# Uniform Bounds for Periods of Endomorphisms of Varieties

## TL;DR

This paper establishes explicit uniform upper bounds on the primitive periods of periodic points for endomorphisms of projective varieties over p-adic fields, extending understanding of dynamical behavior in arithmetic geometry.

## Contribution

It provides the first explicit bounds for primitive periods of endomorphisms on varieties over p-adic fields, using Fakhruddin's method.

## Key findings

- Derived explicit upper bounds for primitive periods of periodic points.
- Extended the application of Fakhruddin's method to varieties over p-adic fields.

## Abstract

Suppose $X$ is a projective variety defined over a finite extension $K$ of $\mathbb{Q}_p$ and suppose $X$ admits a model $\mathcal{X}$ defined over the ring of integers $R$ of $K$. Let $f:{X}\rightarrow {X}$ be an endomorphism of $X$ defined over $K$ that can be extended to an endomorphism of $\mathcal{X}$ defined over $R$. We apply a method of Fakhruddin to prove an explicit upper bound for the primitive period of periodic points defined over $R$.