The modularity of the Barth--Nieto quintic and its relatives
The moduli space of (1,3)polarized abelian surfaces with full level-2 structure is birational to a double cover of the Barth-Nieto quintic. Barth and Nieto have shown that these varieties have Calabi-Yau models Z and Y, respectively. In this paper we apply the Weil conjectures to show that Y and Z are rigid and we prove that the Lfunction of their common third étale cohomology group is modular, as predicted by a conjecture of Fontaine and Mazur. The corresponding modular form is the unique normalized cusp form of weight 4 for the group~Γ1(6). By Tate's conjecture, this should imply that Y, the fibred square of the universal elliptic curve~S1(6), and Verrill's rigid Calabi-Yau~𝒵A3, which all have the same Lfunction, are in correspondence over ℚ. We show that this is indeed the case by giving explicit maps.
Copyright © 2001 by Walter de Gruyter GmbH & Co. KG
Articles in the same Issue
- Near hexagons with four points on a line
- SPG systems and semipartial geometries
- On threefolds admitting a bielliptic curve as abstract complete intersection
- The modularity of the Barth--Nieto quintic and its relatives
- Finite presentations for the mapping class group via the ordered complex of curves
Articles in the same Issue
- Near hexagons with four points on a line
- SPG systems and semipartial geometries
- On threefolds admitting a bielliptic curve as abstract complete intersection
- The modularity of the Barth--Nieto quintic and its relatives
- Finite presentations for the mapping class group via the ordered complex of curves
