Descent and étale-Brauer obstructions for 0-cycles

2022/02/11 by Francesca Balestrieri, Jennifer Berg, Balestrieri, Francesca +1
Mathematics · #11G35 #14C25 #14G05 #14G12 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2202.08120

openalex publication_date 2022/02/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

For 0-cycles on a variety over a number field, we define an analogue of the classical descent set for rational points. This leads to, among other things, a definition of the étale-Brauer obstruction set for 0-cycles, which we show is contained in the Brauer-Manin set and is compatible with Suslin's singular homology of degree 0. We then transfer some tools and techniques used to study the arithmetic of rational points into the setting of 0-cycles. For example, we extend the strategy developed by Y. Liang, relating the arithmetic of rational points over finite extensions of the base field to that of 0-cycles, to torsors. We give applications of our results to study the arithmetic behaviour of 0-cycles for Enriques surfaces, torsors given by (twisted) Kummer varieties, universal torsors, and torsors under tori.

Related

• Arithmetic of rational points and zero-cycles on Kummer varieties
• Abelianized Descent Obstruction for 0-Cycles
• Applications of the fibration method for zero-cycles to the Brauer-Manin obstruction to the existence of zero-cycles on certain varieties
• Zero-cycles on varieties over p-adic fields and Brauer groups
• Product of Brauer--Manin obstruction for 0-cycles over number fields and function fields