Modular forms for the even modular lattice of signature (2,10)

Freitag, Eberhard; Manni, Riccardo

doi:10.1090/S1056-3911-07-00460-2

Modular forms for the even modular lattice of signature $(2,10)$

Authors: Eberhard Freitag and Riccardo Salvati Manni
Journal: J. Algebraic Geom. 16 (2007), 753-791
DOI: https://doi.org/10.1090/S1056-3911-07-00460-2
Published electronically: June 7, 2007
MathSciNet review: 2357689
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Abstract | References | Additional Information

Abstract: We consider the principal congruence subgroup $\Gamma [2]$ of level two inside the orthogonal group of an even and unimodular lattice of signature $(2,10)$. Using Borcherds’ additive lifting construction, we construct a $715$-dimensional space of singular modular forms. This space is the direct sum of the one-dimensional trivial and a $714$-dimensional irreducible representation of the finite group $\operatorname {O}(\mathbb {F}^{12}_2)$ (even type). It generates an algebra whose normalization is the ring of all modular forms. We define a certain ideal of quadratic relations. This system appears as a special member of a whole system of ideals of quadratic relations. At least some of them have geometric meaning. For example, we work out the relation to Kondo’s approach to the modular variety of Enriques surfaces.

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Additional Information

Eberhard Freitag
Affiliation: Mathematisches Institut, Im Neuenheimer Feld 288, D69120 Heidelberg, Germany
MR Author ID: 69160
Email: freitag@mathi.uni-heidelberg.de

Riccardo Salvati Manni
Affiliation: Dipartimento di Matematica, Piazzale Aldo Moro, 2, I-00185 Roma, Italy
Email: salvati@mat.uniroma1.it

Received by editor(s): November 7, 2005
Received by editor(s) in revised form: March 10, 2006
Published electronically: June 7, 2007