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

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 that the Tate--Shafarevich group is finite, generalizing a result of Sa\"idi 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.