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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.

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