2023/01/31 by Konstantin Ardakov, Laurent Berger, Ardakov, Konstantin +1 · 1 citation
Mathematics · #11F80 #11S15 #11S20 #11S31 #13F20 #13J07 #14G22 #22E50 #46S10 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2301.13650
openalex publication_date 2023/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is motivated by an open question in p-adic Fourier theory, that seems to be more difficult than it appears at first glance. Let L be a finite extension of ℚp with ring of integers oL and let ℂp denote the completion of an algebraic closure of ℚp. In their work on p-adic Fourier theory, Schneider and Teitelbaum defined and studied the character variety \mathfrakX. This character variety is a rigid analytic curve over L that parameterizes the set of locally L-analytic characters λ: (oL,+) → (ℂp^×,×). One of the main results of Schneider and Teitelbaum is that over ℂp, the curve \mathfrakX becomes isomorphic to the open unit disk. Let ΛL(\mathfrakX) denote the ring of bounded-by-one functions on \mathfrakX. If μ∈ oL [ [oL] ] is a measure on oL, then λ↦ μ(λ) gives rise to an element of ΛL(\mathfrakX). The resulting map oL [ [oL] ] → ΛL(\mathfrakX) is injective. The question is: do we have ΛL(\mathfrakX) = oL [ [oL] ]? In this paper, we prove various results that were obtained while studying this question. In particular, we give several criteria for a positive answer to the above question. We also recall and prove the ``Katz isomorphism'' that describes the dual of a certain space of continuous functions on oL. An important part of our paper is devoted to providing a proof of this theorem which was stated in 1977 by Katz. We then show how it applies to the question. Besides p-adic Fourier theory, the above question is related to the theory of formal groups, the theory of integer valued polynomials on oL, p-adic Hodge theory, and Iwasawa theory.