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

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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Published 15 February 2017