Defect series and nonrational Hilbert modular threefolds

1994 ◽  
Vol 300 (1) ◽  
pp. 77-88 ◽  
Author(s):  
H. G. Grundman
Keyword(s):  
Hilbert Modular ◽  
Modular Threefolds

scholarly journals Nonrational Hilbert Modular Threefolds

2000 ◽  
Vol 83 (1) ◽  
pp. 50-58 ◽  
Author(s):  
H.G Grundman
Keyword(s):  
Hilbert Modular ◽  
Modular Threefolds

scholarly journals Minimal Siegel modular threefolds

1998 ◽  
Vol 123 (3) ◽  
pp. 461-485 ◽  
Author(s):  
VALERI GRITSENKO ◽  
KLAUS HULEK
Keyword(s):  
Difficult Problem ◽  
Kodaira Dimension ◽  
Abelian Surfaces ◽  
Starting Point ◽  
Hilbert Modular Surfaces ◽  
Hilbert Modular ◽  
Kummer Surfaces ◽  
Torelli Theorem ◽  
Ramification Locus ◽  
Modular Threefolds

The starting point of this paper is the maximal extension Γ*t of Γt, the subgroup of Sp4(ℚ) which is conjugate to the paramodular group. Correspondingly we call the quotient [Ascr ]*t=Γ*t\ℍ2 the minimal Siegel modular threefold. The space [Ascr ]*t and the intermediate spaces between [Ascr ]t=Γt\ℍ2 which is the space of (1, t)-polarized abelian surfaces and [Ascr ]*t have not yet been studied in any detail. Using the Torelli theorem we first prove that [Ascr ]*t can be interpreted as the space of Kummer surfaces of (1, t)-polarized abelian surfaces and that a certain degree 2 quotient of [Ascr ]t which lies over [Ascr ]*t is a moduli space of lattice polarized K3 surfaces. Using the action of Γ*t on the space of Jacobi forms we show that many spaces between [Ascr ]t and [Ascr ]*t possess a non-trivial 3-form, i.e. the Kodaira dimension of these spaces is non-negative. It seems a difficult problem to compute the Kodaira dimension of the spaces [Ascr ]*t themselves. As a first necessary step in this direction we determine the divisorial part of the ramification locus of the finite map [Ascr ]t→[Ascr ]*t. This is a union of Humbert surfaces which can be interpreted as Hilbert modular surfaces.

On the classification of Hilbert modular threefolds

1991 ◽  
Vol 72 (1) ◽  
pp. 297-305 ◽  
Author(s):  
H. G. Grundman
Keyword(s):  
Hilbert Modular ◽  
Modular Threefolds

scholarly journals Hilbert Modular Threefolds of Arithmetic Genus One

2002 ◽  
Vol 95 (1) ◽  
pp. 72-76
Author(s):  
H.G. Grundman ◽  
L.E. Lippincott
Keyword(s):  
Arithmetic Genus ◽  
Hilbert Modular ◽  
Genus One ◽  
Modular Threefolds

scholarly journals On Hilbert modular threefolds of discriminant 49

2012 ◽  
Vol 19 (4) ◽  
pp. 923-947
Author(s):  
Lev A. Borisov ◽  
Paul E. Gunnells
Keyword(s):  
Hilbert Modular ◽  
Modular Threefolds

scholarly journals Families of abelian surfaces with real multiplication over Hilbert modular surfaces

1982 ◽  
Vol 88 ◽  
pp. 17-53 ◽  
Author(s):  
G. van der Geer ◽  
K. Ueno
Keyword(s):  
Endomorphism Ring ◽  
Symplectic Group ◽  
Abelian Surface ◽  
Quadratic Relation ◽  
Holomorphic Map ◽  
Real Multiplication ◽  
Compact Complex Surfaces ◽  
Hilbert Modular Surfaces ◽  
Hilbert Modular ◽  
Abelian Functions

Around the beginning of this century G. Humbert ([9]) made a detailed study of the properties of compact complex surfaces which can be parametrized by singular abelian functions. A surface parametrized by singular abelian functions is the image under a holomorphic map of a singular abelian surface (i.e. an abelian surface whose endomorphism ring is larger than the ring of rational integers). Humbert showed that the periods of a singular abelian surface satisfy a quadratic relation with integral coefficients and he constructed an invariant D of such a relation with respect to the action of the integral symplectic group on the periods.

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