The modularity of the BarthNieto quintic and its relatives
Abstract
The moduli space of (1,3)polarized abelian surfaces with full level2 structure is birational to a double cover of the BarthNieto quintic. Barth and Nieto have shown that these varieties have CalabiYau 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 \Gamma_1(6). By Tate's conjecture, this should imply that Y, the fibred square of the universal elliptic curve S_1(6), and Verrill's rigid CalabiYau Z_{A_3}, which all have the same Lfunction, are in correspondence over Q. We show that this is indeed the case by giving explicit maps.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 October 2000
 arXiv:
 arXiv:math/0010049
 Bibcode:
 2000math.....10049H
 Keywords:

 Algebraic Geometry;
 Number Theory
 EPrint:
 30 pages, Latex2e