Ann. of Math. (2) 102 (1975), no. 3, 517--533.
Comments
The results in this article are easy for a prime $\ell\neq p$. The point is that both the zeta function
of a surface and the $\ell$-part of its Brauer group can be described by the same cohomology groups,
namely, the étale groups $H^i(X,\mu_{\ell^n})$, which were well understood by the 1970s. By contrast,
the $p$-part the Brauer group is described by the flat cohomology groups $H^i(X,\mu_{p^n})$ and the
zeta function by the crystalline cohomology groups, neither of which were very well understood. So I had to understand
the two cohomologies, and work out the relation between them. Of course, this is all much better understood now.
The condition $p\neq 2$
The only reason I had to assume that the characteristic $p$ is odd in the
paper is that, at the time, Bloch's paper (listed as a preprint) was the only
reference for what is now called the de Rham-Witt complex and it requires $p$
to be odd. Illusie's paper Ann ENS 1979 doesn't require condition, so if you
change the reference from Bloch to Illusie, you can drop the condition
"$p$ odd".
Alternatively, you can deduce the main theorem (4.1) of the paper from Theorem
0.4b of my AJM 1986 paper, which doesn't assume $p$ odd.
In more detail: I use the condition p odd only in Theorem 2.1 of the paper.
The proof of that theorem used my flat duality theorem for a surface, which
used Bloch's paper, which assumes p odd. Illusie didn't require p odd, so if
you replace the reference to Bloch by a reference to Illusie you can drop the
condition from my paper 1976a (Ann ENS) and hence from my 1975 paper.
However, the argument in my paper only proves that the pairing on the Brauer
group is skew-symmetric (not alternating), so it only gives that the order of
the Brauer group is a square or twice a square (if finite) --- for a proof
that it is always a square, see Liu,
Qing; Lorenzini, Dino; Raynaud, Michel On the Brauer group of a surface.
Invent. Math. 159 (2005), no. 3, 673--676.
In my papers 1986a and 1988b I prove more general results than in my 1975
paper without the condition $p$ odd (by then Illusie's paper on de Rham-Witt was available).
Conjecture (d) of Tate's 1966 Bourbaki talk
This says that, for a surface $V$ fibred over a curve $C$, the full conjecture
of Birch and Swinnerton-Dyer holds for the Jacobian $A$ of the generic fibre
if and only if the Artin-Tate conjecture holds for the surface $V$.
Ultimately, Conjecture (d) was proved by combining the following two statements:
- The Artin-Tate conjecture holds for a surface $V$ over a finite field if
and only if $\mathrm{Br}(V)(\ell)$ is finite for some prime $\ell$ (Milne 1975a,
Theorems 4.1, 6.1);
- The full conjecture of Birch and Swinnerton-Dyer holds for an abelian
variety $A$ over a global field of nonzero characteristic if and only if
$TS(A)(\ell)$ is finite for some $\ell$ (Kato,
Kazuya; Trihan, Fabien, On the conjectures of Birch and Swinnerton-Dyer in
characteristic $p>0$. Invent. Math. 153 (2003), no. 3, 537--592).
In the situation of Conjecture (d), $\mathrm{Br}(V)(\ell)$ is finite if and
only if $TS(A)(\ell)$ is finite. For $\ell\neq p$, which is all that
is needed, this is an easy calculation in étale cohomology, which
was known to Artin and Tate in the mid 1960s --- see Tate's 1966 Bourbaki talk.
Of course, this was known to all the experts once the two statements had been proved.
Erratum
p.525 The statement of (4.1d$^{\prime}$) should read: $\mathrm{NS}%
(X)\otimes\mathbb{\hat{Z}}$ (the hat is missing). (From Timo Keller).
Caution
The results of this paper are often mistakenly credited to others by
careless authors. Apart from the usual sloppiness, this seems to result from a misperception that a torsion group is finite if each
$\ell$-primary component, $\ell\neq p$, is finite. Here are a few
examples.
Grothendieck 1966, Le groupe de Brauer III, p.169, credits Artin and Tate
with proving that the Tate conjecture for a surface over a finite field
implies the finiteness of the Brauer group of the surface. This was, in fact,
first proved in my 1975 paper.
Artin and Swinnerton-Dyer (1973)1 claim in their introduction to prove the
finiteness of the Tate-Shafarevich group of the general elliptic curve in their family;
only by reading the paper, do you discover that they don't. What they do prove is
the Tate conjecture for
the elliptic $K3$-surface --- to deduce the finiteness of the Tate-Shafarevich group,
you have to apply the theorem in my 1975 paper. Since few mathematicians read beyond the introduction,
this has led to Artin being credited with results he didn't prove every time
he gets another award or honor.2
Nygaard 1983, p.213. Says that, for an elliptic elliptic $K3$-surface, the Tate conjecture is
equivalent to the finiteness of the Tate-Shafarevich group. True, but the proof requires the results
of my 1975 paper.
Sugiyama, Ken-ichi, 2004, J. Differential Geom. 68, 73--98, mistakenly credits
Tate with a major result of my paper.
Coates et al., JAlg, 2009, p.658, mistakenly credit Artin and Tate with a
result that was not known before my 1975 paper.
I should mention that Tate himself is scrupulous about giving credit for these
results, for example, in this talk on the Millenium Prizes.
1. Artin, M.; Swinnerton-Dyer, H. P. F. , The Shafarevich-Tate conjecture for pencils of elliptic curves on K3 surfaces. Invent. Math. 20 (1973), 249--266..↩
2. See for example the citation for the Wolf prize,
which made no sense in its orginal form, and little sense in its "corrected" form.
Artin himself claims (wisely) to have not read the citation.↩