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

Abstract

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 $\lambda :(o_F,+)\to ({\mathbf{C}}_p^{\times },\times )$. 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.