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Computing the Brauer–Manin obstructions

Published online by Cambridge University Press:  07 July 2004

MARTIN BRIGHT
Affiliation:
Faulkes Institute for Geometry, Cambridge University, Cambridge. e-mail: mjbright@liverpool.ac.ukhpfs100@newton.cam.ac.uk
SIR PETER SWINNERTON–DYER
Affiliation:
Faulkes Institute for Geometry, Cambridge University, Cambridge. e-mail: mjbright@liverpool.ac.ukhpfs100@newton.cam.ac.uk

Abstract

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 $\hbox{Br}^0(V)$.

Type
Research Article
Copyright
2004 Cambridge Philosophical Society

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