Finite Orbits in Surfaces with a Double Elliptic Fibration and Torsion Values of Sections
| Autor: | Corvaja, Pietro, Tsimerman, Jacob, Zannier, Umberto |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We consider surfaces with a double elliptic fibration, with two sections. We study the orbits under the induced translation automorphisms proving that, under natural conditions, the finite orbits are confined to a curve. This goes in a similar direction of (and is motivated by) recent work by Cantat-Dujardin, although we use very different methods and obtain related but different results. As a sample of application of similar arguments, we prove a new case of the Zilber-Pink conjecture, namely Theorem 1.5, for certain schemes over a 2-dimensional base, which was known to lead to substantial difficulties. Most results rely, among other things, on recent theorems by Bakker and the second author of `Ax-Schanuel Type'; we also relate a functional condition with a theorem of Shioda on ramified sections of the Legendre scheme. For one of our proofs, we also use recent height inequalities by Yuan-Zhang. Finally, in an appendix, we show that the Relative Manin-Mumford Conjecture over the complex number field is equivalent to its version over the field of algebraic numbers. Comment: 29 pages, comments welcome! |
| Databáze: | arXiv |
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