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Brauer groups and Tate-Shafarevich groups. (English) Zbl 1029.11026
Let \(X_k\) be a proper, smooth and geomerically connected curve over a global field \(k\), and let \(A\) be the Jacobian variety of \(X_k\). Let \(X\) be a 2-dimensional proper, regular model of \(X_k\). After the work of A. Grothendieck, J. Milne gave a connection between the Tate-Shafarevich group of \(A\) and the Brauer group of \(X\) under the assumption that the index \(\delta_v\) of \(X_{k_v}\) equals 1 for all prime \(v\) of \(\pi\). In this paper, the author generalizes Milne’s formula when the \(\delta_v\)’s are no longer equal to 1 and thereby answers partially a question posed by Grothendieck. For the proof, the author generalizes Milne’s methods and employs the Albanese-Picard pairing of Poonen and Stoll instead of the Cassels-Tate pairing used by Milne. In an appendix, the compatibility of the Cassels-Tate pairing with the Albanese-Picard pairing of Poonen and Stoll is verified.

11G35 Varieties over global fields
14K15 Arithmetic ground fields for abelian varieties
14G25 Global ground fields in algebraic geometry