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The graded ring of quaternionic modular forms of degree 2. (English) Zbl 1132.11025
Summary: In the 1960’s J. Igusa [Am. J. Math. 86, 392–412 (1964; Zbl 0133.33301)] described the graded ring of Siegel modular forms of degree 2. In the sequel a few more concrete examples of graded rings of modular forms with respect to paramodular groups or Hermitian modular groups were given, e.g. by E. Freitag [Sitzungsber. Heidelberger Akad. Wiss., Math.-Naturw. Kl. 1967, 1. Abh. 1–50 (1967; Zbl 0156.09203)], H. Aoki and T. Ibukiyama [Int. J. Math. 16, No. 3, 249–279 (2005; Zbl 1068.11030)] or T. Dern and the author [Manuscr. Math. 110, No. 2, 251–272 (2003; Zbl 1038.11030)].
Moreover, E. Freitag and C. F. Hermann [Adv. Math. 152, 203–287 (2000; Zbl 0974.11028)] investigated some modular varieties of low dimension. In particular they discussed the modular variety $$X(\Gamma(\mathcal O))$$ where $$\Gamma(\mathcal O)$$ denotes the extended modular group of degree 2 over the Hurwitz quaternions. They described it as a major problem to determine the graded ring of modular forms in terms of generators and relations.
In this paper we determine this graded ring completely. Surprisingly it turns out to be a polynomial ring in 7 algebraically independent indeterminates given by Siegel-Eisenstein series of weight up to 24. The proof consists of applications of the results by Freitag and Hermann [loc. cit.] and the results on Hermitian modular forms with respect to $$\mathbb Q(\sqrt{-3})$$ in [Dern-Krieg, loc. cit.]. Additionally we have to put more emphasis on the commutator subgroup of the modular group calculated in [the author and S. Walcher, Commun. Algebra 26, No. 5, 1409–1417 (1998; Zbl 0902.11021)] and have to find new descriptions of the Borcherds products constructed in [Freitag and Hermann, loc. cit.]. Finally we consider the theta series from [Freitag and Hermann, loc. cit.]. We show that 6 generators can also be chosen as polynomials in these theta series, where one additionally needs the Siegel-Eisenstein series of weight 6.

##### MSC:
 11F55 Other groups and their modular and automorphic forms (several variables) 11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
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##### References:
 [1] Aoki, H., Ibukiyama, T.: Simple graded rings of Siegel modular forms, differential operators and Borcherds products. Int. J. Math. 16, 249–279 (2005) · Zbl 1068.11030 [2] Bachoc, C., Nebe, G.: Classification of two genera of 32-dimensional lattices of rank 8 over the Hurwitz order. Exp. Math. 6, 151–162 (1997) · Zbl 0886.11021 [3] Burkhardt, H.: Untersuchungen aus dem Gebiet der hyperelliptischen Modulfunctionen II, III. Math. Ann. 38, 161-224 (1890), Math. Ann. 40, 313–343 (1892) · JFM 22.0488.01 [4] Coble, A.: Point sets and allied Cremona groups III. Trans. Amer. Math. Soc. 18, 331–372 (1917) [5] Dern, T., Krieg, A.: Graded rings of Hermitian modular forms of degree 2. Manuscr. Math. 110, 251–272 (2003) · Zbl 1038.11030 [6] Freitag, E.: Modulformen zweiten Grades zum rationalen und Gaußschen Zahlkörper. Sitzungsber. Heidelb. Akad. Wiss., Math.-Naturwiss. Kl. 3–49 (1967) · Zbl 0156.09203 [7] Freitag, E., Hermann, C.F.: Some Modular Varieties of Low Dimension. Adv. Math. 152, 203–287 (2000) · Zbl 0974.11028 [8] Freitag, E., Salvati Manni, R.: Modular forms for the even unimodular lattice of signature (2, 10). Preprint, Heidelberg, 2005 · Zbl 1128.11024 [9] Freitag, E., Salvati Manni, R.: Some Modular Varieties of Low Dimension II. Preprint, Heidelberg, 2005 · Zbl 1186.11033 [10] Hunt, B.: The Geometry of some Arithmetic Quotients. Lect. Notes Math. 1637, Springer-Verlag, Berlin-Heidelberg-New York, 1996 · Zbl 0904.14025 [11] Igusa, J.-I.: On Siegel modular forms of genus two (II). Am. J. Math. 86, 392–412 (1964) · Zbl 0133.33301 [12] Klöcker, I.: Modular Forms for the Orthogonal Group O(2,5). PhD thesis, Aachen, 2005 (to appear) [13] Krieg, A.: Modular Forms on Half-Spaces of Quaternions. Lect. Notes Math 1143, Springer-Verlag, Berlin-Heidelberg-New York, 1985 · Zbl 0564.10032 [14] Krieg, A.: The Maaß Space on the Half-Space of Quaternions of Degree 2. Math. Ann. 276, 675–686 (1987) · Zbl 0613.10027 [15] Krieg, A.: The Maaß Space and Hecke Operators. Math. Z. 204, 527–550 (1990) · Zbl 0683.10024 [16] Krieg, A.: The Maaß Spaces for Hermitian Modular Forms of Degree 2. Math. Ann. 289, 663–681 (1991) · Zbl 0713.11033 [17] Krieg, A.: The Maaß space for the non-trivial multiplier system over the Hurwitz quaternions. Arch. Math. 70, 211–218 (1998) · Zbl 0912.11020 [18] Krieg, A., Walcher, S.: Multiplier systems for the modular group on the 27-dimensional exceptional domain. Comm. Algebra 26, 1409–1417 (1998) · Zbl 0902.11021 [19] Nagaoka, S.: Eisenstein series on quaternion half-space. J. Fac. Sci. Technol., Kinki Univ. 28, 41–48 (1992) · Zbl 0765.11025
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