Determination of All Q-Rational CM-Points in the Moduli Space of Principally Polarized Abelian Surfaces

On Isogenous Principally Polarized Abelian Surfaces

We study a relationship between two genus 2 curves whose jacobians are isogenous with kernel equal to a maximal isotropic subspace of p-torsion points with respect to the Weil pairing. For p = 3 we find an explicit relationship between the set of Weierstrass points of the two curves extending the classical results of F. Richelot (1837) and G. Humbert (1901) in the case p = 2.

The moduli space of (1,11)-polarized abelian surfaces is unirational

We prove that the moduli space A 11 of …1; 11†-polarized Abelian surfaces with level structure of canonical type is birational to Klein's cubic hypersurface in P4. Therefore, A 11 is unirational but not rational, and there are no G11-cusp forms of weight 3. The same methods also provide an easy proof of the rationality of A 9 . Mathematics Subject Classi¢cation (2000). 14K10. Key words. Abelian...

The Uniformization of the Moduli Space of Principally Polarized Abelian 6-folds

Introduction 1 1. Kanev’s construction and Prym-Tyurin varieties of E6-type 7 2. The E6 lattice 11 3. Degenerations of Jacobians and Prym varieties 13 4. Degenerations of Prym-Tyurin-Kanev varieties 15 5. The global geometry of the Hurwitz space of E6-covers 20 6. The Prym-Tyurin map along boundary components of Hur 30 7. Ordinary Prym varieties regarded as Prym-Tyurin-Kanev varieties 39 8. The...

Moduli of CM abelian varieties

We discuss CM abelian varieties in characteristic zero, and in positive characteristic. An abelian variety over a finite field is a CM abelian variety, as Tate proved. Can it be CM-lifted to characteristic zero? Does there exist an abelian variety of dimension g > 3 not isogenous with the Jacobian of an algebraic curve? Can we construct algebraic curves, say over C, where the Jacobian is a

Local positivity of principally polarized abelian threefolds