An Approach to the p - adic Theory of Jacobi Forms

The theory of p-adic modular forms was developed by J.-P. Serre [8] and N. Katz [5]. This theory is by now considered classical. Investigation of p-adic congruences for modular forms of half-integer weight was carried out by N. Koblitz [6] and led him to deep conjectures. It seems natural to search for p-adic properties of other types of automorphic forms. In this paper we use the Serre approach to p-adic theory of modular forms in order to develop p-adic theory of Jacobi forms. The starting point of [8] was the definition of p-adic modular forms via their q-expansions as p-adic limits of usual modular forms. It turned out possible to define weight of p-adic modular form as an element of p-adic Lie group X = Homcont(Z∗ p,Z p). One can obtain this element as a limit of the weights of the modular forms used in the definition of p-adic modular form. This phenomenon was explained in [5] from the point of view of algebraic geometry. Although the geometric interpretation of Jacobi forms seems vague, we show in this paper that an analogous statement is true for Jacobi forms. We state the basic properties of p-adic Jacobi forms in Theorems 1 and 2. In Corollary 1 we consider sequences of Jacobi-Eisenstein series as examples of p-adic Jacobi forms. As a number-theoretic application we obtain p-adic limit formulae for the average number (with appropriate weights) of representations of an integer by inequivalent unimodular quadratic forms (Corollary 2). Now we introduce some notations concerning p-adic analysis. Let p be a prime number. In order to avoid some technical difficulties we assume p = 2. The p-adic Lie group X0 = Homcont(Z∗ p,C p) appears as the domain of definition of the non-Archimedian zeta-function [7]. It contains the subgroup X = Homcont(Z∗ p,Z p). Each element of X could be regarded as a pair (t, u), where t ∈ Zp and u ∈ Z/(p − 1)Z. An element k ∈ X is a pair if k ∈ 2X, i.e., its second component u is a pair. (The structure of X is described in [8].) We will often identify natural numbers with their images in X via the natural inclusion. We