Rational points on even‐dimensional Fermat cubics

We show that even-dimensional Fermat cubic hypersurfaces are rational over any field of characteristic not equal to three, by constructing explicit rational parameterizations with polynomials of low degree. As a byproduct of our rationality constructions, we obtain estimates for the number of their rational points over a number field and exhibit a class of quadro-cubic Cremona correspondences in even-dimensional projective spaces.