Ramified descent

Abstract

We investigate the "ramified descent problem": which adelic points of a smooth geometrically connected variety $X$ defined over a number field $K$ can be approximated by points that lift to a (twist of a) given ramified cover? We show that the natural descent set corresponding to the problem defines an obstruction to Hasse Principle and weak approximation. Furthermore, we introduce a Brauer-Manin obstruction to the problem. This obstruction can be purely transcendental (and non-trivial) even for abelian covers, which answers in the negative a question posed by Harari at a 2019 workshop. Moreover, the counterexample we produce is also an explicit example of transcendental obstruction to weak approximation for a quotient $SL_n/G$, with $G$ constant metabelian.