Dynamique analytique sur $\mathbf{Z}$. II : Écart uniforme entre Lattès et conjecture de Bogomolov-Fu-Tschinkel
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866915440409706496 |
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| author | Poineau, Jérôme |
| author_facet | Poineau, Jérôme |
| contents | We prove that the mutual energy (or the intersection product in the sense of Arakelov theory) of two dynamical systems associated to Lattès morphisms over $\mathbf{\bar Q}$ is uniformly bounded below and deduce a proof of a conjecture of Bogomolov-Fu-Tschinkel: the number of common images of torsion points of two non-isomorphic elliptic curves over $\mathbf{C}$ by a standard morphism to the projective line is uniformly bounded. The proof crucially relies on the theory of Berkovich spaces over $\mathbf{Z}$ and on an original argument allowing to obtain a global estimate from a central estimate (over a trivially valued field). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_01574 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Dynamique analytique sur $\mathbf{Z}$. II : Écart uniforme entre Lattès et conjecture de Bogomolov-Fu-Tschinkel Poineau, Jérôme Number Theory Dynamical Systems 11G05, 11G50, 37P50, 37P15, 14G22 We prove that the mutual energy (or the intersection product in the sense of Arakelov theory) of two dynamical systems associated to Lattès morphisms over $\mathbf{\bar Q}$ is uniformly bounded below and deduce a proof of a conjecture of Bogomolov-Fu-Tschinkel: the number of common images of torsion points of two non-isomorphic elliptic curves over $\mathbf{C}$ by a standard morphism to the projective line is uniformly bounded. The proof crucially relies on the theory of Berkovich spaces over $\mathbf{Z}$ and on an original argument allowing to obtain a global estimate from a central estimate (over a trivially valued field). |
| title | Dynamique analytique sur $\mathbf{Z}$. II : Écart uniforme entre Lattès et conjecture de Bogomolov-Fu-Tschinkel |
| topic | Number Theory Dynamical Systems 11G05, 11G50, 37P50, 37P15, 14G22 |
| url | https://arxiv.org/abs/2207.01574 |