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- Title
On Tate–Shafarevich groups of one-dimensional families of commutative group schemes over number fields.
- Authors
Harari, David; Szamuely, Tamás
- 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.
- Subjects
ABELIAN groups; ABELIAN varieties; ELLIPTIC curves; FINITE fields; TORUS
- Publication
Mathematische Zeitschrift, 2022, Vol 302, Issue 2, p935
- ISSN
0025-5874
- Publication type
Article
- DOI
10.1007/s00209-022-03080-x