Unlikely intersection problems for restricted lifts of p-th power

TL;DR

This paper explores the properties of a special class of morphisms called lifts of p-th powers within perfectoid spaces, applying them to major conjectures in number theory such as Manin-Mumford, Mordell-Lang, and Tate-Volch.

Contribution

It introduces a new class of morphisms called lifts of p-th powers and applies perfectoid space theory to significant conjectures in arithmetic geometry.

Findings

Established properties of lifts of p-th powers in perfectoid spaces.

Connected these morphisms to the proof strategies for key conjectures.

Provided new insights into the structure of rational points on algebraic varieties.

Abstract

I define a morphism on $\mathbb{C}_p$ called a lift of $p$-th power if its natural restriction to the residue field of $\mathbb{C}_p$ is a $p$-th power of some morphism. This definition generalizes from the lift of Frobenius. In this paper the theory of perfectoid spaces is applied on the morphism to the Manin-Mumford conjecutre, the Mordell-Lang conjecture and the Tate-Volch conjecture.