A Converse Theorem for Hilbert-Jacobi Forms

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

“…The converse theorem for GL(n) automorphic representations is a great achievement of several authors: H. Jacquet and R. Langlands [9] when n = 2, H. Jacquet, I. Piatetski-Shapiro and J. Shalika [10] if n = 3, J. Cogdell and I. Piatetski-Shapiro [3] for general n. The first named author found in [13] the analogue of the classical Hecke converse theorem for Jacobi cusp forms. The same result for Hilbert-Jacobi forms was established by K. Bringmann and S. Hayashida [2].…”

“…μ≡0 (2) f μ (4mτ ) and F 2m (τ ) − F m (τ ) = 2m μ=1 μ≡1 (2) f μ (4mτ ) are cusp forms of weight k − 1 2 and character γ = * *…”

“…For completeness sake, we observe that the function g(τ, z) introduced in Lemma 6 is in J k,mβNM 2 ,χψ 2 …”

“…As in the classical case, it is possible to twist the Fourier series f (τ, z) and g(τ, z) by a Dirichlet character ψ mod M in a sensible way. Namely, f ψ (τ, z) = 2mM μ=1 f ψ,μ (τ )Θ mM,μ (Mβτ, βz) (2) with f ψ,μ (τ ) = ∞ D=1 ψ( D+βμ 2 4m )c μ (D) exp ( πiD 2m τ ), and…”