The finiteness of the Tate–Shafarevich group over function fields for algebraic tori defined over the base field
Year: 2024
Type: article
Abstract: 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 Ш ( T , V ) = ker ( H 1 ( K , T ) → ∏ v ∈ 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 Ш ( T , V ) is finite in the following situations: (1) k is finitely generated and X ( k ) ≠ ∅ ; (2) k is a number field.
Source: Comptes Rendus Mathématique
DOI: 10.5802/crmath.588
Authors: Andrei Rapinchuk, Igor Rapinchuk
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
Funder: National Science Foundation
OpenAlex ID: W4402610202