# Irrationality of threefolds via Weil's conjectures

## TL;DR

This paper introduces an arithmetic approach using Weil's conjectures and reduction modulo p to prove the irrationality of smooth projective 3-folds, demonstrated through cubic threefolds and their intermediate Jacobians.

## Contribution

It presents a novel arithmetic method for establishing the irrationality of threefolds by analyzing point counts over finite fields, linking algebraic geometry with number theory.

## Key findings

- Counterexample to the isomorphism of intermediate Jacobian and Jacobian of a curve for a cubic threefold.
- Intermediate Jacobians of certain cubic 3-folds are Prym varieties with maximal point counts over finite fields.
- The method provides a new perspective on irrationality proofs using Weil's conjectures and reduction techniques.

## 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.