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arXiv · 2110.13127

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

David Harari·Tamás Szamuely
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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.

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David Harari, Tamás Szamuely. 2021-10-25. On Tate--Shafarevich groups of one-dimensional families of commutative group schemes over number fields. https://arxiv.org/abs/2110.13127

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