1982f Comparison of the Brauer group with the Tate-S(h)afarevic(h) group
J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 735--743 (1982).

Comments

The duality theorems for abelian varieties have been extended to $p$ --- see the addenda for the second edition of my book Arithmetic Duality Theorems and González-Avilès and Tan 20071 . Therefore the power of $p$ can be dropped from both 1.2 and 1.3.

In the paper, I assumed that the indices of the curves over the local fields all equal $1$ (condition 1.1b). González-Avilès 20032 weakens this to "the local indices are relatively prime". He also verifies that two pairings, which I assumed to be equal in the paper, are in fact equal.

Let $A$ be an abelian variety over a global field $K$ of nonzero characteristic. When $A$ is a Jacobian (as in the paper), then the results of the paper and Milne 1975a show that the following are equivalent (see 1.6):

  1. The $L$-series $L(A,s)$ of $A$ has a zero at $s=1$ of order equal to the rank of $A(K)$;
  2. for some prime $l$ ($l=p$ allowed), the $l$-primary component of the Tate-Shafarevich group of $A$ is finite;
  3. the Tate-Shafarevich group of $A$ is finite and the conjecture of Birch and Swinnerton-Dyer (i.e., Conjecture B of Tate's Bourbaki talk, Exp. 306) is true for $A$.
Bauer (1992)3 proves this for an abelian variety $A$ with good reduction at every prime of $K$ (with $l=p$ in (b)), and Kato and Trihan (2003)4 prove the equivalence of (b) and (c) for an arbitrary abelian variety. On combining the theorem of Kato and Trihan with the main theorem of Milne 1975a, one obtains a proof of conjecture (d) of Tate 1975: the conjecture of Artin and Tate holds for a surface fibred over a curve if and only if the conjecture of Birch and Swinnerton-Dyer holds for the Jacobian of the generic fibre (because both are equivalent to the finiteness of some $l$-component of the Brauer, or Tate-Shafarevich, group).
1. González-Avilès, Cristian D.; Tan, Ki-Seng. A generalization of the Cassels-Tate dual exact sequence. Math. Res. Lett. 14 (2007), no. 2, 295--302. ↩
2. González-Avilès, Cristian D., Brauer groups and Tate-Shafarevich groups. J. Math. Sci. Univ. Tokyo 10 (2003), no. 2, 391--419. ↩
3. Bauer, Werner, On the conjecture of Birch and Swinnerton-Dyer for abelian varieties over function fields in characteristic $p>0$ Invent. Math. 108 (1992), no. 2, 263--287. ↩
4.Kato, Kazuya; Trihan, Fabien On the conjectures of Birch and Swinnerton-Dyer in characteristic $p>0$ Invent. Math. 153 (2003), no. 3, 537--592. ↩