On Rigid, Hard and Soft Problems and Results in Arithmetic Geometry

TL;DR

This paper reviews various problems and results in arithmetic geometry, focusing on the classification of solutions to quadratic forms over rings of integers, and discusses the concept of rigidity within this context.

Contribution

It introduces a framework distinguishing soft, hard, and rigid problems in arithmetic geometry and reviews recent novel results related to quadratic forms and rigidity.

Findings

Analysis of solutions to quadratic forms over local and global fields

Introduction of a new framework for classifying problems in arithmetic geometry

Discussion of recent advances in rigidity concepts

Abstract

Rigid, hard and soft problems and results in arithmetic geometry are presented. "Soft" and "hard" in our paper are limited to the framework of solutions of quadratic forms over rings of integers of local and global fields, the Hardy-Littlewood-Kloosterman method. Next we consider the notion of rigidity. In the framework we give review of some novel results in the aria.