\(p\)-adic aspects of Jacobi forms (Q1355081)

The article concerns \(p\)-adic aspects of Jacobi forms. The author follows Serre's approach to \(p\)-adic modular forms, and defines the \(p\)-adic Jacobi forms (of a given weight) to be a limit of complex Jacobi forms [Definition 2.3; see also \textit{P. Guerzhoy}, Int. Math. Res. Not. 1994, No. 1, 31-39 (1994; Zbl 0869.11039)]. It turns out that \(p\)-adic Jacobi forms for \(\Gamma_0(p)\) are also forms for \(SL(2,\mathbb{Z})\) (theorem 2.4). The author associates to every Jacobi form with integral coefficients a measure on \(\mathbb{Z}_p\) with values in the \(p\)-adic ring of Katz's modular forms (theorem 3.1). It yields an injection and provides examples of \(p\)-adic analytic families of modular forms.