Arithmetic differential geometry in the arithmetic PDE setting, I: connections

TL;DR

This paper develops an arithmetic analogue of Riemannian geometry using Fermat quotient operations to define concepts like geodesics and connections in a p-adic setting, laying groundwork for further geometric invariants.

Contribution

It introduces a novel framework for arithmetic differential geometry, establishing existence and uniqueness of key geometric structures using arithmetic derivatives.

Findings

Existence of arithmetic geodesics

Construction of Levi-Civita and Chern connections

Foundation for arithmetic curvature theory

Abstract

This is the first in a series on papers developing an arithmetic PDE analogue of Riemannian geometry. The role of partial derivatives is played by Fermat quotient operations with respect to several Frobenius elements in the absolute Galois group of a $p$-adic field. Existence and uniqueness of geodesics and of Levi-Civita and Chern connections are proved in this context. In a sequel to this paper a theory of arithmetic Riemannian curvature and characteristic classes will be developed.