# Algebraic Geometry Working Seminar

Tuesday, September 27, 2022 — 12:30 PM EDT

Xiao Zhong, Department of Pure Mathematics, University of Waterloo

"$p$-Adic interpolation of orbits under rational maps"

Rivera-Letelier’s characterization of possible analytic uniformizations of $p$-adic analytic maps has played an important role within arithmetic dynamics over the past fifteen years. The characterization is given by a trichotomy of indifferent, attracting and superattracting cases near a fixed point of a map.

In this talk, we present that if we are only interested in the orbit of a rational map on a point $c$ of $\mathbb{P}^1$ over a characteristic zero global field, we could always $p$-adically interpolate the orbit in the sense similar to the indifferent case of the trichotomy. This is done by working with a finitely generated field extension of $\mathbb{Q}$ and choosing suitable primes for embedding into local fields. We also present an application to the dynamical Mordell-Lang conjecture.

This project is a joint work with Prof. Jason P. Bell (arxiv: 2202.01673).

This seminar will be held jointly online and in person:

• Room: MC 5403
• Zoom information: Meeting ID: 817 1030 9714; Passcode: 063438
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