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 Q₏. Schneider and Teitelbaum defined and studied the character variety X, which is a rigid analytic curve over F that parameterizes the set of locally F -analytic characters (o₅, +) (C₏^, ). Determining the structure of the ring ₅ (X) of bounded-by-one functions on X defined over F seems like a difficult question. Using the Katz isomorphism, we prove that if F= Q₏ℂ, then ₅ (X) = o₅ [o₅] if and only if the o₅ -module of integer-valued polynomials on o₅ is generated by a certain explicit set. Some computations in SageMath indicate that this seems to be the case.