The zeta functions of the Fano surfaces of cubic threefolds

TL;DR

This paper presents an algorithm for computing the zeta function of Fano surfaces associated with smooth cubic threefolds over finite fields, including examples with supersingular reduction.

Contribution

It introduces a novel algorithm for calculating zeta functions of Fano surfaces of cubic threefolds over finite fields.

Findings

Algorithm successfully computes zeta functions for given examples.

Identifies Fano surfaces with supersingular reduction.

Provides explicit examples of such surfaces.

Abstract

We give an algorithm to compute the zeta function of the Fano surface of lines of a smooth cubic threefold $F$ into $\mathbb{P}^4$ defined over a finite field. We obtain some examples of Fano surfaces with supersingular reduction.