On the Brauer group of a surface
The authors establish the following theorem in full generality: Let \(X\) be a smooth proper surfaces over a finite field \(k\) of characteristic \(p\). Suppose that the Brauer group of \(X\) has finite \(l\)-part for some prime number \(l\). Then the order of the Brauer group is a square. Here the assumption on the \(l\)-part is needed to ensure that the Brauer group is finite. The story of this remarkable theorem starts with results of \textit{J. Tate} [in: Dix Exposés Cohomologie Schémas, Adv. Stud. Pure Math. 3, 189--214 (1968; Zbl 0199.55604)], who constructed an antisymmetric pairing on the prime-to-\(p\) part of \(\text{ Br}(X)\), whose radical equals the subgroup of divisible elements. If the entire Brauer group is finite, and furthermore carries an alternating nondegenerate pairing, it would follow that the order of the Brauer group must be a square. Instead of looking for such a pairing, the authors follow a completely different line of reasoning. The basic steps are as follows: First, they modify the surface \(X\) so that there is a fibration \(X\rightarrow\mathbb P^1\) with smooth generic fiber. Care must be taken because we are over a finite field, but things work well due to results of \textit{O. Gabber} [Geom. Funct. Anal. 11, 1192--1200 (2001; Zbl 1072.14513)]. Next, they examine the generic fiber \(X_K/K\) and its Jacobian \(A_K/K\), where \(K\) is the function field of the projective line. Using results \textit{K. Kato} and \textit{F. Trihan} [Invent. Math. 153, 537--592 (2003; Zbl 1046.11047)], they observe, as Saper pointed out, that the conjecture of Birch-Swinnerton-Dyer on the \(L\)-function of the abelian variety \(A_K/K\) is equivalent to the conjecture of Artin-Tate on the Brauer group of the surface \(X\). The equvialence of these two conjectures was itself a conjecture, sometimes called conjecture d) of Artin-Tate. In an earlier paper [Invent. Math. 157, 455--518 (2004; Zbl 1060.14037)], the authors showed, assuming that the Brauer group of \(X\) or equivalently the Shafarevich-Tate group of \(A_K/K\) is finite, that conjecture d) of Artin-Tate holds for \(X\) if and only if the numerical condition \[ | \text{Ш}(A_K)| \prod_\nu\delta_\nu\delta_\nu'=\delta^2| \text{Br}(X)| \] is true, and that the left-hand side then must be a square. The latter statement depends on results of \textit{B. Poonen} and \textit{M. Stoll} [Ann. Math. (2)150, 1109--1149 (1999; Zbl 1024.11040)]. In the preceding equation, the \(\nu\) are the various places of \(K\), and \(\delta_\nu,\delta'_\nu\) are index and period of the induced curves \(X_{K_\nu}\).
scientific article; zbMATH DE number 2156035
scientific article