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 2007
1
. 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):
- The $L$-series $L(A,s)$ of $A$ has a zero at $s=1$ of order equal to the
rank of $A(K)$;
- for some prime $l$ ($l=p$ allowed), the $l$-primary component of the
Tate-Shafarevich group of $A$ is finite;
- 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.
↩