The bilinear form of the Brauer group of a surface
Let \(X\) be a proper smooth surface over a finite field \({\mathbb{F}}\) of characteristic \(p\). Let \({\text{Br}}(X)\) denote the Brauer group of \(X\). If \({\text{Br}}(X)\) is a torsion group, one has the direct sum decomposition \({\text{Br}}(X)=\sum_{\ell}{\text{Br}}(X)(\ell)\) where \(\ell\) runs through all primes and \({\text{Br}}(X)(\ell)\) stands for the \(\ell\)-torsion subgroup. Let \({\text{Br}}(X)(\ell)_{\text{{div}}}\) denote the maximal divisible subgroup of \({\text{Br}}(X)(\ell)\). \textit{J. Tate} [in: Sém. Bourbaki 1965/66, Exposé 306 (1966; Zbl 0199.55604)] proved that if \(\ell\neq p\), the quotient \({\text{Br}}(X)(\ell)_{\text{{nd}}}:={\text{Br}}(X)(\ell)/{\text{Br}} (X)(\ell)_{\text{{div}}}\) is finite and it is equipped with a non-degenerate skew-symmetric bilinear form with values in \({\mathbb{Q}}/{\mathbb{Z}}\). Further, Tate conjectured that the bilinear form on \({\text{Br}}(X)(\ell)_{\text{{nd}}}\) is alternating. It is easily seen that if \(\ell\neq 2\) and \(\ell\neq p\), Tate's form is alternating on \({\text{Br}}(X)(\ell)_{\text{{nd}}}\) and the order \(\#{\text{Br}}(X)(\ell)_{\text{{nd}}}\) is a square. The problem remains open when \(\ell=p\neq 2\), and indeed, the main result of this paper addresses this case. Let \(\overline X =X\times_{\text{{Spec}}{\mathbb{F}}} {\text{Spec}}(\overline{\mathbb{F}})\) be the lift of \(X\) to the algebraic closure \(\overline{\mathbb{F}}\) of \({\mathbb{F}}\). \textbf{ {Theorem}}. Assume that \(p\neq 2\). Then, on the subgroup \({\text{Ker}}({\text{Br}}(X)(2)_{\text{{nd}}}\to {\text{Br}}(\overline X)(2)_{\text{{nd}}})\) Tate's skew-symmetric bilinear form is alternating, and \(\#{\text{Br}}(X)(2)_{\text{{nd}}}\) is a square. (Here the morphism is induced by the covering morphism \(\overline X\to X\).) \textbf{ {Corollary}}. Let \(\Gamma=\overline{\mathbb{F}}/{\mathbb{F}}\) be the Galois group of \({\mathbb{F}}\). If the group \(H^3(\overline X_{\text{et}},{\mathbb{Z}}_2(1))\) has no non-trivial \(\Gamma\)-invariant element of finite order, then Tate's skew symmetric bilinear form is alternating for every prime \(\ell\neq p\). In particular, if \(\overline X\) is either a rational surface, a ruled surface, an abelian surface, a K3 surface, or a complete intersection in a projective space, then Tate's form is alternating. The main tool used in the proof of the theorem is Wu's formula in étale cohomology. \textbf{ {Theorem}}. Let \(Y\) be a proper smooth variety of dimension \(n\) over an algebraically closed field with characteristic \(p\neq 2\). Let \(Sq^r : H^2(Y_{\text{et}},{\mathbb{Z}}/2{\mathbb{Z}})\to H^{r+s}(Y_{\text{et}}, {\mathbb{Z}}/2{\mathbb{Z}})\) denote the \(r\)-th Steenrod squaring operator, and \(v_s\in H^s(Y_{\text{et}},{\mathbb{Z}}/2{\mathbb{Z}})\) be the \(s\)-th Wu class of \(Y\). Then \(v_s=0\) if \(s\) is odd, and \(\sum_{r=0}^s Sq^{2r}(v_{2(s-r)})=\bar c_s(Y)\) for every non-negative integer \(s\). (Here \(\bar c_s(Y)\) denotes the reduction modulo \(2\) of the \(s\)-th Chern class of \(Y\)).
scientific article; zbMATH DE number 931981