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

TL;DR

This paper provides a numerical criterion related to the structure of functions on a $p$-adic character variety, connecting $p$-adic Fourier theory with integer-valued polynomials and offering computational evidence.

Contribution

It establishes a criterion linking the structure of bounded functions on a $p$-adic character variety to the generation of integer-valued polynomials, advancing understanding in $p$-adic Fourier theory.

Findings

For $F=\mathbf{Q}_{p^2}$, the ring of bounded functions equals a power series ring if and only if a certain set generates the module of integer-valued polynomials.

Computational evidence suggests the criterion holds in specific cases.

The paper connects $p$-adic Fourier theory with properties of integer-valued polynomials.

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.