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)
Full Text:

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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.