2014/05/31 by Qing Liu, Fei Xu
Computer Science · Mathematics · #Abelian group #Abelian variety #Algebraic Geometry and Number Theory #Algebraic number #Algebraic number field #Conjecture #Field (mathematics) #Function (biology) #Function field #Geometry #Global field #Mathematical analysis #Mathematics #Meromorphic and Entire Functions #Polynomial and algebraic computation #Pure mathematics #Torus #math.AG #math.NT #msc:11G35 #msc:14F22 #msc:14G05 #msc:14G25
paper · pdf · doi:10.1007/s00208-014-1107-6
published as Math. Ann., 363 (2015), 701-731 · 25 pages. Minor corrections. To appear in Math. Ann
arxiv created 2014/09/18 · openalex publication_date 2015/02/09 · arxiv updated 2016/09/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let F be a global field. In this work, we show that the Brauer-Manin condition on adelic points for subvarieties of a torus T over F cuts out exactly the rational points, if either F is a function field or, if F is the field of rational numbers and T is split. As an application, we prove a conjecture of Harari-Voloch over global function fields which states, roughly speaking, that on any rational hyperbolic curve, the local integral points with the Brauer-Manin condition are the global integral points. Finally we prove for tori over number fields a theorem of Stoll on adelic points of zero-dimensional subvarieties in abelian varieties.