Rational points on even dimensional Fermat cubics

TL;DR

This paper proves that even-dimensional Fermat cubic hypersurfaces are rational over fields with characteristic not equal to three, providing explicit parametrizations, and explores implications for rational points and Cremona transformations.

Contribution

It introduces explicit low-degree polynomial parametrizations demonstrating the rationality of even-dimensional Fermat cubics, a novel constructive approach.

Findings

Explicit rational parametrizations for even-dimensional Fermat cubics.

Estimates on the number of rational points over number fields.

Identification of quadro-cubic Cremona correspondences.

Abstract

We show that even dimensional Fermat cubic hypersurfaces are rational over any field of characteristic different from three by producing explicit rational parametrizations given by polynomials of low degree. As a byproduct of our rationality constructions we get estimates on the number of their rational points over a number field, and a class of quadro-cubic Cremona correspondences of even dimensional projective spaces.