JACOBI FORMS AND A TWO-VARIABLE p-ADIC L-FUNCTION

Introduction. Consider a Jacobi form φ(τ, z) = ∑ n,r c(n, r)q ζ whose Fourier coefficients c(n, r) are algebraic numbers. Let p be an odd prime. In this paper we associate to φ a Λ-adic p-ordinary form in the sense of [4]. The construction comes from the map Dν introduced in [2], Theorem 3.1. This map associates to a Jacobi form a family of modular forms parametrised by ν. We obtain the two-variable p-adic interpolation of special values of symmetric squares of elliptic cusp forms as an application of our construction. This result is closely connected to [6], Theorem I. However, our approach is rather different and much more explicit. Our methods of p-adic interpolation are based on the methods and results of A.A.Panchishkin [10], [11], [12], [13]. We use the technique of Jacobi forms instead of the technique of modular forms of half integral weight used in [13], [6]. It yields considerable simplifications. In particular, we do not need nor the calculation of the trace of certain modular forms of higher level, nor the technique of non-holomorphic modular forms. Our methods might be generalised to the case of Siegel modular forms. The one-variable p-adic interpolation of the special values of the standard L-function (namely, the interpolation along the cyclotomic line) in this case is constructed by A.A.Panchishkin [10]. Recently S.Böcherer and C.-G.Schmidt [1] obtained a result of such type using a different method in a more general setting. The paper of J.Tilouine and E.Urban [15] extends the H.Hida’s theory of ordinary Λ-adic forms to the case of Siegel modular forms. One might expect that our construction is applicable to this case and would yield the two-variable interpolation of special values of the standard L-functions associated with Siegel modular forms. The content of the paper is as follows. In the first section we construct a Λ-adic p-ordinary modular form associated to a Jacobi form (theorem 1 in the text). Some basic facts from p-adic analysis are also recalled in this section. The main references here are [10] and [4]. The Jacobi Eisenstein series E M is introduced in the second section. This