
TL;DR
This paper studies the ramified descent problem over number fields, introducing a Brauer-Manin obstruction that can be transcendental and non-trivial, providing counterexamples to the Hasse principle and weak approximation.
Contribution
It introduces a new Brauer-Manin obstruction to the ramified descent problem, including transcendental cases, and provides explicit counterexamples for certain algebraic quotients.
Findings
Brauer-Manin obstruction can be transcendental and non-trivial.
Counterexamples to Hasse principle and weak approximation are constructed.
Obstruction applies to abelian covers and specific algebraic quotients.
Abstract
We investigate the "ramified descent problem": which adelic points of a smooth geometrically connected variety defined over a number field 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 , with constant metabelian.
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