SIMPLE GRADED RINGS OF SIEGEL MODULAR FORMS, DIFFERENTIAL OPERATORS AND BORCHERDS PRODUCTS

Abstract

In this paper, we show that the graded ring of Siegel modular forms of Γ0(N) ⊂ Sp(2,ℤ) has a very simple unified structure for N = 1, 2, 3, 4, taking Neben-type case (the case with character) for N = 3 and 4. All are generated by 5 generators, and all the fifth generators are obtained by using the other four by means of differential operators, and it is also obtained as Borcherds products. As an appendix, examples of Euler factors of L-functions of Siegel modular forms of Sp(2,ℤ) of odd weight are given.