A Carlitz module analogue of the Grunwald--Wang theorem

A Grunwald-Wang type theorem for abelian varieties

Let A be an abelian variety over a number field k. We show that weak approximation holds in the Weil-Châtelet group of A/k but that it may fail when one restricts to the n-torsion subgroup. This failure is however relatively mild; we show that weak approximation holds outside a finite set of primes which is generically empty. This proves a conjecture of Lang and Tate that can be seen as an analog of the Grunwald-Wang theorem in class field theory. The methods apply, for the most part, to arbitrary finite Galois modules and so may be of interest in their own right.

关于广义Grunwald-Wang问题中的缺陷

On the defect in the generalized Grunwald--Wang problem

研究广义Grunwald-Wang定理中特殊情况下障碍群的有限性和有界性，通过有理函数域和仿射直线点的离散赋值构造反例，证明该群可能无限且阶无界。

关于$n$次幂的Grunwald-Wang定理的一个推广

A Generalization of the Grunwald-Wang Theorem for $n^{th}$ Powers

本研究将n次幂的Grunwald-Wang定理从单个有理数推广至基数不超过n的最小素因子q的有理数有限子集，证明了几乎所有素数p下子集在Q_p中含n次幂等价于含完全n次幂（偶数n有例外），且上界q最优。

Grunwald–Wang定理，一个有效版本

Grunwald- Wang Theorem, an Effective Version

本文建立Grunwald–Wang定理的有效版本，依据域、指数、素点集和局部导子范数界定全局特征导子，并给出一个依赖GRH与两个无条件界。

超有限域上多项式环的高次互反律及 Grunwald–Wang 定理的类比

Higher reciprocity law and An analogue of the Grunwald--Wang theorem for the ring of polynomials over an ultra-finite field

本文研究有限域非主超积上的多项式环，建立显式高次互反律，并据此证明特征 0 与特征 p 下 Grunwald–Wang 定理的类比。

Stable sets of primes in number fields

We define a new class of sets -- stable sets -- of primes in number fields. For example, Chebotarev sets $P_{M/K}(σ)$, with $M/K$ Galois and $σ\in \Gal(M/K)$, are very often stable. These sets have positive (but arbitrary small) Dirichlet density and generalize sets with density 1 in the sense that arithmetic theorems like certain Hasse principles, the Grunwald-Wang theorem, the Riemann's existence theorem, etc. hold for them. Geometrically this allows to give examples of infinite sets $S$ with arbitrary small positive density such that $\Spec \mathcal{O}_{K,S}$ is algebraic $K(π,1)$ (for all …