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 be a field of characteristic zero containing a primitive fifth root of unity. Let 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 is a subgroup of . We find that the intermediate Jacobian of is isogenous to the product of an elliptic curve and the self-product of an abelian surface with real multiplication by . We give explicit models of some algebraic curves related to the construction of as a Prym variety. This includes a two parameter family of curves of genus 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