Integer-valued polynomials and $p$-adic Fourier theory
Laurent Berger, Johannes Sprang
TL;DR
This paper addresses whether the ring of bounded-by-one, $F$-defined functions on the Schneider–Teitelbaum character variety $\mathfrak{X}$ equals the natural Lubin–Tate/Meixner-type power series ring $o_F[[o_F]]$ for $F=\mathbf{Q}_{p^2}$. Using the Katz isomorphism, the authors reduce the problem to a concrete, verifiable criterion: $\Lambda_{\mathbf{Q}_{p^2}}(\mathfrak{X})=o_F[[o_F]]$ if and only if the $o_F$-module of integer-valued polynomials on $o_F$ is generated by an explicit set, equivalently $\operatorname{Pol}(o_F[[X]]^{\psi=0})=\operatorname{Int}$. They prove a density variant $\operatorname{Pol}(o_F[[X]]^{\psi=0})+\pi\operatorname{Int}=\operatorname{Int}$ and provide numerical evidence (via SageMath) that the main equality holds for $F=\mathbf{Q}_{p^2}$, along with a practical method to verify it. The numerical section reinforces the theoretical criterion by showing finite verification bounds for small primes and giving a constructive approach to compute these polynomials. Overall, the work connects $p$-adic Fourier theory with arithmetic of integer-valued polynomials and offers a computational route to a long-standing structural question in the field.
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 $λ: (o_F,+) \to (\mathbf{C}_p^\times,\times)$. Determining the structure of the ring $Λ_F(\mathfrak{X})$ 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 $Λ_F(\mathfrak{X}) = o_F [\![o_F]\!]$ 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.