Some modular varieties of low dimension. (English) Zbl 0974.11028
From the introduction of the paper: Generally spoken it is hard to find projective models for modular varieties. So it is reasonable to modify the question as follows:
Find arithmetic groups \(\Gamma \subset O(2,n)\) such that the Bailey Borel compactification \(X(\Gamma)\) admits a fine covering \(X(\Gamma) \rightarrow \mathbb{P}^n(\mathbb{C})\) of known degree and such that the ramification divisor can explicitly be described as a linear combination of quadratic divisors.
In this paper we propose some examples for further study.
Find arithmetic groups \(\Gamma \subset O(2,n)\) such that the Bailey Borel compactification \(X(\Gamma)\) admits a fine covering \(X(\Gamma) \rightarrow \mathbb{P}^n(\mathbb{C})\) of known degree and such that the ramification divisor can explicitly be described as a linear combination of quadratic divisors.
In this paper we propose some examples for further study.
Reviewer: Chandrashekhar B.Khare (Mumbai)
References:
| [1] | Barth, W.; Nieto, I., Abelian surfaces of type (1, 3) and quartic surfaces with 16 skew lines, J. Algebraic Geom., 3, 173-222 (1994) · Zbl 0809.14027 |
| [2] | R. Borcherds, Automorphic forms with singularities on Grassmannians, preprint, 1997;, http://www.dpmms.cam.ac.uk/home/emu/reb/.my-home-page.html; R. Borcherds, Automorphic forms with singularities on Grassmannians, preprint, 1997;, http://www.dpmms.cam.ac.uk/home/emu/reb/.my-home-page.html |
| [3] | R. Borcherds, The Gross-Zagier theorem in higher dimensions, preprint, 1997;, http://www.dpmms.cam.ac.uk/home/emu/reb/.my-home-page.html; R. Borcherds, The Gross-Zagier theorem in higher dimensions, preprint, 1997;, http://www.dpmms.cam.ac.uk/home/emu/reb/.my-home-page.html |
| [4] | Conway, J. H.; Sloane, N. J.A., Sphere Packings, Lattices and Groups. Sphere Packings, Lattices and Groups, Grundlehren der mathematischen Wissenschaften, 290 (1988), Springer-Verlag: Springer-Verlag New York · Zbl 0634.52002 |
| [5] | Elstrodt, J.; Grunewald, F.; Mennicke, J., Vahlens group of Clifford matrices and spin groups, Math. Z., 196, 369-390 (1987) · Zbl 0611.20027 |
| [6] | Freitag, E., Siegelsche Modulformen. Siegelsche Modulformen, Grundlehren der Mathematischen Wissenschaften, 254 (1983), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York · Zbl 0498.10016 |
| [7] | Freitag, E., Modulformen zweiten Grades zum rationalen und Gaußschen Zahlkörper, Sitzungsber. Heidelb. Akad. Wiss., 1, 1-49 (11967) · Zbl 0156.09203 |
| [8] | Freitag, E., Fortsetzung automorpher Funktionen, Math. Ann., 177, 95-100 (1968) · Zbl 0155.40302 |
| [9] | Freitag, E.; Kiehl, R., Algebraische Eigenschaften der lokalen Ringe in den Spitzen der Hilbertschen Modulgruppe, Invent. Math., 24, 121-148 (1974) · Zbl 0304.32018 |
| [10] | van Geemen, B., Projective models of Picard modular varieties, Classification of Irregular Varieties, Proceedings Trento. Classification of Irregular Varieties, Proceedings Trento, Lecture Notes in Mathematics, 1515 (1990), Springer-Verlag: Springer-Verlag New York/Berlin, p. 68-99 · Zbl 0793.14010 |
| [11] | van der Geer, On the geometry of a Siegel modular threefold, Math. Ann., 260, 317-350 (1982) · Zbl 0473.14017 |
| [12] | van der Geer, Hilbert modular forms for the field \(Q(6)\), Math. Ann., 233, 163-179 (1978) · Zbl 0357.10014 |
| [13] | V. A. Gritsenko, and, V. V. Nikulin, Automorphic forms and Lorentzian Kac-Moody algebras, Part II, preprint, RIMS Kyoto Univ. RIMS-1122, 1966; alg-geom/9611028.; V. A. Gritsenko, and, V. V. Nikulin, Automorphic forms and Lorentzian Kac-Moody algebras, Part II, preprint, RIMS Kyoto Univ. RIMS-1122, 1966; alg-geom/9611028. |
| [14] | Gritsenko, V. A., Modular forms and moduli spaces of Abelian and K3-surfaces, St. Petersburg Math. J., 6, 1179-1208 (1995) · Zbl 0847.14020 |
| [15] | Hermann, F., Some modular varieties related to \(P^4\), (Barth; Hulek; Lange, Abelian Varieties (1995), Walter de Gruyter Verlag: Walter de Gruyter Verlag Berlin/New York) · Zbl 0842.14012 |
| [16] | Hunt, B., The Geometry of Some Special Arithmetic Quotients. The Geometry of Some Special Arithmetic Quotients, Lecture Notes in Math., 1637 (1996), Springer-Verlag: Springer-Verlag Berlin/Heidelberg · Zbl 0904.14025 |
| [17] | Knöller, F. W., Multiplizitäten “unendlich-ferner” Spitzen, Mh. Math., 88, 7-26 (1979) · Zbl 0433.14025 |
| [18] | Krieg, A., Modular Forms on Half-Spaces of Quaternions. Modular Forms on Half-Spaces of Quaternions, Lecture Notes in Math., 1143 (1985), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York · Zbl 0564.10032 |
| [19] | A. Krieg, and, S. Walcher, Multiplier systems for the modular group on the 27-dimensional exceptional domain, Arbeitsbericht, Lehrstuhl A für Mathematik, Rheinisch-Westfälische Technische Hochschule, Aachen, 1996.; A. Krieg, and, S. Walcher, Multiplier systems for the modular group on the 27-dimensional exceptional domain, Arbeitsbericht, Lehrstuhl A für Mathematik, Rheinisch-Westfälische Technische Hochschule, Aachen, 1996. · Zbl 0902.11021 |
| [20] | K. Matsumoto, Theta functions on the classical bounded symmetric domain of type \(I_{2, 2}K\); K. Matsumoto, Theta functions on the classical bounded symmetric domain of type \(I_{2, 2}K\) |
| [21] | Nikulin, V. V., Integral symmetric bilinear forms and some of their applications, Math. USSR Izv., 14 (1980) · Zbl 0427.10014 |
| [22] | Satake, I., Algebraic Structures of Symmetric Domains. Algebraic Structures of Symmetric Domains, Kanô Memorial Lectures, 4 (1980), Iwanami Shoten: Iwanami Shoten Tohyo · Zbl 0483.32017 |
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