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

2018/05/22 by Liang, Yongqi
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1805.08851

Abstract

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 = ℚ or K is one of several quadratic number fields. Over ℚ, we give an explicit example.

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