Brauer Groups and Tate-Shafarevich Groups

J. Math. Sci. Univ. Tokyo
Vol. 10 (2003), No. 2, Page 391--419.

Gonzalez-Aviles, Cristian D.
Brauer Groups and Tate-Shafarevich Groups
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Let $X_K$ be a proper, smooth and geometrically connected curve over a global field $K$. In this paper we generalize a formula of Milne relating the order of the Tate-Shafarevich group of the Jacobian of $X_K$ to the order of the Brauer group of a proper regular model of $X_K$. We thereby partially answer a question of Grothendieck.

Keywords: Brauer groups, Tate-Shafarevich groups, Jacobian variety, index of a curve, period of a curve, Cassels-Tate pairing

Mathematics Subject Classification (1991): 11G35, 14K15, 14G25
Mathematical Reviews Number: MR1987138

Received: 2001-05-01