Pure and Applied Mathematics Quarterly

Volume 12 (2016)

Number 1

Special Issue: In Honor of Eduard Looijenga, Part 2 of 3

Guest Editor: Gerard van der Geer

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

Pages: 141 – 164

DOI: http://dx.doi.org/10.4310/PAMQ.2016.v12.n1.a5

Authors

Bert van Geemen (Dipartimento di Matematica, Università di Milano, Italy)

Takuya Yamauchi (Mathematical Institute, Tohoku University, Aoba-Ku, Sendai, Japan)

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 $\mathrm{Aut}(X)$. 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 $\mathbf{Q}(\sqrt{5})$. 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.

Keywords

cubic threefolds, intermediate Jacobian, elliptic curves, and abelian surfaces with real multiplication

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