Frobenius提升的代数动力学
Algebraic dynamics of the lifts of Frobenius
摘要：
本文研究了系数在p-adic域中的射影空间的自同态的代数动力学，其正特征约化是Frobenius。特别是，我们证明了一个版本的动态Manin-Mumford猜想和动态Mordell-Lang猜想的相干后向轨道，这样的自同态。在这种情况下，我们也给出了一个新的动力学版本的Tate-Voloch猜想的证明。我们的方法是基于P. Scholze提出的完美空间理论。在附录中，我们证明了在定义域上的某些技术条件下，射影簇上Frobenius的一个极化提升的动力系统可以嵌入到射影空间的一个自同态的动力系统中。
英文摘要：
We study the algbraic dynamics for endomorphisms of projective spaces with coefficients in a p-adic field whose reduction in positive characteritic is the Frobenius. In particular, we prove a version of the dynamical Manin-Mumford conjecture and the dynamical Mordell-Lang conjecture for the coherent backward orbits for such endomorphisms. We also give a new proof of a dynamical version of the Tate-Voloch conjecture in this case. Our method is based on the theory of perfectoid spaces introduced by P. Scholze. In the appendix, we prove that under some technical condition on the field of definition, a dynamical system for a polarized lift of Frobenius on a projective variety can be embedding into a dynamical system for some endomorphism of a projective space.
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