The finiteness of the Tate–Shafarevich group over function fields for algebraic tori defined over the base field
Let $K$ be a field and $V$ be a set of rank one valuations of $K$. The corresponding Tate–Shafarevich group of a $K$-torus $T$ is $\Sha (T, V) = \ker (H^1(K, T) \rightarrow \prod _{v\,\in \,V} H^1(K_v, T))$. We prove that if $K = k(X)$ is the function field of a smooth geometrically integral quasi-p...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2024-09-01
|
| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.588/ |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1825206217597779968 |
|---|---|
| author | Rapinchuk, Andrei Rapinchuk, Igor |
| author_facet | Rapinchuk, Andrei Rapinchuk, Igor |
| author_sort | Rapinchuk, Andrei |
| collection | DOAJ |
| description | Let $K$ be a field and $V$ be a set of rank one valuations of $K$. The corresponding Tate–Shafarevich group of a $K$-torus $T$ is $\Sha (T, V) = \ker (H^1(K, T) \rightarrow \prod _{v\,\in \,V} H^1(K_v, T))$. We prove that if $K = k(X)$ is the function field of a smooth geometrically integral quasi-projective variety over a field $k$ of characteristic 0 and $V$ is the set of discrete valuations of $K$ associated with prime divisors on $X$, then for any torus $T$ defined over the base field $k$, the group $\Sha (T, V)$ is finite in the following situations: (1) $k$ is finitely generated and $X(k) \ne \emptyset $; (2) $k$ is a number field. |
| format | Article |
| id | doaj-art-2f080f520a734acf87e0778fcb4997da |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-09-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-2f080f520a734acf87e0778fcb4997da2025-02-07T11:22:28ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-09-01362G773974910.5802/crmath.58810.5802/crmath.588The finiteness of the Tate–Shafarevich group over function fields for algebraic tori defined over the base fieldRapinchuk, Andrei0Rapinchuk, Igor1Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USADepartment of Mathematics, Michigan State University, East Lansing, MI 48824, USALet $K$ be a field and $V$ be a set of rank one valuations of $K$. The corresponding Tate–Shafarevich group of a $K$-torus $T$ is $\Sha (T, V) = \ker (H^1(K, T) \rightarrow \prod _{v\,\in \,V} H^1(K_v, T))$. We prove that if $K = k(X)$ is the function field of a smooth geometrically integral quasi-projective variety over a field $k$ of characteristic 0 and $V$ is the set of discrete valuations of $K$ associated with prime divisors on $X$, then for any torus $T$ defined over the base field $k$, the group $\Sha (T, V)$ is finite in the following situations: (1) $k$ is finitely generated and $X(k) \ne \emptyset $; (2) $k$ is a number field.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.588/ |
| spellingShingle | Rapinchuk, Andrei Rapinchuk, Igor The finiteness of the Tate–Shafarevich group over function fields for algebraic tori defined over the base field Comptes Rendus. Mathématique |
| title | The finiteness of the Tate–Shafarevich group over function fields for algebraic tori defined over the base field |
| title_full | The finiteness of the Tate–Shafarevich group over function fields for algebraic tori defined over the base field |
| title_fullStr | The finiteness of the Tate–Shafarevich group over function fields for algebraic tori defined over the base field |
| title_full_unstemmed | The finiteness of the Tate–Shafarevich group over function fields for algebraic tori defined over the base field |
| title_short | The finiteness of the Tate–Shafarevich group over function fields for algebraic tori defined over the base field |
| title_sort | finiteness of the tate shafarevich group over function fields for algebraic tori defined over the base field |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.588/ |
| work_keys_str_mv | AT rapinchukandrei thefinitenessofthetateshafarevichgroupoverfunctionfieldsforalgebraictoridefinedoverthebasefield AT rapinchukigor thefinitenessofthetateshafarevichgroupoverfunctionfieldsforalgebraictoridefinedoverthebasefield AT rapinchukandrei finitenessofthetateshafarevichgroupoverfunctionfieldsforalgebraictoridefinedoverthebasefield AT rapinchukigor finitenessofthetateshafarevichgroupoverfunctionfieldsforalgebraictoridefinedoverthebasefield |