The Distribution Relation and Inverse Function Theorem in Arithmetic Geometry

TL;DR

This paper develops explicit versions of the inverse function theorem in arithmetic geometry, utilizing distribution relations and Newton iteration, applicable uniformly across families of varieties and maps.

Contribution

It introduces two explicit inverse function theorems in arithmetic geometry, one based on distribution inequalities and the other on Newton iteration, enhancing uniform applicability.

Findings

Established distribution and separation inequalities for arithmetic varieties.

Developed a Newton iteration-based inverse function theorem.

Provided explicit, uniform inverse function results for families of varieties.

Abstract

We study arithmetic distribution relations and the inverse function theorem in algebraic and arithmetic geometry, with an emphasis on versions that can be applied uniformly across families of varieties and maps. In particular, we prove two explicit versions of the inverse function theorem, the first via general distribution and separation inequalities that may be of independent interest, the second via a careful implementation of classical Newton iteration.