The Ring of Jacobi Forms

M‐strings, elliptic genera and N = 4 string amplitudes

“…Notice, however, that (4.16) differs from the Hecke transformations introduced in [35], explaining the modular properties of the partition functions discussed in the previous sections.…”

“…5, f g (τ, ν, β = 1) is a weak Jacobi form of weight 2g − 2 and index 1, we observe that the structure of (4.9) is very reminiscent of a Hecke transformation. Indeed, for ∈ N define the Hecke transform of a weight k Jacobi form Φ k (τ, z) = m,n h(m, n)q m z n in the following manner [35] T Φ k (τ, z) = k−1 ad= a>0 15) while for = 0 we use the definition [31]…”

“…which is a holomorphic modular form of weight 1 2 of SL(2, Z). It can be related to the Jacobi theta functions through [52] With these quantities, we can finally introduce the standard weak Jacobi forms φ −2,1 (z; y) and φ 0,1 (z; y) of index 1 and weight −2 and 0 respectively (see [35])…”

On Elliptic Genera and Foliations

“…(ii) If n > 0, then Lemma 4.5 shows that H(t, τ ) is a holomorphic Jacobi form of negative index. By [EZ,Theorem 1.2], H(t, τ ) must be zero.…”

On integral kernels for Dirichlet series associated to Jacobi forms

“…Important examples of the former are theta functions in two variables and Fourier-Jacobi coefficients of Siegel modular forms over Sp 2 (Z). Even though some Jacobi forms have been around for a long time, the systematic study of these objects is more recent (see [4] for a good introduction to the subject).…”