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arXiv · 2112.05557

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

Haowen Zhang
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Abstract

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.

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Haowen Zhang. 2021-12-10. Weak approximation for homogeneous spaces over some two-dimensional geometric global fields. https://arxiv.org/abs/2112.05557

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