The finiteness of the Tate-Shafarevich group over function fields for algebraic tori defined over the base field

2023/07/03 by Rapinchuk, Andrei S., Rapinchuk, Igor A.
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2307.01185

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 Sha(T , V) = ker(H¹(K , T) → ∏_{v ∈ V} H¹(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) ≠ ∅; (2) k is a number field.

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