On intermediate Jacobians of cubic threefolds admitting an automorphism of order five

TL;DR

This paper studies cubic threefolds with a fivefold automorphism, revealing their intermediate Jacobians decompose into products involving elliptic curves and abelian surfaces with real multiplication, with explicit models provided.

Contribution

It demonstrates that such threefolds have intermediate Jacobians isogenous to a product of an elliptic curve and a special abelian surface, and constructs explicit algebraic curve models.

Findings

Intermediate Jacobian decomposes into elliptic and abelian surface factors.

Explicit models of related algebraic curves are constructed.

The abelian surface has real multiplication by \\mathbb{Q}(\\sqrt{5}).

Abstract

Let $k$ be a field of characteristic zero containing a primitive fifth root of unity. Let $X/k$ be a smooth cubic threefold with an automorphism of order five, then we observe that over a finite extension of the field actually the dihedral group $D_5$ is a subgroup of ${\rm Aut}(X)$. We find that the intermediate Jacobian $J(X)$ of $X$ is isogenous to the product of an elliptic curve $E$ and the self-product of an abelian surface $B$ with real multiplication by $\mathbb{Q}(\sqrt{5})$. We give explicit models of some algebraic curves related to the construction of $J(X)$ as a Prym variety. This includes a two parameter family of curves of genus 2 whose Jacobians are isogenous to the abelian surfaces mentioned as above.