Holomorphic maps sharing preimages over finitely generated fields
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912702272634880 |
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| author | Pakovich, Fedor |
| author_facet | Pakovich, Fedor |
| contents | Let $ R$ be a compact Riemann surface, and let $ P: R \to \mathbb P^1(\mathbb C) $ and $ Q: R \to \mathbb P^1(\mathbb C) $ be holomorphic maps. In this paper, we investigate the following problem: under what conditions do the preimages $ P^{-1}(K) $ and $ Q^{-1}(K) $ coincide for some infinite set $K$ contained in $\mathbb P^1(k)$, where $k$ is a finitely generated subfield of $\mathbb C$ (e.g., a number field)? Equivalently, we study holomorphic correspondences that admit an infinite completely invariant set contained in $\mathbb P^1(k)$. We show that if such a set exists, then there is a holomorphic Galois covering $Θ: R_0 \to \mathbb P^1(\mathbb C)$, where $R_0$ has genus zero or one, such that $ P $ and $ Q $ are ``compositional left factors" of $ Θ.$ We also consider a more general equation $ P^{-1}(K_1) = Q^{-1}(K_2),$ where $K_1$ and $K_2$ are infinite subsets of $\mathbb P^1(k)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_08506 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Holomorphic maps sharing preimages over finitely generated fields Pakovich, Fedor Number Theory Algebraic Geometry Complex Variables Let $ R$ be a compact Riemann surface, and let $ P: R \to \mathbb P^1(\mathbb C) $ and $ Q: R \to \mathbb P^1(\mathbb C) $ be holomorphic maps. In this paper, we investigate the following problem: under what conditions do the preimages $ P^{-1}(K) $ and $ Q^{-1}(K) $ coincide for some infinite set $K$ contained in $\mathbb P^1(k)$, where $k$ is a finitely generated subfield of $\mathbb C$ (e.g., a number field)? Equivalently, we study holomorphic correspondences that admit an infinite completely invariant set contained in $\mathbb P^1(k)$. We show that if such a set exists, then there is a holomorphic Galois covering $Θ: R_0 \to \mathbb P^1(\mathbb C)$, where $R_0$ has genus zero or one, such that $ P $ and $ Q $ are ``compositional left factors" of $ Θ.$ We also consider a more general equation $ P^{-1}(K_1) = Q^{-1}(K_2),$ where $K_1$ and $K_2$ are infinite subsets of $\mathbb P^1(k)$. |
| title | Holomorphic maps sharing preimages over finitely generated fields |
| topic | Number Theory Algebraic Geometry Complex Variables |
| url | https://arxiv.org/abs/2511.08506 |