arXiv subjects

Yisheng Tian

Publications and source records attributed to Yisheng Tian.

7 recordsLinked to original sources

arXiv2026

Cohomological Obstructions for Varieties over $p$-adic Function Fields

We introduce various cohomological obstructions for smooth integral varieties over $p$-adic function fields. We show that the unramified obstruction is the finest one among obstructions arising from arithmetic dualities. We also construct an explicit example whose unramified obstruction gives a proper obstruction while Manin obstruction does not. Finally, we compare the unramified obstruction with the descent obstruction.

math.NT↗

arXiv2026

A Simpler Approach to a Descent Conjecture of Wittenberg

A descent conjecture of Wittenberg [Wit24, Conjecture 3.7.4] predicts that if all the twists of a rationally connected torsor over a smooth base satisfy weak approximation with Brauer-Manin obstruction, then so does the base. We give an alternative proof of Wittenberg's conjecture for certain torsors under connected linear groups via Cao's descent formula.

math.AG↗

arXiv2024

Strong Approximation and Hasse Principle for Integral Quadratic Forms over Affine Curves

We extend some parts of the representation theory for integral quadratic forms over the ring of integers of a number field to the case over the coordinate ring $k[C]$ of an affine curve $C$ over a general base field $k$. By using the genus theory, we link the strong approximation property of certain spin groups to the Hasse principle for representations of integral quadratic forms over $k[C]$ and derive several applications. In particular, we give an example where a spin group does not satisfy strong approximation.

math.NT↗

arXiv2024

Trivialité des groupes de Whitehead réduits avec applications à l'approximation faible et l'approximation forte

We prove some triviality results for reduced Whitehead groups and reduced unitary Whitehead groups for division algebras over a henselian discrete valuation field whose residue field has virtual cohomological dimension or separable dimension $\le 2$. These results are applied to show strong approximation for isotropic absolutely almost simple simply connected groups of type A. In particular, such a group defined over the function field of a nonreal curve $C/k$ satisfies strong approximation if the base field $k$ is a number field, a $p$-adic field, $\mathbb{C}(\!(t)\!)$ or a two-variable function field over $\mathbb{R}$.

math.NT↗

arXiv2022

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.

math.NT↗

arXiv2019

Poitou--Tate sequence for complex of tori over $p$-adic function fields

We complete the picture of local and global arithmetic duality theorems for short complexes of finite Galois modules and tori over $p$-adic function fields. In view of the duality theorems, we deduce a $12$-term Poitou--Tate exact sequence which relates global Galois cohomology groups to restricted topological products of local Galois cohomology groups.

math.NT↗

arXiv2019

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.

math.NT↗