### 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 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 Q(√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.

Original language | English |
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Pages (from-to) | 141-164 |

Number of pages | 24 |

Journal | Pure and Applied Mathematics Quarterly |

Volume | 12 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2016 Jan 1 |

### Keywords

- Abelian surfaces with real multiplication
- Cubic threefolds
- Elliptic curves
- Intermediate Jacobian

### ASJC Scopus subject areas

- Mathematics(all)