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

Abstract

We generalize Weil's converse theorem to Jacobi cusp forms of weight k, index m and Dirichlet character χ over the group Γ 0 (N ) Z 2 . Then two applications of this result are given; we generalize a construction of Jacobi forms due to Skogman and present a new proof for several known lifts of such Jacobi forms to half-integral weight modular forms.

Nonvanishing modulo ℓof Fourier coefficients of Jacobi forms

“…One could treat twists of Jacobi forms by χ(n) or χ(r) in a similar way. The latter, for example, has been considered in [MO11].…”

An Analogue of Weil's Converse Theorem for Harmonic Maass Forms of Polynomial Growth

“…In our context, the meaning of a converse theorem is best illustrated by the Weil's converse theorem for modular forms over congruence subgroups Γ 0 (N) [23], which is a very significant generalization of the corresponding Hecke's theorem for N = 1. Other results of this kind are Maass' converse theorem for Maass waveforms of level 1 [15], its generalization to Γ 0 (N) by Neururer and Oliver [20], converse theorems for Jacobi forms [16,17], Siegel modular forms [14], and Maass Jacobi forms [10]. The converse theorem for GL n is a great achievement of several authors through a string of papers [5,12,13].…”