A threefold violating a local-to-global principle for rationality

Abstract

In this note we construct an example of a smooth projective threefold that is irrational over $\mathbb{Q}$ but is rational at all places. Our example is a complete intersection of two quadrics in ${\mathbb{P}}^5$, and we show it has the desired rationality behavior by constructing an explicit element of order $4$ in the Tate--Shafarevich group of the Jacobian of an associated genus $2$ curve.

7 pages. v2: the example is now unconditional