A Converse Theorem for Hilbert-Jacobi Forms

Abstract

Doi and Naganuma (see [6]) constructed a lifting map from elliptic modular forms to Hilbert modular forms in the case of a real quadratic field with narrow class number one. A Converse Theorem for Hilbert modular forms was one of their basic tools. This gives rise to the question of constructing a lifting map in the case of Jacobi forms. Here we do the first step in this direction and prove a Converse Theorem for Hilbert-Jacobi forms. Studying the connection between functions that satisfy certain transformation laws and the functional equation of their associated L-functions has value on its own and a long history. In a celebrated paper (see [9]), Hecke showed that the automorphy of a cusp form with respect to SL2(Z) is equivalent to the functional equation of its associated L-functions. That only one functional equation is needed is in a way atypical and highly depends on the fact that SL2(Z) is generated by the matrices ( 1 1 0 1 ) and ( 0 −1 1 0 ). This situation already changes if one considers cusp forms with respect to a subgroup of SL2(Z) which have a character. In this case the functional equation of twists is required (see [18]). Hecke’s work has inspired an astonishing number of people and a lot of generalizations of his “Converse Theorem” have been made, e.g. generalizations to Hilbert modular forms as mentioned above (see [6]), Siegel modular forms (see [1], [10]) or Jacobi forms (see [14],[15]). Maass showed an analogue of Hecke’s result for nonholomorphic modular forms (see [13]). He proved that these correspond to certain L-functions in quadratic fields. An outstanding generalization of a Converse Theorem for GL(n) was done by Jacquet and Langlands for n = 2 (see [11]), Jacquet, Piatetski-Shapiro, and Shalika for n = 3 (see [12]) and Cogdell and Piatetski-Shapiro for general n (see [5]). In this paper, we prove a Converse Theorem for Hilbert-Jacobi cusp forms over a totally real number field K of degree g := [K : Q] with discriminant DK and narrow class number 1. The case g = 1, i.e., Jacobi forms over Q as considered by Eichler and Zagier (see [7]), is treated in two interesting papers by Martin (see [14] and [15]). To describe our result, we consider functions φ(τ, z) from H × C into C that have a Fourier expansion with certain conditions on the Fourier coefficients (see (3.4),(3.5), and (3.6)). We show that φ is a Hilbert-Jacobi cusp form (for the definition see Section 2) if