Integer-valued polynomials and $p$-adic Fourier theory
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915242723770368 |
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| author | Berger, Laurent Sprang, Johannes |
| author_facet | Berger, Laurent Sprang, Johannes |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_18053 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Integer-valued polynomials and $p$-adic Fourier theory Berger, Laurent Sprang, Johannes Number Theory 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. |
| title | Integer-valued polynomials and $p$-adic Fourier theory |
| topic | Number Theory |
| url | https://arxiv.org/abs/2502.18053 |