Arithmetic Dynamics on Moduli Spaces of Holomorphic Curves
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| Format: | Recurso digital |
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2025
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| author | SÉRGIO DE ANDRADE, PAULO |
| author_facet | SÉRGIO DE ANDRADE, PAULO |
| contents | This paper explores the emerging field of arithmetic dynamics on moduli spaces of holomorphic curves, specifically focusing on the spaces $mathcal{M}_{g,n}$. We propose a framework for constructing rational self-maps on these moduli spaces derived from natural geometric operations on families of curves. The primary objective is to investigate the arithmetic properties of orbits of points under the iteration of these maps, particularly for points defined over a number field $K$. We introduce a canonical height function analogous to the Néron-Tate height and establish its fundamental properties, including the Northcott finiteness theorem for preperiodic points. A central result of this study is an arithmetic equidistribution theorem, which describes the asymptotic distribution of Galois orbits of points with positive canonical height. This theorem provides a powerful tool for analyzing the statistical behavior of iterated geometric constructions. The methodology combines techniques from algebraic geometry, Diophantine geometry, complex dynamics, and non-archimedean (Berkovich) geometry. We conclude by discussing potential applications to Diophantine problems concerning rational points on subvarieties of moduli spaces and formulate conjectures that extend classical problems in arithmetic dynamics to this broader geometric context. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17690381 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Arithmetic Dynamics on Moduli Spaces of Holomorphic Curves SÉRGIO DE ANDRADE, PAULO This paper explores the emerging field of arithmetic dynamics on moduli spaces of holomorphic curves, specifically focusing on the spaces $mathcal{M}_{g,n}$. We propose a framework for constructing rational self-maps on these moduli spaces derived from natural geometric operations on families of curves. The primary objective is to investigate the arithmetic properties of orbits of points under the iteration of these maps, particularly for points defined over a number field $K$. We introduce a canonical height function analogous to the Néron-Tate height and establish its fundamental properties, including the Northcott finiteness theorem for preperiodic points. A central result of this study is an arithmetic equidistribution theorem, which describes the asymptotic distribution of Galois orbits of points with positive canonical height. This theorem provides a powerful tool for analyzing the statistical behavior of iterated geometric constructions. The methodology combines techniques from algebraic geometry, Diophantine geometry, complex dynamics, and non-archimedean (Berkovich) geometry. We conclude by discussing potential applications to Diophantine problems concerning rational points on subvarieties of moduli spaces and formulate conjectures that extend classical problems in arithmetic dynamics to this broader geometric context. |
| title | Arithmetic Dynamics on Moduli Spaces of Holomorphic Curves |
| url | https://doi.org/10.5281/zenodo.17690381 |