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: https://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 D5 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.

Keywords

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

Published 15 February 2017