Finiteness of the Tate-Shafarevich group over function fields for groups of multiplicative type

2026/07/16 by Igor A. Rapinchuk, Avinash Roy
#math.NT #math.AG

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Abstract

Let K = k(X) be the function field of a smooth geometrically integral variety X of dimension ≥ 2 over a field k of characteristic 0 and V be the set of discrete valuations of K associated with the prime divisors on X. We show that if D is a k-defined group of multiplicative type, then the corresponding Tate-Shafarevich group Sha(D,V) = ker (H¹(K,D) → ∏_{v ∈ V} H¹(K_v, D) ) is finite in the following situations: (1) k is finitely generated and X(k) ≠ ∅; (2) k is a number field. This complements previous work of Harari and Szamuely, which considered the case where X is a curve.

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