Arithmetic differential geometry in the arithmetic PDE setting II: curvature and cohomology

TL;DR

This paper develops an arithmetic PDE framework for curvature and cohomology, extending previous work on connections to include characteristic classes in an arithmetic setting.

Contribution

It introduces arithmetic analogues of curvature and characteristic classes within an arithmetic PDE framework, advancing the geometric understanding in number theory.

Findings

Arithmetic curvature analogues are formulated.

Arithmetic characteristic classes are constructed.

The framework extends previous connection theories in arithmetic geometry.

Abstract

This is the second paper in a series devoted to developing an arithmetic PDE analogue of Riemannian geometry. In Part 1 arithmetic PDE analogues of Levi-Civita and Chern connections were introduced and studied. In this paper arithmetic analogues of curvature and characteristic classes are developed.