| Summary: | 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 𝔛. This character variety is a rigid analytic curve over L that parameterizes the set of locally L-analytic characters λ:(oL,+)→(ℂ×p,×). One of the main results of Schneider and Teitelbaum is that over ℂp, the curve 𝔛 becomes isomorphic to the open unit disk. Let ΛL(𝔛) denote the ring of bounded-by-one functions on 𝔛. If μ∈oL[[oL]] is a measure on oL, then λ↦μ(λ) gives rise to an element of ΛL(𝔛). The resulting map oL[[oL]]→ΛL(𝔛) is injective. The question is: do we have ΛL(𝔛)=oL[[oL]]?
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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.
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