Certain K3 surfaces parametrized by the Fibonacci sequence violate the Hasse principle (Q2409615)

The goal of the article under review is to construct families of K3 surfaces that are given as the smooth intersection of three quadrics in \(\mathbb{P}_\mathbb{Q}^5\) and that are counterexamples to the Hasse principle explained by the Brauer-Manin obstruction. A starting point of the construction is work of \textit{J.-L. Colliot-Thelene} et al. [J. Reine Angew. Math. 320, 150--191 (1980; Zbl 0434.14019)] (Proposition 7.1). Apart from their main results on the Hasse principle for certain algebraic varieties, they show that the (singular) projective variety \(V\subset \mathbb{P}_{\mathbb{Q}}^5\) which is defined as the intersection of the two quadrics \[ u_1^2-5v_1^2=2xy,\quad u_2^2-5v_2^2=2(x+20y)(x+25y) \] violates the fine Hasse principle. That is \(V\) has non-singular local points in all completions of \(\mathbb{Q}\) but no \(\mathbb{Q}\)-rational point. This violation can be explained by a Brauer-Manin obstruction. \textit{D. Coray} and \textit{C. Manoil} [Acta Arith. 76, No. 2, 165--189 (1996; Zbl 0877.14005)] (Proposition 5.1) use this example to show that the K3 surface \(S\subset \mathbb{P}_\mathbb{Q}^5\) given as the smooth intersection \[ \begin{aligned} U^2 & =XY+5Z^2\\ V^2& =13X^2+950XY+32730 Y^2+670 Z^2\\ W^2 & =-X^2-134XY-654Y^2+134Z^2 \end{aligned} \] is a counterexample to the Hasse principle. By the first two equations, \(S\) is contained in the threefold \(V\), which is already known to violate the Hasse principle. The third equation is engineered in a way that \(S\) is smooth and such that \(S\) has a real point and \(S(\mathbb{Q}_p)\neq \emptyset\) for all primes \(p\). In a previous work [Pac. J. Math. 274, No. 1, 141--182 (2015; Zbl 1341.14008)] (Theorem 1.2), the author generalized the example of the threefold \(V\) by Colliot-Thélène, Coray and Sansuc to a family of singular threefolds in \(\mathbb{P}_{\mathbb{Q}}^5\), which are given as the intersection of two quadrics and that violate the Hasse principle. Concretely, these examples are given by systems of equations \[ \begin{aligned} \mathcal{Y}: u_1^2-pv_1^2 & =2xy\\ u_2^2-pv_2^2 & = 2(x+4pb^2y)(x+p^2d^2y), \end{aligned} \] for parameters \(p,b,d\) that satisfy a number of local conditions. In particular, there exist infinitely many such triples that satisfy the required conditions and in this case \(\mathcal{Y}\) is a counterexample to the Hasse principle explained by the Brauer-Manin obstruction, and contains no zero-cycle of odd degree over \(\mathbb{Q}\). In the work under review, the author takes up the idea of Coray and Manoil and constructs \(K3\)-surfaces contained in one of the threefolds \(\mathcal{Y}\), that violate the Hasse principle and contain no zero-cycle of odd degree over \(\mathbb{Q}\). These examples are obtained as the smooth intersection of the singular threefold \(\mathcal{Y}\subset \mathbb{P}^5\) with another quadric of the form \[ \mathcal{Q}:\quad v_2^2=\lambda x^2+\mu xy + \nu y^2+v_1^2, \] for parameters \(\lambda,\mu,\nu\) that satisfy a number of algebraic conditions. For the admissible tuple \((p,b,d)=(5,1,1)\) and \(n\in \mathbb{Z}_{\geq 0}\), the author constructs a concrete family of K3 surfaces \(\mathcal{K}_n=\mathcal{Y}\cap \mathcal{Q}\) violating the Hasse principle and containing no zero-cycle of odd degree over \(\mathbb{Q}\), with parameters \((\lambda_n,\mu_n,\nu_n)\) given by \[ \begin{aligned} \lambda_n & =F_{2n}^2/536 - 1099/14472\\ \mu_n & =55F_{2n}^2/536- 7669/1608\\ \nu_n & =-352 F_{2n}^2/268 + 71219/7236.\end{aligned} \] Here \(F_n\) denotes the \(n\)th Fibonacci number, i.e., \(F_0=0\), \(F_1=1\), and \(F_{n+2}=F_{n+1}+F_n\) for \(n\geq 0\).