Integer-valued polynomials and $p$-adic Fourier theory

The goal of this paper is to give a numerical criterion for an open question in p -adic Fourier theory. Let F be a finite extension of \mathbf{Q}_{p} . Schneider and Teitelbaum defined and studied the character variety \mathfrak{X} , which is a rigid analytic curve over F that parameterizes the set of locally F -analytic characters \lambda\colon (o_{F},+) \to (\mathbf{C}_{p}^{\times},\times) . Determining the structure of the ring \Lambda_{F}(\mathfrak{X}) of bounded-by-one functions on \mathfrak{X} defined over F seems like a difficult question. Using the Katz isomorphism, we prove that if F= \mathbf{Q}_{p^2} , then \Lambda_{F}(\mathfrak{X}) = o_{F} [[o_{F}]] if and only if the o_{F} -module of integer-valued polynomials on o_{F} is generated by a certain explicit set. Some computations in SageMath indicate that this seems to be the case.