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

Bert van Geemen, Takuya Yamauchi

Research output: Contribution to journalArticle

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.

Original languageEnglish
Pages (from-to)141-164
Number of pages24
JournalPure and Applied Mathematics Quarterly
Volume12
Issue number1
DOIs
Publication statusPublished - 2016 Jan 1

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

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