Failure of the Hasse principle on general \(K3\) surfaces (Q2852319)

Let \(X\) be a smooth projective geometrically integral variety over a number field \(k\). Let \(X(k)\) be the set of \(k\)-rational points and \(X(\mathbf{A}_k)\) the set of adelic points of \(X\). For any subset \(S\) of the cohomological Brauer group \(\mathrm{Br}(X):=H^2_{et}(X,\,\mathbb{G}_m)\), an intermediate set \(X(k)\subseteq X(\mathbf{A}_k)^S\subseteq X(\mathbf{A}_k)\) can be defined via the Brauer-Manin pairing. One says that there is a Brauer-Manin obstruction (relative to \(S\)) to the Hasse principle for \(X\) if \(X(\mathbf{A}_k)^S=\emptyset\) yet \(X(\mathbf{A}_k)\neq\emptyset\).NEWLINENEWLINEInside the Brauer group \(\mathrm{Br}(X)\) one has the subgroup \(\mathrm{Br}_1(X)\) of algebraic classes, defined as the kernel of the natural map \(\mathrm{Br}(X)\to\mathrm{Br}(\bar{X})\), where \(\bar{X}=X\times_k\bar{k}\) for a fixed algebraic closure \(\bar k\) of \(k\). Elements of \(\mathrm{Br}(X)\) lying outside \(\mathrm{Br}_1(X)\) are called transcendental. While the study of algebraic Brauer-Manin obstructions abounds in the literature, obstructions coming from transcendental Brauer classes are still mysterious. For curves (and surfaces of negative Kodaira dimension) there are no transcendental classes. A threefold with a transcendental Brauer-Manin obstruction to the Hasse principle was constructed by \textit{D. Harari} [Lond. Math. Soc. Lect. Note Ser. 235, 75--87 (1996; Zbl 0926.14009)].NEWLINENEWLINEThe paper under review answers the natural question: can transcendental Brauer classes obstruct the Hasse principle on a surface? Of course the question is for surfaces of nonnegative Kodaira dimension, of which \(K3\) surfaces are some of the simplest examples. Previously, it was known that there may be transcendental obstructions to \textit{weak approximation} on \(K3\) surfaces: \textit{O. Wittenberg} [Arithmetic of higher-dimensional algebraic varieties. Proceedings of the workshop on rational and integral points of higher-dimensional varieties, 2002. Prog. Math. 226, 259--267 (2004; Zbl 1173.11336)], \textit{E. Ieronymou} [J. Inst. Math. Jussieu 9, No. 4, 769--798 (2010; Zbl 1263.14023)] and \textit{T. Preu} [Transcendental Brauer-Manin obstruction for a diagonal quartic surface. Ph. D. thesis, Universität Zürich (2010),NEWLINENEWLINE\url{http://www.math.uzh.ch/fileadmin/user/preu/publikation/preuThesis.pdf}] have found examples with elliptic fibration structure (and hence geometric Picard rank \(>1\)); examples with geometric Picard number 1 were exhibited in a joint work of the authors and Varilly.NEWLINENEWLINEIn the present paper, the authors construct a \(K3\) surface \(X\) over \(\mathbb{Q}\) and a transcendental Brauer class \(\alpha\) on \(X\) which gives rise to a Brauer-Manin obstruction to the Hasse principle. The construction is inspired from Hodge-theoretic work of \textit{B. van Geemen} [Adv. Math. 197, No. 1, 222--247 (2005; Zbl 1082.14040)]. That the example has geometric Picard number 1 is verified using a recent theorem of \textit{A.-S. Elsenhans} and \textit{J. Jahnel} [Algebra Number Theory 5, No. 8, 1027--1040 (2011; Zbl 1243.14014)]. Curiously enough, one of the most delicate steps is a computational subtlety: one has to factor an integer with 318 digits, whose smallest prime divisor turns out to have 66 digits.