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 Γ 0 (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.

Spectrum of dyons and black holes in CHL orbifolds using Borcherds lift

“…Let us assume the seed to be a weak Jacobi form 3 . Now, proposition (6.1) in [28] states that the space of weak Jacobi forms of even weight is generated as linear combinations of two weak forms φ −2,1 and φ 0,1 which in turn are given in terms of elementary theta functions by…”

“…In this section we summarise the construction of Hecke operators and the multiplicative lift, following [28]. Let us define ∆ N (t) as…”

“…Choose the complete set of cusps {s} of Γ 0 (N) represented by the set of representative matrices {g s }. Let As usual, one can show that c s (n, l) depends only on 4n − l 2 and l mod 2 so we write c s (n, l) = c s,l (4n−l 2 ) following the notation in [28]. In general n ∈ h −1 s Z need not be an integer.…”

Structure theorems for vector valued Siegel modular forms of degree 2 and weight detk ⊗Sym(10)

“…We can compute Fourier coefficients of χ 35 by the Wronskian given in [1] . Since χ 5 is the Saito-Kurokawa lift of a Jacobi theta series (see [15], [6]), we can easily compute Fourier coefficients of χ 5 .…”

Twisted Elliptic Genus for K3 and Borcherds Product

“…We set the Hecke operator V n on φ ∈ J k,m (Γ 0 (N)) defined as [1,6] We have (V n φ) (z; τ ) ∈ J k,mn (Γ 0 (N)). The representatives of cosets are given by cusps of Γ 0 (ord(g)) as [1] …”