Computing the Brauer–Manin obstructions
Martin Bright, Sir Peter Swinnerton-Dyer · Mathematical Proceedings of the Cambridge Philosophical Society · 2004
Let $V$ be a complete nonsingular projective surface defined over an algebraic number field $k$ , such that the Picard variety of $V$ is trivial and Pic( $\bar{V}$ ) is torsion-free. Since our main interest is in necessary conditions for $V(k)$ not to be empty, we shall further assume that $V(k_v)$ is non-empty for every completion $k_v$ of $k$ . We do not assume that $\bar{V}$ is rational, and indeed the case which primarily interests us is when $V$ is a K3 surface. Our objective is to describe effective ways of computing the Brauer–Manin obstructions to the existence of points on $V$ defined over $k$ . Write \[ \hbox{Br}^0 (V) = \{\hbox{Ker}(\hbox{Br}(V)\longrightarrow\hbox{Br}(\bar{V}))\}/\{{\rm Im}({\rm Br}(k)\longrightarrow\hbox{Br}(V))\};\] then it is known that $\hbox{Br}^0(V)$ is isomorphic to ${\rm H}^1(k, \hbox{Pic}(\bar{V}))$ , though to the best of our knowledge there is in general no known algorithm for computing either $\hbox{Br}^0(V)$ or this isomorphism. It is generally believed that $\hbox{Br}^0(V)$ contains all the information about the Brauer group Br $(V)$ which is useful in this context. Most of this paper is concerned with computing groups isomorphic to $\hbox{Br}^0(V)$ , and with describing in terms of these groups the Brauer–Manin obstructions coming from elements of $\hbox{Br}^0(V)$ .