On Tate--Shafarevich groups of one-dimensional families of commutative group schemes over number fields

2021/10/25 by David Harari, Harari, David, Tamás Szamuely +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2110.13127

openalex publication_date 2021/10/25 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

Given a smooth geometrically connected curve C over a field k and a smooth commutative group scheme G of finite type over the function field K of C we study the Tate--Shafarevich groups given by elements of H¹(K,G) locally trivial at completions of K associated with closed points of C. When G comes from a k-group scheme and k is a number field (or k is a finitely generated field and C has a k-point) we prove that the Tate--Shafarevich group is finite, generalizing a result of Saïdi and Tamagawa for abelian varieties. We also give examples of nontrivial Tate--Shafarevich groups in the case when G is a torus and prove other related statements.

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