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.
References
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References
- D. Allcock and E. Freitag, Cubic surfaces and Borcherds products, Commentarii Math. Helv. Vol. 77, Issue 2, 270–296 (2002) MR 1915042 (2004c:14067)
- W. L. Baily and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Annals of Math. 84, No. 3, 442–528 (1966) MR 0216035 (35:6870)
- R. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132, 491–562 (1998) MR 1625724 (99c:11049)
- R. Borcherds, The Gross-Kohnen-Zagier theorem in higher dimensions, Duke Math. J. 97, 219–233 (1999) MR 1682249 (2000f:11052)
- J. Bruinier and M. Kuss, Eisenstein series attached to lattices and modular forms on orthogonal groups, Manuscr. Math. 106, 443–459 (2001) MR 1875342 (2003g:11048)
- J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, Springer-Verlag, New York, Grundlehren der mathematischen Wissenschaften 290 (1988) MR 920369 (89a:11067)
- J. Fay, On the Riemann-Jacobi formula, Nachr. Akad. Wiss. Göttingen, II. Math.-Phys. Kl. 1979, 61–73 (1979) MR 568803 (81g:14020)
- E. Freitag and C. F. Hermann, Some modular varieties in low dimension, Advances in Math. 152, 203–287 (2000) MR 1764105 (2001j:11023)
- E. Freitag, Modulformen zweiten Grades zum rationalen und Gaußchen Zahlkörper, Sitzungsberichte der Heidelberger Akademie der Wissenschaften, 1 Abh. (1967)
- E. Freitag, Some modular forms related to cubic surfaces, Kyungpook Math. J. 43, No. 3, 433–462 (2003). MR 2003489 (2004h:11044)
- E. Freitag, Comparison of different models of the moduli space of marked cubic surfaces, Proceedings of Japanese-German Seminar, Ryushi-do, edited by T. Ibukyama and W. Kohnen, 74–79 (2002)
- E. Freitag and R. Salvati-Manni, Some modular varieties of low dimension II, preprint (2004)
- J. Igusa, On the graded ring of theta-constants, Amer. J. Math. 86, 219–246 (1964) MR 0164967 (29:2258)
- M. Kneser, Erzeugung ganzzahliger orthogonaler Gruppen durch Spiegelungen, Math. Ann. 255, 453–462 (1981) MR 618179 (82i:10026)
- S. Kondo, The moduli space of Enriques surfaces and Borcherds products, J. Algebraic Geometry 11, 601–627 (2002) MR 1910262 (2003m:14055)
- S. Kondo, The moduli space of $8$ points on $P^1(\mathbb {C})$ and automorphic forms, preprint (2005)
- K. Koike, The projective embedding of the configuration space $X(2,8)$, preprint (2005)
- K. Matsumoto and M. Yoshida, Configuration space of $8$ points on the projective line and a $5$-dimensional Picard modular group, Compositio Math. 86, 265–280 (1993) MR 1219628 (94c:14038)
- V. V. Nikulin, Integral symmetric bilinear forms and some of their applications, Math. USSR Izvestija 14, No. 1 (1980)
- N. Scheithauer, Moonshine for Conway’s group, Habilitationsschrift, Heidelberg (2004)
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