Irrationality of generic cubic threefold via Weil's conjectures

Abstract

An arithmetic method of proving the irrationality of smooth projective 3-folds is described, using reduction modulo [Formula: see text]. It is illustrated by an application to a cubic threefold, for which the hypothesis that its intermediate Jacobian is isomorphic to the Jacobian of a curve is contradicted by reducing modulo 3 and counting points over appropriate extensions of [Formula: see text]. As a spin-off, it is shown that the 5-dimensional Prym varieties arising as intermediate Jacobians of certain cubic 3-folds have the maximal number of points over [Formula: see text] which attains Perret's and Weil's upper bounds.