On geometric Brauer groups and Tate-Shafarevich groups

2020/12/03 by Yanshuai Qin, Qin, Yanshuai
Mathematics · #14F22 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2012.01681

openalex publication_date 2020/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a smooth projective variety over a finitely generated field K of characteristic p>0. We proved that the finiteness of the ℓ-primary part of Br(X_K^s)^G_K for a single prime ℓ≠ p will imply the finiteness of the prime-to-p part of Br(X_K^s)^G_K, generalizing a theorem of Tate and Lichtenbaum for varieties over finite fields. For an abelian variety A over K, we proved a similar result for the Tate-Shafarevich group of A, generalizing a theorem of Schneider for abelian varieties over global function fields.

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