Cara Mullen : Critical Orbit Structure of $f_c(z)=z^2+c$ over $\mathbf{C}_p$

Posted by Abel Castillo , part of the Graduate Number Theory Seminar.

At

Nov. 11, 2013, 3 p.m.

In

SEO 427

Abstract

Fix a prime $p>2$ and consider $f_c(z)=z^2+c$ with $c\in\mathbb{C}_p$. How is the critical portrait of $f_c$ (over $\mathbb{C}_p$) related to the critical portrait of the reduction map $\bar{f_c}$ (over the residue field $k=\mathcal{O}_{\mathbb{C}_p}/{\mathfrak{m}_p}=\overline{\mathbb{F}_p})$? We will show that because none of the intermediate iterates ``get close" to 0 (in the non-archimedean sense), the critical point still has exact period $n$. We will then examine an alternative proof that can be generalized to all rational functions with periodic points.