Regular Functions on Formal-Analytic Arithmetic Surfaces
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911312380952576 |
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| author | Goodman, Samuel |
| author_facet | Goodman, Samuel |
| contents | In this paper, we show that for a broad class of pseudoconvex formal-analytic arithmetic surfaces over $\text{Spec}(\mathbb{Z})$, those which admit a nonconstant monic such regular function, that a conjecture of Bost-Charles that the ring of regular functions has continuum cardinality is implied by a purely complex-analytic conjecture. Under the conjecture, a Fekete-Szego-type approximation argument produces a polynomial "large" relative to the regular function, which in turn yields continuum many distinct regular functions. We also introduce a formula for the pushforward by a holomorphic function of the equilibrium Green's functions for our bordered Riemann surface with boundary, a formula which has constant term related to Arakelov degree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_07098 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regular Functions on Formal-Analytic Arithmetic Surfaces Goodman, Samuel Complex Variables Number Theory In this paper, we show that for a broad class of pseudoconvex formal-analytic arithmetic surfaces over $\text{Spec}(\mathbb{Z})$, those which admit a nonconstant monic such regular function, that a conjecture of Bost-Charles that the ring of regular functions has continuum cardinality is implied by a purely complex-analytic conjecture. Under the conjecture, a Fekete-Szego-type approximation argument produces a polynomial "large" relative to the regular function, which in turn yields continuum many distinct regular functions. We also introduce a formula for the pushforward by a holomorphic function of the equilibrium Green's functions for our bordered Riemann surface with boundary, a formula which has constant term related to Arakelov degree. |
| title | Regular Functions on Formal-Analytic Arithmetic Surfaces |
| topic | Complex Variables Number Theory |
| url | https://arxiv.org/abs/2512.07098 |