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

2021/12/10 by Zhang, Haowen
#11E72 #14G12 #14G27 #14M17 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2112.05557

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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