On the Brauer group

S. G. Tankeev

Izvestiya. Mathematics , Tome 64 (2000) no. 4, pp. 787-806

Résumé

For an arithmetic model $X$ of a Fermat surface or a hyperkahler variety with Betti number $\operatorname{b}_2(V\otimes\bar k)>3$ over a purely imaginary number field $k$, we prove the finiteness of the $l$-components of $\operatorname{Br}'(X)$ for all primes $l\gg 0$. This yields a variant of a conjecture of M. Artin. If $V$ is a smooth projective irregular surface over a number field $k$ and $V(k)\ne\varnothing$, then the $l$-primary component of $\operatorname{Br}(V)/{\operatorname{Br}(k)}$ is an infinite group for every prime $l$. Let $A^1\to M^1$ be the universal family of elliptic curves with a Jacobian structure of level $N\geqslant 3$ over a number field $k\supset\mathbb Q(e^{2\pi i/N})$. Assume that $M^1(k)\ne\varnothing$. If $V$ is a smooth projective compactification of the surface $A^1$, then the $l$-primary component of $\operatorname{Br}(V)/{\operatorname{Br}(\overline M^1)}$ is a finite group for each sufficiently large prime $l$.