

Non-invariance of weak approximation properties under extension of the ground field

For rational points on algebraic varieties defined over a number field $K$, we study the behavior of the property of weak approximation with Brauer-Manin obstruction under extension of the ground field. We construct K-varieties accompanied with a quadratic extension $L/K$ such that the property holds over $K$ (conditional on a conjecture) while fails over $L$. The result is unconditional when $K = \mathbb{Q}$ or $K$ is one of several quadratic number fields. Over $\mathbb{Q}$, we give an explicit example.

几何全局域上的递归性与弱逼近

Reocurrence and weak approximation over geometric global fields

该研究在两类几何全局域上证明了递归定理，部分替代Chebotarev定理，并将其应用于带有限稳定子的SLₙ齐性空间的弱逼近研究。

上同调稳定性与弱逼近

Homological stability and weak approximation

研究Fano纤维空间的上同调稳定性与弱逼近，证明了投影包和曲面包的上同调稳定性。

Weak approximation on del Pezzo surfaces of degree 1

We study del Pezzo surfaces of degree 1 of the form w^2 = z^3 + Ax^6 + By^6 in the weighted projective space P_k(1,1,2,3), where k is a perfect field of characteristic not 2 or 3 and A,B \in k^*. Over a number field, we exhibit an infinite family of (minimal) counterexamples to weak approximation amongst these surfaces, via a Brauer-Manin obstruction.

The defect of weak approximation for homogeneous spaces. II

Let X be a homogeneous space of a connected linear algebraic group G' over a number field k, containing a k-point x. Assume that the stabilizer of x in G' is connected. Using the notion of a quasi-trivial group, recently introduced by Colliot-Thélène, we can represent X in the form X=G/H, where G is a quasi-trivial k-group and H is a connected k-subgroup of G. Let S be a finite set of places of k. Applying results of [B2], we compute the defect of weak approximation for X with respect to S in terms of the biggest toric quotient T of H. In particular, we show that if T splits over a metacyclic …

整体域上约化群的弱逼近缺陷

The defect of weak approximation for a reductive group over a global field

该研究针对整体域上的约化群，提出利用其代数基本群计算弱逼近缺陷的方法，解决了相关代数计算问题。

Weak Approximation over Function Fields of Curves over Large or Finite Fields

Let $K=k(C)$ be the function field of a curve over a field $k$ and let $X$ be a smooth, projective, separably rationally connected $K$-variety with $X(K)\neq\emptyset$. Under the assumption that $X$ admits a smooth projective model $π: \mathcal{X}\to C$, we prove the following weak approximation results: (1) if $k$ is a large field, then $X(K)$ is Zariski dense; (2) if $k$ is an infinite algebraic extension of a finite field, then $X$ satisfies weak approximation at places of good reduction; (3) if $k$ is a nonarchimedean local field and $R$-equivalence is trivial on one of the fibers $\mathca…