Algebraic families of hyperelliptic curves violating the Hasse principle (Q2352464)

Denote by \(X\) an algebraic variety defined over \(\mathbb{Q}\). Hasse's local-global principle says that if \(X\) possesses rational points over \(\mathbb{R}\) and \(\mathbb{Q}_{p}\) for each prime number \(p\) then it possesses rational points over \(\mathbb{Q}\). Many examples have shown that this is not the case. Manin defined a pairing between families of local rational points on \(X\) and the Brauer group \(\mathrm{Br}(X)\) such that non-orthogonality of local points with \(\mathrm{Br}(X)\) prevents \(X\) from possessing \(\mathbb{Q}\)-points. Therefore one can make use of such a pairing to construct plenty of varieties violating the Hasse principle, and so does this paper. Using the method in [J. Am. Math. Soc. 13, No. 1, 83--99 (2000; Zbl 0951.11022)], \textit{J.-L. Colliot-Thélène} and \textit{B. Poonen} explicitly constructed an \(1\)-parameter algebraic family of genus \(1\) curves over \(\mathbb{Q}\) violating the Hasse principle [\textit{B. Poonen}, J. Théor. Nombres Bordx. 13, No. 1, 263--274 (2001; Zbl 1046.11038)]. In this paper, for any integer \(g>5\) that is not divisible by \(4\), the author produces an \(1\)-parameter algebraic family of genus \(g\) hyperelliptic curves over \(\mathbb{Q}\) violating the Hasse principle. The method is summarised as follows. For each \(g\), Coray and Manoil constructed a single hyperelliptic curve violating the Hasse principle [\textit{D. Coray} and \textit{C. Manoil}, Acta Arith. 76, No. 2, 165--189 (1996; Zbl 0877.14005)]. The proof of such a statement uses a smooth embedding of the curve to a certain \(3\)-fold violating the Hasse principle constructed by \textit{J.-L. Colliot-Thelene} et al. [J. Reine Angew. Math. 320, 150--191 (1980; Zbl 0434.14019)]. One is going to make these constructions work as a family. First of all, one produces a \(3\)-parameter family of \(3\)-folds containing the \(3\)-fold mentioned above and a \(6\)-parameter family of genus \(g\) hyperelliptic curves containing the curve mentioned above. Second, one finds out a set of sufficient conditions on the parameters such that under such conditions these hyperelliptic curves can be smoothly embedded into the \(3\)-folds and such that the \(3\)-folds and the curves can be proved violate the Hasse principle. Finally, one shows that the values of parameters satisfying those sufficient conditions can be given by some carefully chosen rational functions. Then the construction gives the desired \(1\)-parameter algebraic family of curves. By the way, a theorem of \textit{M. Amer} [Quadratische Formen über Funktionenkörpern, thesis, Johannes Gutenberg University, Mainz (1976)] and \textit{A. Brumer} [C. R. Acad. Sci., Paris, Sér. A 286, 679--681 (1978; Zbl 0392.10021)] implies that the curves constructed in this paper do not possess any zero-cycle of odd degree. The result in a preprint of Bhargava-Gross-Wang says that, for all \(g\geq1\) and over \(\mathbb{Q}\), among genus \(g\) hyperelliptic curves having rational points locally everywhere, there is a positive proportion that have no \(\mathbb{Q}\)-rational points.