Ramified descent
Julian Lawrence Demeio
TL;DR
This work studies ramified descent for open varieties and shows that the ramified descent set $X( obreak ext{A}_K)^{ obreak ext{λ}}$ obstructs both the Hasse principle and weak approximation on the compactification $X$. It introduces a Brauer–Manin obstruction to ramified descent, $ ext{Br}_{ obreak ext{λ}}^{ ext{ram}}X$, which can be transcendental and may strictly enlarge the algebraic obstruction; this yields negative answers to Harari's question on the compatibility of ramified descent with Brauer obstructions. The paper further provides explicit computations in the setting $U=SL_n/H$ with $H$ metabelian, establishing a concrete transcendental obstruction to weak approximation for such quotients and giving explicit counterexamples where $ ext{Br}_aX=0$ but $X( obreak ext{A}_K)^{ ext{Br}X} eq X( obreak ext{A}_K)$. The methods combine ramified descent, Čech–to–étale cohomology, and a detailed analysis of unramified Brauer groups, yielding new phenomena beyond the unramified Brauer–Manin obstruction and enriching our understanding of local-global principles in arithmetic geometry.
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.