Bounded functions on the character variety
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, Sc...
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| Materiálatiipa: | Journal article |
| Giella: | English |
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Mathematisches Institut der Universität Münster
2024
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| _version_ | 1826314365446389760 |
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| author | Ardakov, K Berger, L |
| author_facet | Ardakov, K Berger, L |
| author_sort | Ardakov, K |
| collection | OXFORD |
| description | 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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| first_indexed | 2024-04-09T03:57:28Z |
| format | Journal article |
| id | oxford-uuid:219487e1-7af8-494b-9103-f9739ec39046 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-09-25T04:31:25Z |
| publishDate | 2024 |
| publisher | Mathematisches Institut der Universität Münster |
| record_format | dspace |
| spelling | oxford-uuid:219487e1-7af8-494b-9103-f9739ec390462024-08-29T10:58:45ZBounded functions on the character varietyJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:219487e1-7af8-494b-9103-f9739ec39046EnglishSymplectic ElementsMathematisches Institut der Universität Münster2024Ardakov, KBerger, LThis 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]]? <br> 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. |
| spellingShingle | Ardakov, K Berger, L Bounded functions on the character variety |
| title | Bounded functions on the character variety |
| title_full | Bounded functions on the character variety |
| title_fullStr | Bounded functions on the character variety |
| title_full_unstemmed | Bounded functions on the character variety |
| title_short | Bounded functions on the character variety |
| title_sort | bounded functions on the character variety |
| work_keys_str_mv | AT ardakovk boundedfunctionsonthecharactervariety AT bergerl boundedfunctionsonthecharactervariety |