Certain K3 surfaces parametrized bythe Fibonacci sequenceviolate the Hasse principle

Abstract

For a prime $p \equiv 5 \pmod {8}$ satisfying certain conditions, we show that there exist an infinitude of K3 surfaces parameterized by certain solutions to Pell's equation $X^2 - pY^2 = 4$ in the projective $5$-space that are counterexamples to the Hasse principle explained by the Brauer-Manin obstruction. Further, these surfaces contain no zero-cycle of odd degree over~$\mathbb{Q} $. As an illustration for the main result, we show that the prime $p = 5$ satisfies all of the required conditions in the main theorem, and hence, there exist an infinitude of K3 surfaces parameterized by the Fibonacci sequence that are counterexamples to the Hasse principle explained by the Brauer-Manin obstruction.