关于曲线乘积上相对平展系统的动力 Mordell--Lang 猜想

中文摘要 AI 辅助

我们在复数域 \\(\mathbb C\\) 上建立了动力 Mordell--Lang 猜想，该猜想针对的是在光滑射影曲线有限乘积的任意自态射上相对平展的自态射。特别地，我们为光滑射影曲线乘积的任意自态射以及相对平展的斜积建立了该猜想。算术输入是我们关于 \\(\overline{\mathbb Q}\\) 上分裂自态射的无条件定理，其证明通过将问题归结为临界点局部周期的同时 Hasse 原理，并分析联合树状塔中的局部惯性。我们将这一算术输入与相对平展性、保持游荡基坐标的同时代数特化、指定约化的 \\(p\\) 进嵌入以及 \\(p\\) 进插值相结合，从而在复情形下得到该结果。

英文摘要

We establish the dynamical Mordell--Lang conjecture over \(\mathbb C\) for endomorphisms that are relatively étale over arbitrary endomorphisms of finite products of smooth projective curves. In particular, we establish the conjecture for arbitrary endomorphisms of product of smooth projective curves and relatively étale skew products. The arithmetic input is our unconditional theorem for split endomorphisms over \(\overline{\mathbb Q}\), proved by reducing to a simultaneous Hasse principle for the local periods of critical points and analyzing local inertia in joint arboreal towers. We combine this arithmetic input with relative étaleness, simultaneous algebraic specialization preserving wandering base coordinates, prescribed-reduction \(p\)-adic embeddings, and \(p\)-adic interpolation to obtain the result in the complex case.

发表机构

• The University of Memphis（孟菲斯大学）

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