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 \(H^1(K,G)\) 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 finiteness of generalizing a result of Saïdi and Tamagawa for abelian varieties. We also give examples of nontrivial in the case when G is a torus and prove other related statements.
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