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On vector valued Siegel modular forms of degree 2 with small levels. (English) Zbl 1256.11033

Let \(\Gamma=\Gamma_0(N) \subseteq \text{Sp}_2(\mathbb{Z})\) (genus \(2\)) for \(N=2,3\) or \(4\), and consider the space \[ A_{*,2}(\Gamma) = \oplus_{k\in\mathbb{Z}} A_{k,2}(\Gamma), \] where \(A_{k,2}(\Gamma)\) is the space of Siegel modular forms of weight \(\rho = \text{Sym}^2 \otimes \det^k\). The main result of this paper is that \(A_{*,2}(\Gamma)\) is generated by ten specific modular forms, and that these generators can be constructed by differential operators.

MSC:

11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11F60 Hecke-Petersson operators, differential operators (several variables)
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Full Text: Euclid

References:

[1] H. Aoki: Estimating Siegel modular forms of genus 2 using Jacobi forms , J. Math. Kyoto Univ. 40 (2000), 581-588. · Zbl 0972.11034
[2] H. Aoki: Vector valued Siegel modular forms with small levels ; in Automorphic Representations, Automorphic Forms, L-functions, and Related Topics (Kyoto, 2008), Sūrikaisekikenkyūsho Kōkyūroku 1617 , Res. Inst. Math. Sci. (RIMS), Kyoto, 167-177, 2008.
[3] H. Aoki and T: Ibukiyama, Simple graded rings of Siegel modular forms, differential operators and Borcherds products , Internat. J. Math. 16 (2005), 249-279. · Zbl 1068.11030
[4] G. van der Geer: Siegel modular forms and their applications ; in The 1-2-3 of Modular Forms, Universitext Springer, Berlin, 181-245, 2008. · Zbl 1259.11051
[5] W. Eholzer and T. Ibukiyama: Rankin-Cohen type differential operators for Siegel modular forms , Internat. J. Math. 9 (1998), 443-463. · Zbl 0919.11037
[6] S. Hayashida and T. Ibukiyama: Siegel modular forms of half integral weight and a lifting conjecture , J. Math. Kyoto Univ. 45 (2005), 489-530. · Zbl 1122.11028
[7] T. Ibukiyama: On Siegel modular varieties of level \(3\) , Internat. J. Math. 2 (1991), 17-35. · Zbl 0721.11018
[8] T. Ibukiyama: On differential operators on automorphic forms and invariant pluri-harmonic polynomials , Comment. Math. Univ. St. Paul. 48 (1999), 103-118. · Zbl 1007.11023
[9] T. Ibukiyama: Differential operators and structures of vector valued Siegel modular forms ; in Algebraic Number Theory and Related Topics (Kyoto, 2000) Sūrikaisekikenkyūsho Kōkyūroku 1200 , Res. Inst. Math. Sci. (RIMS), 71-81, 2001, (Japanese). · Zbl 0985.11503
[10] J. Igusa: On Siegel modular forms of genus two , Amer. J. Math. 84 (1962), 175-200. · Zbl 0133.33301
[11] J. Igusa: On Siegel modular forms genus two , II, Amer. J. Math. 86 (1964), 392-412. · Zbl 0133.33301
[12] T. Satoh: On certain vector valued Siegel modular forms of degree two , Math. Ann. 274 (1986), 335-352. · Zbl 0571.10028
[13] E. Witt: Eine Identität zwischen Modulformen zweiten Grades , Abh. Math. Sem. Hansischen Univ. 14 (1941), 323-337. · Zbl 0025.01701
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