P-adic Aspects of Jacobi Forms

Abstract

We are interested in understanding and describing the p-adic properties of Jacobi forms. As opposed to the case of modular forms, not much work has been done in this area. The literature includes [3, 4, 7]. In the first section, we follow Serre’s ideas from his theory of p-adic modular forms. We study Jacobi forms whose Fourier expansions have integral coefficients and look at congruences between them. Non-trivial examples are given by Jacobi-Eisenstein series. It turns out that two Jacobi forms need to have the same index and satisfy a condition on the weights in order to be congruent. If we define p-adic Jacobi forms in the natural way in this context, and restrict ourselves to the case of SL2(Z), we obtain a structure theorem for the space of p-adic Jacobi forms for SL2(Z) of a given weight χ ∈ Z ′ p and index m ∈ Z. Another feature is that p-adic Jacobi forms for Γ0(p) are also forms for SL2(Z). This parallels the similar result for modular forms, and it will most probably play an important role in defining some p-adic operators that do not arise directly from complex operators. In the second section, we associate to every Jacobi form with integral coefficients a measure on Zp with values in the p-adic ring of Katz’s generalized modular forms. This is an injection that allows us to interpret Jacobi forms with p-adic coefficients as truly p-adic objects, and this suggests where to look for the adequate “test objects” for a modular p-adic theory. It also provides examples of p-adic analytic families of modular forms.