Generating Jacobi Forms

Abstract

In this paper we explore the relationship between vector-valued modular forms and Jacobi forms and give explicit relations over various congruence subgroups. The main result is that a Jacobi form of square-free index on the full Jacobi group is uniquely determined by any of the associated vector components. In addition, an explicit construction is given to determine the other vector components from this single component. In other words, we give an explicit construction of a Jacobi form from a subset of its Fourier coefficients. This leads to results about how the transformation properties are affected by congruence restrictions on the Fourier expansion.

On the analogue of Weil’s converse theorem for Jacobi forms and their lift to half-integral weight modular forms

“…This result for N = 1 and m square-free was established by Skogman in [17]. His proof is based on explicit formulas for the Γ 0 (2m)-action on the space of theta series μ CΘ m,μ (τ, z).…”

“…Indeed, by elementary arguments we derive from our theorem and basic properties of theta functions the following: (i) a new proof for the construction of distinct Jacobi forms from the theta decomposition of a given one due to H. Skogman [17], (ii) its generalization to higher levels, (iii) a new proof of the lift from Jacobi forms to half-integral weight modular forms used in the argument for the Saito-Kurokawa conjecture presented in [5], and (iv) a family of similar lifts labeled by the divisors of the index of a Jacobi cusp form. More precisely, we get Corollary 1 Let f (τ, z) be a Jacobi cusp form as in part (i) of Theorem 1 with theta series expansion (1).…”

“…(In other words, Conditions II and III satisfied by (17) are |d(n, r)| < C(4mn − Nr 2 ) σ for all n, r and d(n, r) = d(n + λrN + λ 2 mN, r + 2mλ) for all λ ∈ Z. )…”

“…Our first application is both a new proof and a generalization of a theorem established by H. Skogman in [17]. We are referring to Corollary 1, given in the introduction.…”