Effective local differential topology of algebraic varieties over local fields of positive characteristics

TL;DR

This paper develops a framework for analyzing distances and measures on algebraic varieties over local fields of positive characteristic, extending concepts from differential topology to this algebraic setting.

Contribution

It introduces a novel approach to local differential topology for algebraic varieties over positive characteristic local fields, including analogues of key theorems like the implicit function theorem.

Findings

Established analogues of the implicit function theorem.

Analyzed behavior of smooth measures under submersions.

Provided quantitative tools for distances and measures on algebraic varieties.

Abstract

In this paper we provide a framework for quantitative statements on distances and measures when studying algebraic varieties and morphisms of algebraic varieties over local fields. We will concentrate on local fields of the type $\mathbb{F}_\ell((t))$ and work uniformly with respect to finite extensions of $\mathbb{F}_\ell$. In this framework we prove analogues of standard results from local differential topology, including the implicit function theorem and study the behavior of smooth measures under push forward with respect to submersions.