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arXiv · 1705.10395

Irrationality of threefolds via Weil's conjectures

Dimitri Markushevich·Xavier Roulleau
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Abstract

An arithmetic method of proving the irrationality of smooth projective 3-folds is described, using reduction modulo $p$. 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 $\mathbb F_3$. 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 $\mathbb F_q$ which attains Perret's and Weil's upper bounds.

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Dimitri Markushevich, Xavier Roulleau. 2017-09-01. Irrationality of threefolds via Weil's conjectures. https://arxiv.org/abs/1705.10395

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